studies in soliton behaviour
TRANSCRIPT
STUDIES IN SOLITON BEHAVIOUR
I I M I S - m f — 1 0 4 7 1
PETER CORNELIS SCHUUR
HOW, IN FRAMES AT REST
THE TAU GOES WEST
WHILE THE EAST IS WON
BV THE SÖLITOM
STUDIES IN SOLITON BEHAVIOUROMTRENT SOLITONGEDRAG
(MET EEN SAMENVATTING IN HET NEDERLANDS)
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR INDE WISKUNDE EN NATUURWETENSCHAPPEN AAN DERIJKSUNIVERSITEIT TE UTRECHT, OP GEZAG VANDE RECTOR MAGNIFICUS PROF. DR. O.J. DE JONG,VOLGENS BESLUITVAN HET COLLEGE VAN DECANENIN HET OPENBAAR TE VERDEDIGEN OP MAANDAG
9 SEPTEMBER 1985 DES NAMIDDAGS TE 4.15 UUR
DOOR
PETER CORNELIS SCHUUR
GEBOREN OP 31 JANUARI 1950 TE ZWOLLE
DRUKKERIJ ELINKWIJK BV - UTRECHT
PROMOTOR : PROF. DR. IR. W. ECKHAUSCOPROMOTOR: DR. A. VAN HARTEN
TABLE OF CONTENTS
PREFACE
1. Historical remarks 12. 1ST for KdV: the gist of the method 53. Asymptotics for nonzero reflection coefficient:
main purpose of the thesis 74. Brief description of the contents 8References 1 1
ACKNOWLEDGEMENTS 12
CHAPTER 1 : THE EMERGENCE OF SOLITONS OF THE KORTEWEG-DE VRIES
EQUATION FROM ARBITRARY INITIAL CONDITIONS
1 . Introduction 132. Formulation of the problem 153. Analysis of flc and T(- 174. Solution of the Gel'fand-Levitan equation 205. Decomposition of the solution and estimates 216. Analysis of S and gj 24Appendix A: Case C 24Appendix B: Case P 26Appendix C: Generalization to higher KdV equations 29References 33
CHAPTER 2 : ASYMPTOTIC ESTIMATES OF SOLUTIONS OF THE KORTEWEC-DE VRIES
EQUATION ON RIGHT HALF LINES SLOWLY MOVING TO THE LEFT
1. Introduction 342. Preliminaries and statement of the problem
2.1 Direct scattering at t = 0 382.2 Inverse scattering for t > 0 432.3 Statement of the problem 45
3. Analysis of fic and Tc 464. Solution of the KdV initial value problem
in the absence of solitons 605. The operator (I+Td)~i 636. Solution of the Gel'fand-Levitan equation
in the presence of bound states 687. Decomposition of the solution of the KdV problem
when the initial data generate solitons 69References 76
CHAPTER 3 : MULTISOLITON PHASE SHIFTS FOR THE KORTEWEG-DE VRIES
EQUATION IN THE CASE OF A NONZERO REFLECTION COEFFICIENT
1. Introduction 782. Scattering data and their properties 803. Forward and backward asymptotics 824. An explicit phase shift formula 845. An example: the continuous phase shifts
arising from a sech initial function 86References 88
CHAPTER 4 : ON THE APPROXIMATION OF A REAL POTENTIAL IN THE
ZAKHAROV-SHABAT SYSTEM BY ITS REFLECTIONLESS PART
1 . Introduction 892. Construction and properties of the scattering data 903. Simplification of the inverse scattering algorithm 964. Statement of the main result 985. Auxiliary results ]006. Proof of theorem 4.1 105References 107
CHAPTER 5 : DECOMPOSITION AND ESTIMATES OF SOLUTIONS OF THE MODIFIED
KORTEWEG-DE VRIES EQUATION ON RIGHT HALF LINES SLOWLY
MOVING LEFTWARD 109
1. Introduction 1102. Some comments on the asymptotic structure of the
reflectionless part ]|33. Two useful results obtained previously lib4. Estimates of fic(x+y;t) 1175. Estimates of q(x,t)-qd(x,t) 119References 122
CHAPTER 6 : MULTISOLITOW PHASE SHIFTS FOR THE MODIFIED KORTEWEG-DE VRIES
EQUATION IN THE CASE OF A NONZERO REFLECTION COEFFICIENT
1. Introduction 1242. Left and right scattering data and their relationship 1263. Forward and backward asymptotics 1284. An explicit phase shift formula 1305. An example: the continuous phase shifts arising
from a sech initial function 132References 135
CHAPTER 7 : ASYMPTOTIC ESTIMATES OF SOLUTIONS OF THE SINE-GORDON
EQUATION ON RIGHT HALF LTNES ALMOST LINEARLY MOVING
LEFTWARD
1. Introduction 1362. The asymptotic structure of the reflectionless part 1403. A useful result obtained earlier 143
4. Estimates of £2c(x+y;t) 1445. Estimates of q(x,t)-qd(x,t) and q(x,t) 147References 149
CHAPTER 8 : ON THE APPROXIMATION OF A COMPLEX POTENTIAL IN THE
ZAKHAROV-SHABAT SYSTEM BY ITS REFLECTIONLESS PART
1. Introduction 1512. Direct scattering 1533. Inverse scattering 1564. Statement of the main result 1575. First steps to the proof 1586. Proof of theorem 4.] 1647. An application: cmKdV asymptotics 167References 171
CHAPTER 9 : INVERSE SCATTERING FOR THE MATRIX SCHRÓÜINGER EQUATION
WITH NON-HERMITIAN POTENTIAL
1. Introduction 1732. The inverse scattering problem for the matrix
Schrödinger equation 1752.1 Jost functions 1762.2 Scattering coefficients 1782.3 Bound states 1802.4 The Gel'fand-Levitan equation 184
References 189
CHAPTER 10 : UNIFICATION OF THE UNDERLYING INVERSE SCATTERING PROBLEMS
1. Introduction 1902. Jost functions 1913. Scattering coefficients and bound states 1934. Inverse scattering 196References 199
CONCLUDING REMARKS 201
APPENDIX : AN OPEN PROBLEM 2031
References 205 !
iSAMENVATTING 207 j
CURRICULUM VITAE 209 i
PREFACE
For centuries nonlinearity formed a dark mystery.
Nowadays, though things still look rather black, there are a few bright
spots where we may confidently expect steady progress. This thesis deals
with one of these sparkles of hope: the. i-nverse scattering tranafornation.
1. Historical remarks.
Many physical phenomena are nonlinear in nature. More often than not
they can be modelled by nonlinear partial differential equations offering
a wide range of complexity. Until the late sixties of this century the
analyst had, roughly speaking, the choice: approximate or apologize. In f
the past two decades this situation changed, since various powerful j}
nonperturbative mathematical techniques made their entrance. One of these '
is the inverse scattering technique (1ST), also called inverse scattering
transformation or spectral transform.
Its discovery is due to Gardner, Greene, Kruskal and Miura (GGKM for
short) and was first reported in 1967 in their famous two-pagei signal
paper [9]. In this paper GGK>! showed how to obtain the solution u(x,t)
of the Korteweg-de Vries (KdV) initial value problem
(1.1a) u - 6uu + u = O, -t» < x < +°°, t > 0
(1.1b) u(x,0) = uQ(x).
Here and in the sequel a subscript variable indicates partial differen-
tiation, e.g. u = -r— . Equation (1.1a) was first derived by Korteweg andX 0X
de Vries [13] in 1395 in the context of free-surface gravity waves
propagating in shallow water (see [4] for its historical background).
Below we shall discuss the GGKM method in some detail. Here we only
mention its amazing starting point, namely the introduction of the
solution u(x,t) of (1.1) as a potential in the Schro'dinger scattering
problem.
In 1968 Lax [16] put the GGKM method into a framework that clearly
indicated its generality and had a substantial influence on future
developments. In particular, Lax showed that (1.1a) is a member of an
infinite family of nonlinear partial differential equations that can all
be analysed in a similar fashion.
Guided by Lax' generalization of the pioneering work of GGKM, Zakharov
and Shabat [25] were able to solve the initial value problem for another
nonlinear equation of physical importance, the nonlinear Schrödinger
equation (NLS)
(1.2)
To this end they associated (1.2) with a spectral problem based on a
system of two coupled first order ordinary differential equations.
Incidentally, the NLS shows up in the description of plasma waves and
models plane self-focusing and one-dimensional self-modulation.
Subsequently, Tanaka [20], [22] extended and rigorized the direct
and inverse scattering theory for the Zakharov-Shabat system and showed
furthermore that another interesting nonlinear evolution equation could
be solved by this system, namely the modified Korteweg-de Vries equation
(mKdV)
(1.3) ut + 6u>ux + u x x x - 0
which appears in the continuum limit of a one-dimensional lattice with
quartic anharmonicity [5].
Ablowitz, Kaup, Newell and Segur [1], [2] then showed that NLS and
mKdV belong to a large class of nonlinear partial differential equations
that can be solved via a generalized version of the Zakharov-Shabat
scattering problem. Among these newly found integrable equations were
several of physical importance, such as the sine-Gordon equation
r rx 1(1.4) uc = è sin 2 u(x',t)dx'
which arises as an equation for the electric field in quantum optics [15],
though the related forms
(1.A) ' o = sin a andxt
(1.4)" o - o = sin oxx tt
appear more frequently in the literature (cf. [12]).
Herewith the triumphal march of the inverse scattering technique began.
We shall not follow it further but refer to fie survey articles [5], [10],
[15], [17], L18J as well as the many textbooks on solitons ['i\, [b], [7],
[3], [14], [24J currently available. We only mention that several other
classes of physically relevant equations were found to be solvable by
inverse scattering methods. In fact the process of finding new integrable
nonlinear evolution equations has continued until this very day and has
grown out into a major industry. Moreover, 1ST had its spin-off's to other
areas of mathematics, like algebraic and differential geometry, functional
and numerical analysis, etc. Nowadays - as stated in [6] - its applications
range from nonlinear optics to hydrodynamics, from plasma to elementary
particle physics, from lattice dynamics to electrical networks, from
superconductivity to cosmology and geophysics. Moreover, 1ST is <II'VL- lopirn;
into an interdisciplinary subject, since it has recently penetrated in
epidemiology and neurodynamics.
An essential reason for this wide applicability lias not been mentioned
so tar: a dominant feature of nonlinear evolution equations of physical
importance solvable via 1ST is that they admit exact solutions that
describe the propagation and interaction of .' .'' '..'.
At the moment there is no generally accepted mathematical definition
of a soliton. As a working definition of a soliton we mi;;ht take (cf. [5])
that it is a "localized" wave (in the sense of sufficiently rapidly
decaying) which asymptotically preserves its shape and speed upon
interaction with any other such localized wave. However, the concept of
a soliton has 'a great intuitive appeal and is a good illustration of the
fact that a happily chosen terminology is half of the success of a theory.
The soliton was discovered in 1965 by Zabusky and Kruskal [23] while
performing a numerical study of the KdV. Actually, the name "soliton"
was suggested by Zabusky, who originally used the term "solitron" instead
(see [6], pp. 176, 177).
Let us discuss their discovery in some detail. Already Korteweg and
de Vries theirselves knew [13] that the KdV had a special travelling
wave solution, the solitary wave
(1.5) u(x,t) = -2k2sech2[k (x - xQ - 4k^t)], (sech z = -~——)
e + e
where k„ and x„ are constants. Observe, that the velocity of this wave,
4ki, is proportional to its amplitude, 2ki. Now, in [23] Zabusky and
Kruskal considered two waves such as (1.5), with the smallest to the right,
as initial condition to the KdV. They discovered that after a certain
time the waves overlap (the bigger one catches up), but that next the
bigger one separates from the smaller and gradually the waves regain their
initial shape and speed. The only permanent effect of the interaction is
a phase shift, i.e. the center of each wave is at a different position
than where it would have been if it had been travelling alone.
Specifically, the bigger one is shifted to the right, the smaller to the
left. The name soliton was chosen so as to stress this remarkable
particle-like behaviour.
To conclude these introductory remarks, let us not forget to mention
that, although in the past few years soliton interaction has been
observed in various physical systems (see [3]), the first physical
observation of what is now known as the single soliton solution (1.5)
of the KdV already took place in the month of August 1334 by John Scott
Russell, during his celebrated chase on horseback of a huge wave in the
Union Canal, which from Edinburgh, joins with the Forth-Clyde canal and
thence to the two coasts of Scotland. His own report of this experience,
though classical by now, cannot be missed in any true soliton story.
It reads as follows [19]:
"I was observing the motion of a boat which was rapidly drawn
along a narrow channel by a pair of horses, when the boat
suddenly stopped - not so the mass of water in the channel
which it had put in motion; it accumulated round the prow
of the vessel in a state of violent agitation, then suddenly
leaving it behind, rolled forward with great velocity,
assuming the form of a large solitary elevation, a rounded,
smooth and well defined heap of water, which continued its
course along the channel apparently without change of form
or diminution of speed. I followed it on horseback, and over-
took it still rolling on at a rate of some eight or nine miles
an hour, preserving its original figure some thirty feet long
and a foot to a foot and a half in height. Its height
gradually diminished, and after a chase of one or two miles
I lost it in the windings of the channel. Such, in the month
of August 1334, was my first chance interview with that
singular and beautiful phenomenon ...".
2. 1ST for KdV: the gist of the method. |
iTo comfort the reader who is completely new to the subject, let us j
at least give a rough sketch of how 1ST works, referring to [8] for the •
many intricate mathematical details. To this end we indicate here very j
briefly the basic features of the GGKM method, which is the first and
undoubtedly the most fundamental example of an inverse scattering method.
Let us consider the KdV initial value problem (1.1) with u_(x) an
arbitrary real function, sufficiently smooth and rapidly decaying for
x * ±". The surprising discovery of GGKM is now, that the nonlinear
problem (1.1) can be solved in a series of linear steps, schematically
representable in the following diagram
initial functionu(x,0) = uff(x)
solution u(x,t)at t • 0
direct
lution ination space
inverse
Schrödinger
Schrödinger
scatterin'scattering data
at t = 0
time evolution in >spectral space
scattering
f
scattering dataat t • 0
The manipulations suggested by this diagram are the following:
For each t "" 0, introduce the real function u(x,t) as a potential in the
Schrödinger scattering problem
(2.1) + (k2 - u(x,t)h> = 0,
_ 2• 9
— i 2For t = 0, compute the associated bound states —•
K. v 0, right normalization coefficients c. and right reflection
coefficient b (k) (see Chapter 2 for their definition and properties),
in other words, compute the right scattering data 1b (k),> . ,i'. 1 associated
with u_(x). Then, as u(x,t) evolves according to the KdV, its right
scattering date evolve in the following simple way:
(2.2a)
(2.2b) cT(t) =ryexpU»U}, j = 1,2 ÏI
(2.2c) b (k,t) = b (k) exp{8ik 3t}, -™ • k
To recover u(x,t) from these data, one applies the inverse scattering
procedure for the Schrödinger equation found by Gel'f and and Levitan
[11], and defines
N -2i..r
(2.3) .;(•'. ;t) = 2 .Ij [cf(t)]2e J f br(k,t)e2lk'dk.
Next, one solves the Gel'fand-Levitan equation
(2.4) ;-:(y;x,t) + !-'(x+y;t) + >:(x+y+z; t ):•' (z ;x, t )d7. = 00J
with y • O, x £ K, t • 0 . The s o l u t i o n i-;(y;x,t) has the important
p r o p e r t y
(2.5) 3 ( 0 + ; x , t ) = u ( x ' , t ) d x ' , x £ K, t • 0 ,x-'
and so we find that the solution of the KdV problem (1.1) is given by
(2.6) u(x,t) = - ~ B(0+;x,t), x £ JR, t • 0.
Notice that the original problem for the nonlinear partial differential
equation (1.1) is essentially reduced in this way to the problem of
solving a one-dimensional linear integral equation.
Explicit solutions of (2.4) have only been obtained for b = 0. The
solution u,(x,t) of the KdV with scattering data '0,< . ,c . (t) } is called
the pure N-soliton solution associated with u„(x), on account of its
asymptotic behaviour displayed in the following remarkable result due to
Tanaka L21]
(2.7 a) lim supN
a' ' ' p=lx£E '= 0
+ _ 1 J p •• f (XP - ir
lon^~^i V^ p
Thus for large positive time u,(x,t) arranges itself into a parade of N
solitons with the largest one in front and this happens uniformly with
respect to x on E.
3. Asymptotics for nonzero reflection coefficient: main purpose of the
thesis.
As illustrated by the previous section, the inverse scattering method
enables us to obtain rather explicit exact solutions to nonlinear wave
equations and to determine their asymptotic behaviour, which generally
corresponds to a decomposition into solitons. Evidently, the problem of
the asymptotic behaviour evolving from an arbitrary initial condition is
in this way far from exhausted. It is still necessary to determine the
asymptotic properties of the "nonsoliton part" of the solution whose
presence is connected with the reflection coefficient being nonzero. In
this thesis we concern ourselves with this problem.
Rather than to give an elaborate general discussion, let us
illustrate the ideas involved by considering again the KdV problem (1.1).
Suppose un(x) is not a reflectionless potential. Then, in view of the
fact that the linearized version of (1.1a) is a dispersive equation with
associated group velocity v = -3k2 = 0, one expects that for lf.rge time
the soliton part and the dispersive component will separate out, the
dispersive wavetrain moving leftward and the solitons nicely arranging
theirselves into a parade moving to the right similar to that described
by (2.7). However, this is only heuristic reasoning. In fact it is
dangerous reasoning too, since for nonlinear equations there is no such
thing as a superposition principle.
The circumstance that at the time the question of validity of the
above "plausible" conjecture had not been answered in a mathematically
satisfactory way, formed the impetus for the research laid down in the
present thesis.
The main purpose of this thesis is therefore to give a complete and
rigorous description of the emergence of solitons from various (classes
of) nonlinear partial differential equations solvable by the inverse
scattering technique.
Throughout the thesis we focus our attention on coordinate regions
where the dispersive component is sufficiently small, e.g. x • -t for
the (m)KdV problem. The behaviour of the solution in other regions, where
the dispersive waves interact, is not discussed, since entirely different
techniques are needed. For recent results in those regions we refer to [3].
4. Brief description of the contents.
The chapters in this thesis are largely self-explanatory. Only
Chapter 1 forms an exception. We therefore advise the reader new to the
field to start with Chapter 2. In fact, both chapters deal with the KdV.
However, in Chapter t the central ideas of our asymptotic method are
exposed in the simplest nontrivial setting, wheroas Chapter 2 serves to
extend the results of Chapter 1, as well as to supply the details of the
inverse scattering machinery. Also, the discussion of existence and
uniqueness for the KdV initial value problem is postponed to Chapter 2.
In Chapter 1 we present a rigorous demonstration of the emergence of
solitons from the KdV initial value problem with arbitrary initial
function. We show that for any choice of the constants v > 0 and M 5 0
there exists a function a(t) tending to zero as t -*•<*>, such that
(4.1) sup |u(x,t) - u (x,t)| = 0(o(t)), as t •» ••xS-M+vt
The asymptotic analysis given in Chapter 1 is extended in Chapter 2. It
is shown that in the absence of solitons the solution of (1.1) satisfies
(4.2) sup |u(x,t)| =0(t~ 2 / 3), ast-.-»,xa-t1'3
whereas in the general case
(4.3) sup |u(x,t) - u (x,t)| = 0(t~' / 3), ast-»-».xè-t1'3
The emergence of solitons is clearly displayed by the remarkable
convergence result
(4.4) lim suiIim s upt-«» xS-t'/>3
N ,
£, (-2u(x,t) - Z, -2K2sech2U' (x-x -4K 2 t)] = 0P=1 \ P P P P
with x as in (2.7b).P
In Chapter 3, while studying multisoliton solutions of the KdV in the
general case of a nonzero reflection coefficient we derive a new phase
shift formula which shows that each soliton experiences in addition to
the ordinary N-soliton phase shift an extra phase shift to the left
caused by the interaction with the dispersive wavetrain.
In Chapter 4 we consider the question how well a solution of a nonlinear
wave equation is approximated by its soliton part in a more general
setting: we derive an estimate which indicates how well a real potential
in the Zakharov-Shabat system is approximated by its reflectionless part.
Moreover, the associated inverse scattering formalism is simplified
considerably.
In Chapter 5 we use the estimate derived in Chapter 4 to obtain
asymptotic bounds of solutions of the mKdV of the same type as those
found in Chapter 2 for the KdV.
Jsing the results from Chapter 5 we derive in Chapter 6 a general
phase shift formula for the mKdV remarkably similar in form to that
found in the KdV case in Chapter 3; the only difference is, however, that
now the extra phase shift is to the right, i.e. mKdV solitons are
advanced by their interaction with the dispersive wavetrain.
Chapter 7 is devoted to the asymptotic analysis of the sine-Gordon
equation on right half lines almost linearly moving leftward. Again the
estimate found in Chapter 4 is shown to be of vital importance.
In Chapter 8 we study the Zakharov-Shabat system with complex potential
and show that the results obtained in Chapter 4 can be generalized to
this case. As an illustration we investigate the long-time behaviour of
the solution of the complex modified Korteweg-de Vries initial value
problem.
In Chapter 9 we develop an inverse scattering formalism for a higher
order scattering problem than those considered in the rest of this thesis,
namely the matrix Schrödinger equation with non-Hermitian potential.
Finally, in Chapter 10 we show how this matrix Schrödinger problem can be
seen as a synthesis of all the scattering problems discussed in the
present thesis.
In an appendix we illustrate that our methods fail to give any result if
the associated group velocity is not of constant sign. As an example we
discuss the NLS and pose an interesting open problem.
10
References
[ 1) M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, Method forsolving the sine-Gordon equation, Phys. Rev. Lett. 30 (1973),1262-1264.
[ 2] M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, The inversescattering transform - Fourier analysis for nonlinear problems,Stud. Appl. Math. 53 (1974), 249-315.
[ 3] M.J. Ablowitz and H. Segur, Solitons and the Inverse ScatteringTransform, Philadelphia, SIAM, 1981.
[ 4] F. van der Blij, Some details of the history of the Korteweg—de Vriesequation, Nieuw Archief voor Wiskunde 26 (1978), 54-64.
[ 5] R.K. Bullough and P.J. Caudrey, The soliton and its history, in:Solitons, Topics in Current Physics 17, Springer, New York, 1980(edited by the same).
[ 6] F. Calogero and A. Degasperis, Spectral Transform and Solitons,Amsterdam, North-Holland, 1982.
[ 7] R.K. Dodd, J.C. Eilbeck, J.D. Gibbon and H.C. Morris, Solitons andWonlinear Wave Equations, Academic Press, 1982.
[ 3] W. Eckhaus and A. van Harten, The Inverse Scattering Transformationand the Theory of Solitons, North-Holland Mathematics Studies 50,1931 (2nd ed. 1983).
[ 9] C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura, Method forsolving the Korteweg-de Vries equation, Phys. Rev. Lett. 19 (1967),1095-1097.
[10] C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura, Korteweg-deVries equation and generalizations VI, Cornm, Pure Appl. Math. 27(1974), 97-133.
[11] I.M. Gel'fand and B.M. Levitan, On the determination of a differentialequation from its spectral function, Izvest. Akad. Nauk 15 (1951),309-360, AMST 1 (1955), 253-309.
[12] D.J. Kaup and A.C. Newell, The. Goursat and Cauchy problems for thesine-Gordon equation, SIAM J. Appl. Math. 34 (1978), 37-54.
[13] D.J. Korteweg and G. de Vries, On the change of form of long wavesadvancing in a rectangular canal, and on a new type of longstationary waves, Phil. Mag. 39 (1895), 422-443.
[14] G.L. Lamb, Jr. , Elements of Soliton Theory, Wiley-Interscience, 1980.
[15] G.L. Lamb, Jr. and D.W. McLaughlin, Aspects of soliton physics, in:Solitons (Ed. R.K. Bullough and P.J. Caudrey) Topics in CurrentPhysics 17, Springer, New York, 1980.
[16] P.D. Lax, Integrals of nonlinear equations of evolution andsolitary waves, Comm. Pure Appl. Math.21 (1968), 467-490.
[17] R.M. Miura, The Korteweg-de Vries equation: A survey of results,SIAM Review 18 (1976), 412-459.
1 1
[18] A.C. Scott, F.Y.F. Chu and D. McLaughlin, The soliton: a new conceptin applied science, Proc. IEEE 61 (1973), 1443-1483.
[19] J. Scott Russell, Report on waves in: Report of the fourteenthmeeting of the British association for the advancement of science,John Murray, London, 1844, 311-390.
[20] S. Tanaka, Modified Korteweg-de Vries equation and scattering theory,Proc. Japan Acad. 48 (1972), 466-439.
[21] S. Tanaka, On the N-tuple wave solutions of the Korteweg-de Vriesequation, Publ. R.I.M.S. Kyoto Univ. 8 (1972), 419-427.
[22] S. Tanaka, Non-linear Schrödinger equation and modified Korteweg-deVries equation; construction of solutions in terras of scatteringdata, Publ. R.I.M.S. Kyoto Univ. 10 (1975), 329-357.
[23] N.J. Zabusky and M.D. Kruskal, Interactions of "solitons" in acollisionless plasma and the recurrence of initial states, Phys.Rev. Lett. 15 (1965), 240-243.
[24] V.E. Zakharov, S.V. Manakov, S.P. tlovikov and L.P. Pitaievski, Theoryof Solitons. The Inverse Problem Method, Nauka, Moscow, 1980 (inRussian).
[25] V.E.. Zakharov and A.B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in non-linearmedia, Soviet Phys. JETP (1972), 62-69.
IACKNOWLEDGEMENTS i
J wish to focus my thanks on Viktor Eakhaus and Aart van Harten,my thesis super- and advisors, for their stimulating guidanoeand continuous interest in my work and for their pleasant wayof combining moral support with valuable criticism.
To Wilma van Nieuwamerongen I am indebted for skilfully typingthe manuscript.
12
CHAPTER ÖME
THE EMERGEIICE OF SOLITONS OF THE KORTEWEG-DE VRIES EQUATION FROM
ARBITRARY INITIAL CONDITIONS
We study the solution u(x,t) of the Korteweg-de Vries equation
u - &uu + u = 0 evolving from arbitrary real initial conditionst x xxx
u(x,0) = u-Cx), uo(x) decaying sufficiently rapidly as | x] -+ <». Using the
method of the inverse scattering transformation we analyse the Gel'fand-
Levitan equation in all coordinate systems moving to the right and give
a complete and rigorous description of the emergence of solitons.
*)1. Introduction.
The discovery by Gardner, Greene, Kruskal and Miura [8], [9] of a
method of solution for the Korteweg-de Vries equation by the inverse
scattering transformation has led to a rapid and impressive development,
which one can find described for example in [1], [7]. The rapid progress
has produced a wealth of results, however, it has also left certain
questions unanswered.
Let us recall that, by the inverse scattering transformation, the
initial value problem for the (nonlinear) Korteweg—de Vries equation, is
13
reduced to the problem of solving the (linear) Gel'fand-Levitan integral
equation. The initial values u..(x), prescribed for the solution of the
KdV equation, when introduced as a potential in the Schrödinger equation,
provide the scattering data that are needed to define the kernel of the
Gel'fand-Levitan equation. However, explicit solution of that equation
has been obtained only in the case that the reflection coefficient
corresponding to u„(x) is zero. One then has the famous "pure" M-soliton
solution, with N being the number of discrete eigenvalues in the
Schrödinger scattering problem.
If ur,(x) is not a reflectionless potential, then by a heuristic
reasoning one arrives at a conjecture about the behaviour of the solutions,
as follows: one knows that solitons, if present, move to the right, while
the dispersive waves that are expected to be present when the reflection
coefficient is not zero, move to the left. This leads to the expectation
that for large time the two ingredients of the solution will separate
out, and that observers moving with suitable speeds to the right will
eventually see a parade of solitons, each one followed by a decaying
train of dispersive waves, as described in [10]. In spite of attempts
such as [11], the question of validity of this "plausible" conjecture has
not been answered in a mathematically satisfactory way.
In this paper we study the solution u(x,t) of the Korteweg-de Vries
equation u - 6uu + u = 0 with arbitrary real initial conditionst x xxx
u(x,0) = u_(x), u^Cx) decaying sufficiently rapidly as |x| -> « for the
whole of the inverse scattering transformation to hold. We analyse the
Gel 'fand-Levitan equation in all coordinate systems moving to the right
and give a complete and rigorous description of the emergence of solitons.
It will probably not be a surprise to most readers that we find " solitons
emerging if un(x) is a potential that produces N discrete eigenvalues in
the Schrödinger equation, but it may be a surprise to some that this fact
has never been demonstrated mathematically, except for the reflectionless
potentials. We further show that the nature of decay of the dispersive
wave trains behind the solitons is essentially related to properties of
the reflection coefficient b (k), at t = 0, such as differentiability and
See also the more detailed introduction to Chapter 2, especially i'ordetails about existence and uniqueness.
14
behaviour for |k[ -+ »>, or the possibility of extension of b (k) to an
analytic function. These properties are in turn related to properties of
the initial function u~(x).
The problem of relations between properties of potentials uQ(x) and
corresponding reflection coefficients b (k) belongs to the scattering
theory and is not discussed in detail in this paper (see Chapter 2,
subsection 2.1). From the literature it is known that, if u„(x) decays
exponentially for x -+ +•», then b (k) can be extended to an analytic
function on a strip in the upper half plane, as can be seen from [6].
Furthermore, if un(x) and its derivatives up to fourth one decay
algebraically for |x| -+ °> sufficiently fast, then b (k) belongs to
C ( q )(K) for some q and b£m)(k) = 0<|k|~5) as |k| > », m = 0,1 ,... ,q
(see [5]).
We also no not discuss the behaviour of solutions of the KdV equation
for large time viewed in coordin te systems moving to the left, where
the dispersive waves interact. Recent results on that problem (in the
case of no discrete eigenvalues) have been given in [3],
The analysis given in this paper consists of a rather simple
reasoning in an abstract setting, supplemented by hard labour that is
needed to obtain the necessary estimates. The reasoning is developed in
sections 2 to 5, the labour is performed in section 6 and in two
appendices. In the last appendix we show that our method also works in a
more general setting by considering the so-called higher KdV equations.
2. Formulation of the problem.
We consider the Gel'fand-Levitan equation
(2.1) 3(y;x,t) + ft(x+y;t) + S2(x+y+z; t)iUz;x,t)dz = 0,0J
w i t h y > 0 , x £ K, t > 0 ,
( 2 . 2 ) s i ( f . ; t ) = a d ( c ; t ) + ? 2 c ( £ ; t ) ,
N -2K.t',(2.3) f!d(J;;t) = 2 ^ [ c j ( t ) p e J , 0 < KN < . . . - <2 <
15
(2
(2
(2
.4)
.5)
.6)
^c
b
rc.
( 5 ; t )
(k.t)
(t) =
= |
- b
cTeJ
r(k)e
4KUJ
b ( k ,
' t
t )2ik£„e dk,
Here -<2., c , j = 1,2,...,N are the bound states and (right)
normalization coefficients and b (k) is the (right) reflection coefficient
associated with the potential ur/x) in the Schrödinger scattering problem
(see Chapter 2, section 2, for their definition and properties). In the
integral equation (2.1) the unknown g(y;x,t) is a function of the variable
y, whereas x and t are parameters. The solution of the KdV equation is
given by
(2.7) u(x,t) = - Y- (3(0+;x,t).crX
We shall study the solution of (2.1) in moving coordinates in the parameter
space x,t, defined by
(2.8) "x = x - vt, v = 4c2 , c > 0.
In particular we shall examine the behaviour for large positive times,
with x confined to arbitrary half lines x S -M, where M £ 0 is independent
of t. For each c = K. we expect to see a soliton emerging.
We now give the problem an abstract formulation.
Let V be the Banach space of all real continuous and bounded functions %
on (0,»), equipped with the supremum norm
IIgil = sup |g(y) | .
For g £ V we write
(2.9) (Tdg)(y) = j {?d(x+y+z;t)g(z)dz,
(2.10) (T g)(y) = [ oc(x+y+z;t)g(z)dz.
C 0J
T clearly is a mapping of V into V; T will be investigated in the next
section.
Our problem is thus to find an element P € V such that
16
(2.11) (I + Td)3 + Tc@ = -fi,
(2.12) fi = nd + n c,
where I is the identity mapping.
We know the solution (3, ofa
(2.13) (I + T,)6, = -S?,,da a
which yields the pure N-soliton solution of the KdV equation. We intend to
study the full problem as a perturbation of the pure N-soliton case.
3. Analysis of Ü and T .
We consider
/•> i\ „ /•"./ ?- .- 1 f , „ , 2ik(x+y) 8itk(c2+k2) ,.(3.1) Slc(x+4c2 t+y;t, = — b (k)e } e dk.
It should be clear that Q is an oscillatory integral for large t, x ï -M,
tending to zero as t tends to infinity. The precise behaviour depends on
the behaviour of b (k), which in turn is determined by the initial
condition for the KdV equation.
Imposing suitable conditions on b (k) we shall establish that ft is
strongly differentiable in V with respect to x and obeys an estimate of
type
(3.2a) |fic(jc+vt+y;t)[ + |si^(x+vt+y;t) | :- H(y,t), t S tQ, x > -M,
such that
(3.2b) H(y,t) is a monotonically decreasing function of y for fixed t,
(3.2c) o(t) 5 H(z,t)dz + sup H(y,t) < +»,(r O'y-+<»
(3.2d) o(t) ->• 0 as t ->• ».
In (3.2a) we introduced the prime as a quick notation for the derivative
with respect to x.
17
We shall work out in detail two cases of conditions on t>r(k) that are
more or less typical (for a priori knowledge about b (k) as well as the
motivation for these conditions see Chapter 2, subsection 2.1).
Case C There exists an E > 0 such that b (k) is analytic on— — — ^ — f
0 < Im k < € and continuous on 0 s lm k S c, while in that
strip br(k) = o(|k|2) for |k| •+ «.
Case P ( i) b (k) is n S 2 times differentiable on the real axis;b
z . . * , (m) /, 2m+2 \ i, i „ - <( n ) b = o(k ) , |k •+ o°, m = 0,1,...,n-1;
r b b 0 )
(iü) ^ , — ^ , ••., b^" 1 ), (H|kl)b^n) belong1 | | n 2
to L 1(R).
In Case C - treated in Appendix A - one finds by means of contour
integration
(3.3) H(y,t) = Ye~2 f ye" a t, o(t) = ü(e" a t),
where y and a are positive constants.
In Case P - the subject of Appendix B - integration by parts pro 'uces
(3.4) H(y,t) = , o(t) = 0(— l- T),
(-M+y+vt)n tn~'
where again p is a positive constant.
With the result (3.2), examplified by (3.3) and (3.A), we proceed to
investigate the mapping T .
In moving coordinates we have
(3.5) (T g)(y) = j <; fx"+vt+y+z;t)g(z)dz,
which is a continuous function of y, since the integrand is dominated by
H(z,t)|g(z)|. Furthermore,
CO
(3.6) IIT gll i llg» H(z,t)dz S llglla(t).
We have thus established that T is a continuous mapping of V into V with
norm tending to zero for t -* •» uniformly on x * —M.
18
We next claim that T is strongly x-differentiable in V with derivative
f -(3.7) (T';;)(y) = fl'(x+vt+y+z;t)g(z)dz.
c 0J
The proof consists in showing that
(3.3) A, = sup (A, (y+z)-Q'(x+vt+y+z;t))g(z)dzn 0<y<+» 0Jy 0
tends to zero as h + 0, where
(3.9) Ah(y+z)=-£1» (x+vt+y+z+h;t)-U (x+vt+y+z;t)
Clearly \ S J B ( z ) | g ( z ) | d z , with
(3.10) B j / Z ) = SUP \
Since Si is strongly x-dif f erentiable, B (z) tends to zero as h •+ 0. Thus
in virtue of the dominated convergence theorem it suffices to show that
(3.11) B (z) < G(z) with G £ L1(0,»).
Let x S -M+6, 6 > 0. For |h| t' ó one has
(3.12) |Ah(y+z)| = |ic''(x-ö+vt+y+z+Oh+iS;t)| for some •? € (0,1)
S H(y+z+0h+6,t)
i H(z,t).
Consequently B (z) •.-' 2H(z,t) which is in L (0,«).
Finally, we deduce from (3.2) and the above
( 3 . 1 3 ) n i a x i II I I . I l T I I , il ' H . l l r M I j • r ( t ) ,L c c c c J
w h e r e j ( t ) -• 0 a s t • -".
19
4. Solution of the Gel 'fand-Levitan equation.
We consider
(4.1) (I + Td)fc! = -(£3 + T cS).
Since T is an integral operator with degenerate kernel, solutions of
(1 + Td)g = f, f,g £ V
can be studied explicitly. In lemma 6.1 we shall show that the inverse
(I+T,) indeed exists as a mapping of V into V and that furthermore
Il(l+T )~1II is bounded for t •> 0, it 5 -M.
To simplify the notation we shall write
(4.2) (I + T d ) ~1 = S
and we have
(4.3) ItSlI £ a for t > 0, 1 è -M.
We can thus "invert" (4.1) and obtain the equation
(4.4) B = - Sn - ST 3.
It can be easily shown that (4.4) possesses a unique solution p € V.
Indeed, consider the mapping T defined by
(4.5) Tg = f - STcg, f,g € V.
By the results (3.13) and (4.3) one has
(4.6) IIST II s IISIIHT If ï ao(t).
Hence, for sufficiently large t, we find IIST II - 1 and T is a contractive
mapping in the Banach space V. It follows that a unique solution g of
(4.7) g = f - STcg, f,g £ V
exists. Furthermore, one easily obtains an estimate for the solution. In
fact, since
'23
( 4 . 3 ) II gil < IIfII + IIST gil S IIfII + IIST I l l lg l l ,
w e o b v i o u s l y h a v e
(4.9) «g« S , _ 1 „ S T „ llfll.c
5. Decomposition of the solution and estimates.
We write
(5.1) d = Sd + 3c,
with
(5.2) sd = -snd.
In lemnia 6.2 i t will be shown that fi (y;x+vt,t) is uniformly bounded for
t •• 0, x £ -M, y "•• 0. Ue recall that p, produces the pure H-soliton
solution of the KdV equation through the formula
(5.3) u,(x+vt,t) = = B (0+;x'+vt,t).a 3x a
Introducing the decomposition (5.1) into (4.4) we have the equation
(5.4) p + ST p = -Ss> - ST B,.c c c c c d
From the analysis of the preceding section we know that a unique
solution B € V exists. To estimate the solution we proceed as folLows: i
( 5 . 5 ) l i p II < II S T I I I I H II + I I S l l l l ' ! II + II S T I I I I B . I I . 'c c c c c d •
Using (3.13), (4.3) and (4.6) one gets
(5.6) up ii < , ao(t), . (1 + I I B . H ) .c 1 - ao(t) d
Our final result at this stage is that in all moving coordinates
x = x-vt, v ~' 0, in any half line x 't. —M, for large t
21
(5.7a) llBcll S ba(t),
where b is some constant and o(t) + 0 as t •» ».
Furthermore, in the first approximation we have
(5.7b) @c = -S(«c + Tc(5d) + 0(a2(t)).
Unfortunately, the labour is not finished yet. The solution of the
KdV equation is given by
(") .(5.3) u(x+vt,t) = ud(x+vt,t) = a (0 ;x+vt,t).
We thus need estimates of the derivative of (-! with respect to x. To obtain
these estimates we return to equation (5.4). One verifies without
difficulty that both S and 3, are strongly x-differentiable. From
equation (5.4) we then see that g , too, is strongly x-differentiable.
Oifferentiating both sides with respect to x we find
(5.9) 13 + STc3^ = -S{T<;(0c+ed) + 9.'c + T fj)
Jsing section 4 we again conclude that a unique solution 8' exists and
proceed to estimate the solution.
By lemma 6.2 Bj(y;x+vt,t) is uniformly bounded for t > 0, x "• -M, y • 0,
while lemma 6.1 tells us that
(5.10) IIS 'II < a for t > 0, ~x. > -M.
From ( 5 . 9 ) a n d t h e e s t i m a t e s ( 3 . 1 3 ) , ( 4 . 3 ) , ( 4 . 6 ) , ( 5 . 7 a ) and ( 5 . 1 0 ) o n e
f i n d s
( 5 . 1 1 ) II6 Ml S 1 ! ° a ^ ( t ) ( 2 + 2«SdH + ll!?dll + 2 b o ( t ) ) .
T h u s f o r l a r g e t
( 5 . 1 2 ) II6 Ml S B o ( t ) ,
c
where B is some constant. Evidently
22
(5.13) sup •-= 0 (y;x+vt,t
= IIBMI
é Ba(t).
We thus arrive at our final result, which can be stated as follows:
Theorem 5.1. Let u(xst) be the solution of the Korteweg-de Vries -problem
( 5 . 1 4 )r U - 6UU + U = 0, -oo < X < +oo, t > 0j t X XXX ' '
*• u(x,0) = u Q ( x ) ,
where the real initial function un(x) is sufficiently smooth and decays
sufficiently rapidly for |x| •* •» for the whole of the inverse
scattering method to work and to guarantee an estimate of type (3.2).
Then for any choice of the constants v > 0 and M > 0 there is a function
o(t) such that
(5.15) sup |u(x,t) - ud(x,t) | = 0(a(t)) as t •» «.xa-M+vt
Here u,(x,t) is the pure H-soliton solution, N being the number of
discrete eigenvalues corresponding to the potential un(x). The function
o(t) tends to zero as t + « and the exact behaviour of o(t) depends on
properties of the reflection coefficient b (k).
If there exists an c > 0 such thai b (k) is analytic on 0 < lm k * r
and continuous on 0 ï lm k :-v z, while in thai strip b (k) = o(|k|2) for
ik| -»• », then o(t) = 0(e ) , a - 0. •
If b (k) is n ; 2 i f me s differentiab'V /«; '»r r<?i7.' axis, with
b<m)r= o(k 2 m + 2), |k| - », m = 0,),...,n-1 Ji.i
(1 + lk|)1-nbr> (1 + |k|)
2"nbJl),...,b^n-1),(1 + |k|)b^n) belong to
;fe« o(t) = 0(t'"n).
6. Analysis of S and B,.
In this se.ction we present estimates concerning S, 8, and their
x-derivatives that were essential in the previous investigation of the
Gel'fand-Levitan equation. The original proofs as given in Chapter 2,
reference [d], have been improved considerably and are therefore omitted.
Instead we refer to the corresponding proofs occurring in Chapter 2.
Lemma 6.1. I + T, is an invertible operator1 on the Banaoh space V. Writing
(6.1) S = (I + T d)~'
we have
(6.2) llsll, US Ml i a for t > 0, 1 ^ -M,
where a is some constant and S' denotes the strong x-de.rivative of S.
Proof: See Chapter 2, lemma 5.1.
Lemma 6.2. (5, (y ;x+vt,t) and S!(y;x+vt,t) arc uniformly bounded for t > 0,
JC a - M , y •> 0.
i
Proof; This follows from combining Chapter 2, (5.15) with Chapter 2, (5.7). j
Appendix A: Case C.
Assuming that
(A.1) there exists an c > 0 such that b (k) is analytic on
0 ' In t < E and continuous on 0 S lm k S c, while in that
strip b (k) = o(|k|2) for |k| •* »,
we shall derive the estimate
(A.2) |«c(x+4c2t+y;t)| + |i2 (x+4c* t+y;t) | f- y e " 2 ^ . ^ ^ ,
t .> t0, x > -M,
24
where a and y are positive constants.
Remark. The reason why we use no steepest descent method is that generically
b (k), if at all extendable to the upper half plane, has poles on the
imaginary axis, corresponding to i« IK ,..., itc . Working as in (A.1)
we avoid them.
Let us fix c > 0 and choose 0 < r. < c.
Putting w = x+4c2t+y we integrate b (k)e ' e around a rectangle in
the complex k-plane with vertices at -R, R, R+it;, -R+ir. By (A. 1) one has
for r = ±R
,-, I f r + 1 £ v r, \ 2ikw Sitk' | . 2Me f' ,, , . ., -24r 2ts,(A.3) b (k)e e dk S e |b (r+is)|e dslrJ I 0 J
f0J
2Mf= kr • -p- max lb
r(r+ i s )!
MtQ r Oisr:
= o(1) for R - ».Thus, using Cauchy's theorem
,. . , fR , ,, s 2ikw 3 i t k 3 , , fR , ,, . . 2i(k+ic)w 8 i t (k+ir.)3 ,.(A.4) b (k)e e dk = b (k+j.c)e e dk
-RJ r -RJ r
Now the integrand on the right clearly belongs to L (E).
Hence, if we let R -* •» in (A.4) we find that the integral in (3.1) is
Cauchy convergent and equals
(A.5) H (x+4c2t+y;t) =
where the integral on the right converges absolutely.
From (A.5) one easily deduces
(A.6) ]H c&4c z t+y;t) | < y^e~2 c ye~a t ,
—24c];2 twith Y- = - e2cA f°° jb (k+ie) (e °dk and a = 8r (c2-r2) - 0.
Obviously we can differentiate (A.5) with respect to x uniformly in y.
Estimating the derivative one finds
25
(A.7) |^ ^
r» -24ck21with Y 2 = \ e
2 £ M |k+ie||br(k+iE)|e dk.
Hence the proof of (A.2) is complete.
Appendix B: Case P.
Let us demonstrate that the conditions
(B.1) b (k) is n 'i 2 times differentiable on the real axis;
(B.2) b ( m ) = o(k 2 m + 2), |k| * -, m - 0,1,....n-1;
b b 0 )
(B.3) r-—,, T—^=I b<n"1), <i + |k|)b<n)
(1+|k|)n 1 ( H | k | ) n 2
belong to L (E);
guarantee the estimate
(E.4) |fi Cx+vt+y;t)| + | SI' (x+vt+y;t) | Sfi Cx+vt+y;t)| + | SI(x+vt+y;t) | S ,c c (-a+y+vt)n
-M, t 2 t - > —, v = 4c2 , c - 0.
For brevity of writing we put
(B-J' t 3x t' f W *' f
Furthermore we define
(B.6) w = jc + vt + y ,
(B.7) (j> = 2k(x+y)+atk(c2+k2) = 2kw + 8tk3 ,
(B.3) s = * °^j 2 s + 2 y + 2 v t + 24tk2 * °-
Now, since b ,b ,...,b are locally integrable we can integrate by
parts n times to find
26
R , R .(B.9) [ ei+b dk = -ise1* ï <iT)b +
_RJ r *• w r _ R _RJ
where the operator T is defined by
(B.10) Tf = (sf) ( 1 ) = s(1)f + sf(1).
Induction reveals that the t-th iterate of T has the structure
(ii. 11a) T f = s Z ctp f P » with ap = 1, whereas for p
U \ Ii - i . . . i\ s / \ s
^ =p
where a,, » » » are nonnegative integers, independent of s and i.
Applying Leibniz' formula to the identity (2w+24tk2)s = 1, we find
.12) (w+12tk2)s(j) + 24jtks(j"1) + 12 j (j-1 )ts ( j" 2 ) = 0,(£
from which it is easily seen that
05.13)(j)
3 1,2
where M. i s a c o n s t a n t .J
Thus, in view of (B.11) there are constants A„ such that^ P
Returning to (C.9) we extract from (B.2) and (B.14) that
fR • .(B.15) e l l | ) [b - ( i T ) " b ]dk = o ( 1 ) for R - » .
-R J r r
i lence, u s i n g ( B . 3 ) , (B.14) and ( 3 . 1 ) , we o b t a i n
e l(iT)b rdk,
where tlie integral is absolutely convergent. From (H.14) we find the bound
_(B.17) |:.'c(x+vt+y;t)| (-M+y+vt)
1 " f ibr Iwith u. = I_ A dk.
1 n p=0 n,p _ J , | h P
27
Next, we consider the x-derivative of the integrand in (B.16). Since
n , _ .we are led to examine (Tnb )' = £ n (s"a )'b^n~p , in which (s"a )' is
r p=0 n,p r n,pa linear combination of terms of the following form
where l} + 2^ + ... + pi = p, lQ + ^ + . . . + £ = n.
Since s' J = (-2s2) J we obtain from (B.13) and Leibniz' formula
(3.20) |s' J | S M. , where M. is a constant.J (l+|k|)J J
Estimating each term (B.19) by (B.13) and (B.20) one gets
(TnbrV p |bj
where the A are constants.n,P
Finally, combining (B.14), (B.18) and (B.21), we find constants B suchn,p
that
(B.22)1 7i r ' , .n p=ü ,, i I .p-1 ' r
(-M+y+vt) v (1+|k|)
where by (B.3) the right hand side belongs to L (K).
To prove that Ü is strongly x-differentiable we proceed as follows.
For h > M-vtQ let g(h) denote T br with x replaced by x + h. One has
I Si (x+h+vt+y;t) - Q (x+vt+y;t)(B.23) M r—£ — I (eM'Tnb )'dk!11 -J S br
(B.24) Gh = |e2lkh(g(h)-g(0)-hg'(0)) + hg'(0)(e2lkh-l) +
+ g(0)(e2ikh-1-2ikh)|
|S"(0h)|+|hg'(0)|.]e2ikh-1|+|g(0)|.|e2ikh-1-2ikh!
for some 6 £ (0,1).
Examining the x-derivative of (B.I 9) we obtain the estimate
n + 2 ^ n P )03.25)
where the A are constants.n,p
From (B.14), (B.21), (B.24) and (B.25) it is clear that
(B.26) lim -JT- f ( sup G )dk = 0.h+0 l
hl -J Vy<+» VHence, (B.23) yields that fi is strongly x-differentiable, the
derivative being given by
. n r0
(B.27) n'(x+vt+y;t) = — (e *T b )'dk,C TT ' _ ' »
satisfying the obvious bound
(B.28) |fi'(x+vt+y;t)| S
U2
(-M+y+vt)n
n p» |b n P |
=0 Bn I " =T d k '
This completes the proof of (B.4).
i iAppendix C: Generalization to higher KdV equations. [
We now show that the method described in this paper is still working
in a more general setting.
In [2], Appendix 3, the class of evolution equations
(C.1.a) qt + CQ(L*)qx = 0, q(x,0) = qQ(x)
+ _ _ i 32 _ i j
x-*
where C is a ratio of entire functions, is found to be solvable by
29
the inverse scattering transformation associated with the Schrödinger
equation.
Setting q = -ü, c„(z) = -a(-4z) we can rewrite (C.1.a) as
(C.1) u = a(L)u , u(x,0) = un(x)
L = ^ L _ 4 u + 2ux j dy.In this form the class has been investigated by Calogero (see [4] and
subsequent papers). Introducing the solution u(x,t) of (C.1) as a
potential in the Schrödinger equation one obtains the following simple
time evolution of the spectral parameters
(C.2) br(k,t) = br(k)exp{2ik«(-4k2)t},
(C.3) <.(t) = K-J(O) = ^ , j = 1,2,...,N,
(C.4) c'i(t) = crexp|-K.ti(4K2.)t}, j = 1,2... . ,N.
In particular, choosing n(z) = —z we rediscover the KdV equation and its
wellknown time evolution in spectral space (see (2.5) and (2.6)). For
simplicity we shall assume a(z) to be a polynomial. In this case the
equations (C.1) are generally called "higher KdV equations", though, of
course, this appellation is only relevant if •» has degree higher than one.
Let us first consider some special solutions of (C.I).
If b (k) = 0 and N = 1, we find
(C.5) u(x,t) = - 2 K 2 2
(C.7) v1 = - U ( 4 K 2 ) ,
which is immediately recognized as the celebrated single soliton
solution.
If b (k) = 0 and N • 1 one obtains by an exercise in linear algebra
the so-called pure H-soliton solution, which is such that a transformation
to moving coordinates with speed
(C.3) v. = -u(4>-2.), j = 1,2,...,N
30
makes the j-th soliton stationary as 11 •* «•, provided that all v.'s are
different.
In the case b (k) # 0 dispersive waves enter in the solution. The
linearized version of (C.I), reading
(C.9) ut = u ( ^ r ) v u(x,0) = uo(x),
is a dispersive equation with associated group velocity
(CIO) vn = - -LkaHc 2)], k £ R.
Now, if all v.'s have the same sign, while v has the opposite sign (as
J S
is the case for the KdV equation with v. = 4K2. and v = -3k 2), one
expects that for large time the soliton part and the dispersive component
will separate out. In order to convert this expectation into a
mathematical fact we shall impose the following conditions upon the
function a occurring in (C.1):
(C.11a) a is a polynomial,
(C.1 1b) v = - ^ [ku(-k2)] • 0, k 6 R, j
( d i e ) V J = - C X ( 4 K ? ) - 0 , j = 1,2 N» J
i
(C. lid) v. * v. for i * j. !
To reassure the reader let us mention a class of functions meeting these j
requirements j
(C.12) oc(z) = -z , m • 1 an odd integer. ;
For convenience we shall order the solitons so that
(C.13) 0 < vN < ... < v2 <• v,.
We can now generalize theorem 5.1 as follows:
31
Let u(x,t) be the solution of the higher Korteweg-de Vries equation
(C.I), with a as in (C.ll), evolving from an arbitrary real initial
function u.(x) I such that uQ(x) is sufficiently smooth and decay o
sufficiently rapidly for |x| •* °° for the whole of the inverse scattering
method to work and to guarantee an estimate of type (3.2)).
Then fop any choice of the constants v > 0 and M % 0 there is a function
o(t) such that
(C.14) sup |u(x,t) - u,(x,t)| = O(a(t)) as t + «.xê-M+vt
Here u,(x,t) is the pure ii~soliton solution, H being the number of
discrete eigenvalues corresponding to the potential u„(x). The function
a(t) tends to zero as t •+ « and the exact behaviour of o(t) depends
on properties of the reflection coefficient b (k).
If b (k) is n ï 2 times differentiable on th<: real axis, with
b^ m ) = o(k 2 m + 2), |k| - », tn = 0,1,....n-1 and
belona to
L'(IR), then o(t) = 0(t 1~ n).
To prove this we repeat the analysis given in sections 2 to 6 and
Appendix B with the following obvious adaptations:
In (2.3) the ordering is replaced by (C.13). Furthermore (2.5) and
(2.6) are replaced by (C.2) and (C.4). Throughout, the speed 4c2 is
identified with v and case C is left out.
(B.7) becomes :• = 2kw + 2s(k)t; g(k) = ka(-4k2).
(B.8) becomes s = —rvy = — TTK • 0.:• ' 2x+2y+2vt+2gU J(k)t
(2w+24tk-')s = 1 is replaced by (2w+2g^1 (k)t)s = 1.
(11.12) becomes (w+g( ' } (k) t)s (j } = -t ^ ( )g ( r + I ) (k)s ( j" r ) .
32
The reader is Invited to verify in detail that our analysis remains
valid once the alterations summarized above have been carried out.
References
[ 1] M.J. Ablowitz, Lectures on the inverse scattering transform,Stud. Appl. Hath. 53 (1978), 17-94.
[ 2] M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, The inversescattering transform - Fourier analysis for nonlinear problems,Stud. Appl. Math. 53 (1974), 249-315.
[ 3] M.J. Ablowitz and H. Segur, Asymptotic solutions of the ICorteweg-de Vries equation, Stud. Appl. Math. 57 (1977), 13-44.
[ 4] F. Calogero, A method to generate solvable nonlinear evolutionequations, Lett. ÏJuovo Cimento 14 (1975), 443-447.
[ 5] A. Cohen, Existence and regularity for solutions of the Korteweg-de Vries equation, Arch, for Rat. Mech. and Anal. 71 (1979),143-175.
[ ó] P. üeift and E. Trubowitz, Inverse scattering on the line, Comm.Pure Appl. Math. 32 (1979), 121-251.
[ 7] W. Eckhaus and A. van Harten, The Inverse Scattering Transformationand the Theory of Solitons, North-Holland Mathematics Studies 50,1981.
[ 8] C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura, Method forsolving the Korteweg-de Vries equation, Phys. Rev. Lett. 19 (1967),1095-1097.
[ 9] C.S. Gardner, J.M. Greene, M.D. Kruskal and K.M. Miura, Kortewer;-deVries equation and generalizations VI, Comm. Pure Appl. Math. 27(1974), 97-133.
[10] R.M. Miura, The Korteweg-de Vries equation: A survey of results,SIAM Review 18 (1976), 412-459.
[11] H. Segur, The Korteweg-de Vries equation and water waves, J. FluidMech. 59 0973), 721-736.
33
CHAPTER TWO
ASYMPTOTIC ESTIMATES OF SOLUTIONS OF THE KORTEWEG-DE VRIES
EQUATION ON RIGHT HALF LINES SLOWLY MOVING TO THE LEFT
We consider the Korteweg—de Vries equation u - 6uu + u = 0 witht x xxx
arbitrary real initial conditions u(x,0) = u_(x), sufficiently smooth and
rapidly decaying as |x| •-> °». Using the method of the inverse scattering
transformation we analyse the behaviour of the solution u(x,t) in coordinate
regions of the form t -? tQ, x ~- - n - \>T, T = (3t) where u, v and t_ are
nonnegative constants. We derive explicit x and t dependent bounds for the
nonsoliton part of u(x,t). These bounds enable us to prove a convergence
result, which clearly displays the emergence of solitons. Furthermore, they
help us to establish some interesting momentum and energy decomposition
formulae.
1 . Introduction.
We study the Korteweg-de Vries (KdV) problem
(1.1a) ut - 6uux + u x x x = 0, ~» •- x •• +», t - 0
(1.1b) u(x,0) = uQ(x),
where the initial function u-(x) is an arbitrary real function on R, such
34
that
(1.2) un''x) ^s sufficiently smooth and (along with a number of its
derivatives) decays sufficiently rapidly for |x| -<• •» for the
whole of the inverse scattering method to work and to guarantee
certain regularity and decay properties of the right reflection
coefficient, to be stated further on.
To make the discussion less abstract let us quote a definite example from
[4] in which (1.2) is fulfilled:
(1.3a) u_ is of class C3 on E and has a piecewise continuous fourth
derivative,
(1.3b) U Q J ) ( * ) = 0(|x|~M) as |x| •> « for j •• 4,
where M > y. Here y is a constant which is 8 in the generic case (see
(2.13)) but which is 10 in the exceptional case (see (2.14)). In [4] it
is shown by an inverse scattering analysis that condition (1.3) guarantees
the existence of a real function u(x,t), continuous on H*[0,t»), which
satisfies (1.1) in the classical sense.
Let us recall that there is uniqueness of solutions of (1.1) within
the "Lax-class", i.e. the class of functions which, together with a
sufficient number of derivatives vanish for |x| -• °», as discussed in [12].
Whenever, in the sequel, we speak of "the solution" of (1.1) we shall
refer to the solution obtained by inverse scattering (unique within the
Lax-class).
The long-time behaviour of the solution u(x,t) of (1.1) has been
discussed by several authors, the general picture being, that as t •* <"
the solution decomposes into a finite number of solitons moving to the
right and a dispersive wavetrain moving to the left. The emergence of
solitons from initial conditions as arbitrary as (1.2) was demonstrated
rigorously in [8] corresponding to Chapter 1 of this thesis. It was
proven there, that for any choice of the constants v • 0 and M - 0 there
is a function o(t) such that
(1.4) sup |u(x,t) - u (x,t)| = 0(o(t)) as t ->• »>,xS-M+vt
35
where u,(x,t) denotes the pure N-soliton solution (see (5.16)), N being
the number of bound states corresponding to u (x), when introduced as a
potential in the Schrödinger scattering problem. The function a(t) tends
to zero as t + • and the exact behaviour of a(t) depends on properties
of u0.
If u„ decays exponentially for x -*•+•» then u(t) = 0(e ) for some
constant a > 0.
If u„ and its derivatives up to the fourth one decay algebraically
for ]x| •+ •» sufficiently fast, then a(t) = (Kc ) f°r some constant m 2 1.
Earlier results in this direction, though less detailed and not widely
known, were given in [16], where it was shown that
(1.5) lim sup |u(x,t) - u,(x,t)| = 0 for v > 0 arbitrarily fixed.t-H» x>Vt
In fact, in recent years, most of the asymptotic attention was devoted to
the solitonless KdV initial value problem. In this case, the analysis
given in [1] led to the recognition of four distinct asymptotic regions
i. x a 0(t) ii. |x| s 0(t1/3)
III. -x = 0{t1/3(ln t)2/3} IV. -x 'i 0(t),
where 0 denotes positive proportionality. Within each region, the
solution u(x,t) has an asymptotic expansion, characteristic for that
region. However, interesting as they may be, the results are far from
rigorous. Indeed, discussing the matter in their book [2], p. 68 the
authors remark
"Two warnings should be made before we begin the analysis. The
first is that almost none of the results to be described in this
section are known rigorously. These results are formal, and have
great practical value, but proofs of asymptoticity are yet to be
given. The second (related) warning is that some of the existing
literature on this question contains errors".
In this paper we extend the asymptotic analysis given in Chapter 1. We
use the method of the inverse scattering transformation to analyse the
behaviour of the solution u(x,t) of (1.1) - both in the absence of
solitons as well as in the general case - in coordinate regions of the
36
form
1 73(1.6) t È t0, x 2 -c, C = y + vT, T = (3t)
where u, v and t. are nonnegative constants. Here |j is arbitrary, but the
values of v are restricted to 0 S u < v where v is some generic number
connected with the Airy function, its numerical value being 1.39. Further-
more, t„ depends on u, v and the behaviour of u„ as well. Note that (1.6)
covers region I and almost all of region II.
It is shown that in the absence of solitons
(1.7) sup |u(x,t)| =0(t" 2 / 3) as t °°.x>-c
In the general case we improve (1,5) by
(1.8) sup |u(x,t) - ud(x,t)| =0(t"1 / 3) as t •* ».
XÈ-5
This leads to the convergence result (7.17), which clearly displays the
emergence of solitons. Moreover, we construct several remarkably explicit
x and t dependent bounds for the nonsoliton part of the solution valid in
regions (1.6). With the help of these bounds we establish the momentum and
energy decomposition formulae
(••» N( 1 . 9 a ) u ( x , t ) d x = - 4 I . K + 0 ( t "
_ J P = 1 P
( 1 . 9 b ) j u ( x , t ) d x = - | Qf°° l o g ( 1 - | b r ( k ) | 2 ) d k + 0 ( t ~ 1 / 3 ) a s t ^ «
f°° if, N - 1 / 1
(1.9c) u2(x,t)dx =-if ï t ' + 0 ( t " J ) a s t > »_qJ P P
(1.9d) u2(x,t)dx = - | 0 / k2 log(1- |b r(k) |2)dk+0(t" l / J) as t - »,
where -K 2 (K > 0) p = 1,2,...,N are the bound states and b (k) is the
right reflection coefficient associated with the potential un(x) in the
Schrödinger scattering problem.
Let us point out three major differences between the present paper and
[1]. Firstly, it is our main purpose to obtain explicit bounds for the
nonsoliton part of the solution of (1.1), valid in the region (1.6),
whereas in [i] the emphasis lies on the construction of asymptotic
37
expansions of the (solitonless) solution in the various regions. Secondly,
the analysis in [1] requires that u~ decays faster than exponentially as
x -+ +<=. For our results, however, except explicitly stated otherwise, an
algebraic decay rate of type (1.3) is sufficient. The third difference
lies in the fact that we venture to call our results rigorous.
The composition of this paper is as follows. In section 2 we briefly
discuss the direct and inverse scattering problem for the Schrödinger
equation and formulate our problem. In section 3 we isolate certain
properties of the function Q and the operator T , both occurring in the
Gel'fand-Levitan equation. These properties are used in section 4 to
investigate the solution of (1.1) in the absence of solitons. In the sub-
sequent sections it is supposed that solitons are present. In that case
the operator I + T appears in the Gel'fand-Levitan equation. Section 5
is devoted to showing that this operator is invertible. In section 6, the
operator S = (I + T.) is applied to both sides of the Gel'fand-Levitan
equation. It is shown that the resulting equation has a unique solution fi,
representable by a Neumann series. Finally, in section 7, we write R as
the N-soliton state plus a perturbation. This leads to a decomposition of
the solution u(x,t) ot (1.1) into an N-soliton part and a nonsoliton part.
For the nonsoliton part we derive explicit estimates and discuss their
consequences.
The notation used is similar to that in Chapter 1. Since the constants
appearing in this paper have a simple structure, we have traced them all.
In this way the interested reader can obtain numerical estimates in a
practical case.
2. Preliminaries and statement of the problem.
2.1. Direct scattering at t = 0.
Let us briefly review the direct scattering problem for the
Schrödinger equation
\ £• • i / 1 ~ v, is. r\ ''T — " > T ^ " j
38
where u„ is the initial function in (1.1b) and k a complex parameter. For
details we refer to [A], [6] and [7], Our notation closely resembles that
used in [7] .
The results of this subsection are valid for any real function u„(x),
continuous on K, vanishing at infinity and integrable with respect to
x2dx.
For In k i 0 we introduce the Jost functions ijj (x,k) and (Jr„(x,k), two
special solutions of (2.1) uniquely determined by
(2.2a) ij, (x,k) = e~lkxR(x,k), lira R(x,k) = 1, lira R (x,k) = 0r x
X-S--1» K-»-»
(2.2b) i^(x,k) - e l k x L ( x , k ) , lira L(x,k) = 1, lira Lx(x,k) = 0.x-v+°° x->-+w
The functions R, R , R , L, L , L are continuous in (x,k) on RxC and' x' xx' ' x' xx ' +
analytic in k on C + for each x 6 TR. Furthermore, for k real, k * 0, the
pairs i(.^,(x,k), i|) ,(x,-k) and J<r(x,k), V'r(x,-k) constitute fundamental
systems of solutions of equation (2.1).
In particular we have for x £ R, k £ E\{0}
(2
(2
(2
.3a)
.3b)
• 3c)
r+(k)
r (k)
r+(k)i|,£(x,k)
where W[I|J. ,!(/„] = 'i'A'n ~ ^1 ' 2 e n o t e s t n e Wronskian of ifi. and \p~. It is
easily verified that for k £ R\(0}, writing for complex conjugation:
(2.4) r*(k) = r.(-k) r*(k) = r (-k)+ +
(2.5) |r_(k) 2
The representation (2.3c) enables us to extend r_(k) to a function
analytic on Im k > 0 and continuous on lm k J 0, k * 0. One can prove,
that r_(k) has at most finitely many zeros, all simple and on the
imaginary axis. Let us denote them by itc ., j = 1,2,...,N and order
(2.6) K 1 > K 2 > ... •> < N > 0.
It turns out that the eigenvalues of (2.1) (the so-called bound states)
are given by -K? < -<% < ... < ~K*. The associated L2-eigenspaces are
one-dimensional and spanned by the real-valued exponentially decaying
functions iK(x,iK.) , j = 1,2,...,N. In terms of these one defines the*~ J
(right) normalization coefficients
* = [ [•* '-—00''
(2.7)
Note that
(2.8) c* = lira if). (x)e -* ,
where ty. (x) stands for the normalized eigenfunction lliK(. ,1K .) II 2i)i.(x,i<.).
In [6] it is proven that (2.1) has a bound state if and only if i(;-(x,0)
vanishes for some x.
Next, we introduce the following functions for k £ E\{0}
(2.9a) ar = r_ , the (right) transmission coefficient
(2.9b) b = r+r , the (right) reflection coefficient.
The appellation is motivated by the asymptotic behaviour of \j> for
|x| -* co with k £ E\{0} fixed:
(2.10) ar(k)i//r(x,k) « e"l k x + br(k)e
lkx as x •* +»
,. s -ikx*** ar(k)e as x -+ -•».
The growth of u_ permits us to extend a and b in a natural way to
continuous functions on all of R, where they satisfy
(2.11) a*(k) = ar(-k), b*(k) = br(-k)
(2.12) |ar(k)|2 + |br(k)|
2 = 1 and for k £ E\fO} |ar<k) | > 0.
It is shown in [7], that br is an element of L1 D L2(1R), which behaves
as o( |k| ) for k -+ ±».
We shall call the aggregate of quantities {b (k),K.,c.} the (right)
scattering data associated with the potential u_. It is a remarkable fact
that a potential is completely determined by its scattering data.
AO
In general, starting from a given potential u it is not possible
to obtain the scattering data in closed form. An exception constitutes
the potential U Q ( X ) = ~^U + 1)sech2x, X > 0, as can be seen from
Chapter 3, section 5.
Let us examine the behaviour of the reflection coefficient b (k) in
some detail. Following [4], we shall from now on distinguish two cases,
the "generic case" and the "exceptional case". In the generic case, the
Jost functions ty (X,0) and <]i„(x,0) are linearly independent
(2.13) lim 2ikr_(k) = W[if> (x,0),<f/„(x,0)] * 0,k->-0 r
whereas the exceptional case is characterized by
(2.14) lim 2ikr (k) = W[i|i (x,0) ,1>. (x,0) ] = 0,k->0 r Z
imkiOexpressing the linear dependence of <p (x,0) and iK,(x,0). The value of b
at k = 0 will frequently appear in our analysis. Therefore, let us give
it due attention.
In the generic case, we obtain from (2.3-13)
(2.15) br(k) = -1 + ark + o(|k|) as k -+ 0,
where a * 0 is some constant,r
In the exceptional case, there is a constant a„ € R\{0} such thato
(2.16) i r(x,0) = a Q^(x,0).
As a consequence of (2.3-14) both r+ and r_ have a finite limit as k + 0.
Taking k -+ 0 in (2.3a) we find
(2.17) r+(0) + r_(0) = aQ.
Hence, in view of (2.4-5)
(2.18) r+(0) = è(c0 - a"'), r_(0) = J(aQ + a^1),
(2.19) br(0) = — ^ 6 (-1,1).
Observe that b (0) = 0 if and only if we are in the "degenerate exceptional
.'.1
case" characterized by
(2.20) i> (x,0) =±(J(„(x,0).
As an example of an exceptional case consider the potential
u _ = v + v2, v € C (R), where v and v decay rapidly enough. Then
(2.21a) ijj (x,0) = exp v(y)dy , iK(x,0) = exp - v(y)dy
(2.21b) aQ = expf J v(y)dyj, br(0) = tanhf j v(y)dyj.
Since \p„(x,0) does not vanish, there are no bound states in this example.
On the other hand, any potential u„ S 0, u» 4 0, is generic with no bound
states.
In [13] it was supposed that the zeros of b (k) are associated with the
discontinuities of u~(x) and that b (k) would have no zeros for a
perfectly smooth initial condition. The previous example shows that this
supposition is incorrect: e.g., any rapidly decaying odd function v of
class C (K) produces a perfectly smooth u~ with b (0) = 0.
Next, let us mention some regularity and decay properties of b (k). If
u.(x) merely satisfies the conditions stated at the beginning of this
subsection, then we have b (k) = o(|k] ) for k •+ +<*> and no better.
However, if u„(x) has rapidly decaying derivatives, then b (k) has
rapidly decaying derivatives as well. The converse is also true.
Specifically, u~ is in the Schwartz class if and only if b is in the
Schwartz class.
To be less demanding, suppose u 0 satisfies the hypotheses (1.3). Then [4],
lemma 2.4 tells ua
(2.22a) In the generic case, b is of class C on R.
ÏÏ1Ü-2(2.22b) In the exceptional case, b is of class C on R\{0} and of
class C on all of R.
(2.22c) In either case, b^^(k) = 0CIUj~5> as k •+ ±°° for j < M-2.
Here IMI denotes the largest integer strictly less than M. As a final
remark, let us mention that it is straightforward to obtain the following
result from the representation formulae (4.2.4.2-4) in [7]:
42
If u,, satisfies (1.3) and there exists an E. > 0 such that UQ(X) =
= O(exp(~2£gx)) as x ->• +« then for any E^ with 0 < e^ < min(eo,KN) the
function b (k) is analytic on 0 < lm k < E. and continuous onr I — *
0 S Im k i e,, while in that strip b (k) = 0(|k| ) as |k| •+ °°. In
particular, if u- decays faster than exponentially as x + t» and has no
bound states, then b (k) is analytic on the open upper half plane.
In the derivation of the asymptotic expansions given in [1] the
extension of b to the upper half plane plays a crucial role. It is there-
fore surprising that we shall not need any supposition of this kind for
our asymptotic analysis, except for an example given at the end of section
7.
2.2. Inverse scattering for t > 0.
From now on we shall impose stronger conditions on the initial function
u„, than those stated at the beginning of the preceding subsection. In
fact, we shall assume that (1.2) is fulfilled.
As a specific example it is good to keep in mind that, except explicitly
stated otherwise, all the results of this paper are valid if (1.3) holds.
Consequently, there is a unique (within the Lax-class) real function
u(x,t), continuous on Rx[0,<=) and satisfying (1.1b), which is a
classical solution of the KdV equation (1.1a) for all t > 0.
Moreover, for fixed t S 0, the function u(x,t) decays rapidly enough for
|x| + • to fall in the class of potentials discussed in the previous sub-
section.
The important discovery by Gardner-Greene-Kruskal-Miura [9], [10] is the
following:
For each t a 0, introduce u(x,t) as a potential in the Schrödinger
scattering problem
(2.23) ijj + (kz - u(x,t))i|> = 0 , -» < x ••- +»;
then the bound states -K 2 •- -K* < ... < -K* do not change with time,
whereas the associated normalization coefficients and reflection
coefficient change in a simple way
(2.24) cT(t) = cTexpUKU}, j = 1,2,...,N
(2.25) b (k,t) = b (k)exp{8ik3t}, -•» < k < +~.
To determine the solution of the KdV problem for all t > 0, one
exploits the fact that the potential u(x,t) can be recovered from the
scattering data {b (k,t),K.,c.(t) } by solving the inverse scattering
problem. For that purpose we introduce the real functions
(2.26a) «(£;»•) = « dU;t) + a^-.t), f, e R, t > 0
N -2K.C
(2.26b) SJd(S;t) = 2 j|1 [cT(t)]2e j , 0
(2 Joo t
br(k,t)e lkCdk.
Since b (k,t) is in L1 fl C_ (-» < k < +•»), the integral in (2.26c)
converges absolutely and J2 (C;t) belongs to L2 D C- (-•» •' 5 < +»).
Consider now the Gel'fand-Levitan equation (see [7])
(2.27) g(y;x,t) + S2(x+y;t) + 0/™ fi(x+y+z;t)S(z;x,t)dz = 0,
with y > 0, x £ R, t > 0.
In this integral equation the unknown S(y;x,t) is a function of the
variable y, whereas x and t are parameters. For each x € E, t > 0 there
is a unique solution 8(y;x,t) to (2.27) in L2 (0 < y < +»). Furthermore,
g(y;x,t) is real and belongs to L1 D L2 fl C (0 < y < +»), such that both
limits at the boundary of 0 < y < +°° exist . In fact we have
(2.28a) lim S(y;x,t) = 0y-M-00
as well as the important property
(2.28b) B(0+;x,t) = f°° u(x,t)dx, x € E, t > 0.
Herewith the inverse scattering problem is solved, since the solution of
the KdV problem is given by
(2.29) u(x,t) = - ~ B(0+;x,t), x € K, t - 0.
Note that the original problem for the nonlinear partial differential
equation (1.1) is essentially reduced in this way to the problem of solving
a one-dimensional linear integral equation. Explicit solutions of (2.27)
44
have only been obtained for b = 0. On account of its asymptotic
behaviour (cf. (5.21)), the solution ud(x,t) of the KdV equation with
scattering data {0,<.,c.(t)} is called the pure N-soliton solution
associated with u„(x).
2.3. Statement of the problem.
We shall study the solution of (2.27) in parameter regions of the
form
(2.30) t £ t0, x > - u - vT, T = (3t) 1 / 3
where p, v and t„ are nonnegative constants. Here y is arbitrary, but the
values of v are restricted to 0 S v < v where v is some generic numberc c
to be specified later on. Furthermore t_ depends on u, v and u_.
It is essential to give our problem a convenient abstract formulation.
Remarkably enough the function space introduced in Chapter 1 also works in
the more general setting of this chapter.
So, once again, let V denote the Banach space of all real continuous
and bounded functions g on (0,°°), equipped with the supremum norm
and
(2.
write
31)
II g U
(Tdg)
sup0<y<+
(y) = d(x+y+z;t)g(z)dz
(2.32) (T g)(y) = f 'u (x+y+z;t)g(z)dz.c nJ c
0J
As in Chapter 1, it is readily verified that T, maps V into V. In the next
section it is shown that if the right reflection coefficient b (k) satisfies
certain, rather modest regularity and decay conditions, then T is indeed
a raapping of V into V. Nevertheless, there is a fundamental difference
with the situation of Chapter 1. There, IIT II tends to zero as t •* » (cf.
Chapter 1,(3.13)). In the region (2.30), however, T is only a small
operator. In fact, v must be chosen with due care so as to guarantee that
llT II < 1. More so, it is an amusing question whether T is a small
operator in the region (2.30) with >> arbitrary! Plainly, this difference
in the long-time behaviour of llT II has its impact on the technicalities
(see section 6).
In the above notation our problem amounts to analysing the solution of
(2.33a) (I + Td)S + 1 6 = -fi, 8 6 V
(2.33b) S. = «d + «c,
- where I is the identity mapping - in the parameter region (2.30).
As in Chapter 1, we know the solution B. of
(2.34) (I + Td)Pd = -nd,
yielding the pure N-soliton solution of the KdV equation. Despite the
differences in the long-time behaviour of IIT II, the basic thought of the
analysis in this chapter is rln same as in Chapter 1, namely to treat the
full problem (2.33) as a perturbation of the pure N-soliton case (2.34).
3. Analysis of a and T .
Let us examine
(3.1) I2(x+y;t)-l f b (k)e^"VXTy^oiK"Ldk
1/3in the parameter region T = (3t) > 0, x > - v - vT, where v and ~J are
nonnegative constants. Performing the change of variables s = 2kT,
T = (3t) 1 / 3 we find
(3.2) ^ J " br<^)exp
which shows that the asymptotic behaviour of L? for t •+ » is intimately
related to the behaviour of the Airy function
1 T s3
(3.3) Ai(n) = -=- lim exp[ins + i -]ds, ri e E.2" R-» -RJ 3
Of course we can rewrite (3.3) as
(3.4) Ai(n) = lira Re /R exp[ins + i -y]dsR-KO
46
1 . . [E
= — l i m I11 R ~ O-"
scos[ns + -^-]
a form which is also frequently encountered in the literature.
Let us list some wellknown [14] properties of (3.3) that will be used in
the sequel. The function Ai(n) has an analytic continuation to the whole
complex n plane, which satisfies
(3.5) ~ Ai(n) = ') Ai(n), i, e C
For n - 0 one has the estimates
(3.6a) 0 < Ai(n) S w 7
n2
(3.6b) x = fn2 3/2
Together, (3.5) and (3.6) imply that for any constant y • 0
(3.7) a o(v) = sup (|Ai(m)(n)|exp[|o(^ + n)372]} - +»
ffl' 1)5— v I LJ JJ
for 0 i 6 < 1, m = 0,1,2
Let us conclude with an innocent looking property of the Airy function that
plays a key role in this paper
(3.3) Qr Ai(n)dn = .
Starting point for our analysis of in tho parameter region1/3 • r
T = (3t) • 0, x • — p - i-T is the following transcription of (3.2)
f'" s 3
(3.9a) :Mx+y;t) = r b(, s ,,;)oxp [ i- s + i -j-]ds, with
(3.9b) b(k,u) = br(k)e"2ik!', • = 2T , ' = Y^ •
Thanks to this representation our task is n-duced to the examination of
the integral (3.9a) in the parameter region 0, • -..
Basically, this examination is performed in the next two lemmas. These
are then combined to give theorem 3.3 describing the structure of
Throughout the following notational convention is used. If f : IR>.[0, ) - C
is bounded, then we write for u •• 0
k€R
The first lemma isolates certain regularity and decay properties with
respect to n of integrals of type (3.9a).
Lemma 3.1. Let g be of class C2[0,=°). Assume that the derivatives g -1 (s),
j = 0,1,2 satisfy
( 3 . 1 0 ) g ( s ) = 0 ( s ) , g O ) ( s ) = 0 ( s ) , g ( 2 ) ( s ) = 0 ( O
for s -* +°°. Then
fR
(3.11) I(n) = lim g(s)exp[ins + i ^-]dsR-«° 0J
i s well-defined for all n E R and I G C2 (-» < n < +») uit^z
(3.12) l i m ( J - ) ^ I ( n ) = 0 , j = 0 , 1 , 2 .
Furthermore, if instead of (S. 10) we make the isii'onjur assurni'tion
(3.13) g(j)(s) = 0(D for s •* +«•
iten ite following representation holds
(3.14) T — I(n) = lim isg(s)exp[ins + i -^-]ds.
Proof: The idea of the proof is to rewrite I(n) as a nice integral over a
finite interval plus a remainder which can be treated using integration
by parts.
In fact, we claim the following. Let n_ ' 0 be an arbitrary constant. Set
s = / i + n . Then for ri 5 —n one has
(3.15) I(n) = IyM + I 2 (n ) + I 3 (n) , with
(3.16a) I ^ n ) = QSS° g(s)exp [ins + i ^ - ] d s ,
(3.16b) I2(r,) = ^
oo S3
(3.16c) 1'n) = - / G(n,s)exp[iiis + i -=-]ds , whoro sn - s and
^ , ( 2 ) ( S ) .
To prove this it obviously suffices to show that
43
(3.18) limR —
Let us write
(3.19a) * =
rR ig(s)exp[iris + i -y]ds = I2(n)
(3.19b)
Then for s £ s_, n 5 ~n„ one has
(3.20) 0 <-nQ + s
2 S 1.n + sz
Now, integrating by parts twice we find for R > s
R R
so so'gds,
1(3.21) e Tgds = -I>I>É J-
where the operator T is defined by
(3.22) Tg = -^-(^g)-
Clearly,
(3.23a) (Tg)(n,s) = ^ ' ^ ^ g t s )
(3.23b) (T2g)(n,s) = G(n,s)
with G as in (3.17).
Substituting (3.23) into (3.21), taking R ->• » and usini; (3.10) we arrive
at the desired identity (3.13), where the integral defining I3 is
absolutely convergent.
H e r e w i t h I ( ' i ) i s w e l l - d e f i n e d f o r n ' ~ r U . N e x t , l e t u s show t h a t
1 e C2 (-nQ <, n < +») with (3.12).
Note first that I. and I„ behave perfectly. Actually, both I. and I?
belong to C (~1Q - n < +ro) and all derivatives vanish for n -+ +<». As for
I., the first statement follows from dominated convergence, the second
from the Riemann-Lebesgue lemma. As for I_, both statements are evident
from its explicit form (3.16b).
Thus it remains to consider I.. From (3.17) we readily obtain that for
each fixed s 5 s_ the function n> ->• e G(n,s) belongs to C2 (-'U s " '; +m)
with
(3.24) lira (- ) = 0, j = 0,1,2.
Moreover, in view of (3.10-20) there are constants c.(n,>,g), depending
only on r\~ and g, such that for n Ï -rin' s ' s0
(3.25) S c.(no,g)s \ j = 0,1,2.
Since the right hand side of (3.25) clearly belongs to L (s S s < +=°),
we may apply the dominated convergence theorem and conclude that
I £ C2 (-nQ S n < +<*>) with
(3.26) lira (4-)JI3(n) = 0 , j = 0,1,2.
As a consequence I £ C2 (~nn S n < +00) with (3.12), as was to be proven.
Finally let us prove (3.14) for n £ -ru under the assumption (3.13).
By virtue of the dominated convergence theorem it is sufficient to show
that
(3.27) ^- lira [ e^gds = lim -i- f ei<J>gds.3n R— sQ
J R - 3n SQ-1
Now, insert (3.21) into (3.27). Then, by the above it is clear that (3.27)
holds provided that
(3.28) lim ^- [-i^e1* ^ Q (iT)£g] = 0.
S->-+oo
It is an easy matter to show that under the assumption (3.13) condition
(3.28) is indeed satisfied.
To conclude with note that since n was arbitrarily chosen we have in
fact shown that I £ C2 (-•» < n < +•») . Likewise (3.14) under the assumption
(3.13) holds for all n £ E. Herewith the proof of the lemma is completed.D
Let us observe at this point that as a consequence of (3.14) and the
above lemma the first derivative of the Airy function can be represented
as follows
(3.29) Ai(1)(n) = -$- lim f is exptins + i -]ds, n £ E./1T R+<= - R j i
We proceed with H rather general lemma, which provides a good insight in
the structure of integrals of type (3.9a) when considered in the
parameter region e > 0, n i -u. To prove it we use some fruitful ideas
already developed in the proof of lemma 3.1.
50
Lemma 3.2. Let b be any function satisfying
(3.30a) b is of class C 2(K), such that b*(k) = b(-k)
(3.30b) The derivatives b J (k), j = 0,1,2 are bounded on R.
Then
fR(3 .31) j ( c , n ) = -s— lira j b ( e s )exp[ i r i s + i -^-]ds
71 R-H» - R J
is well-defined for all n £ K, e > 0 and belongs to C2 (-<= < n < +t»)
(3.32) lim (J-)Jj(e,n) = 0 , j =0,1,2.on
Furthermore one has the representation
(3.33) J(e,n) = b(0)Ai(n) - ieb(1)(0)Ai(1)(n) + R2<c,n>,
where the remainder term can be estimated as follows.
Let e > 0 be arbitrarily fixed.
Then for n > 0
(3 .34a) | R , ( E , r , ) | £ c 2 l lb ( 2 ) l l 4 " 3 / 2 , Hb (2 )ll = sup | b ( 2 ) (k) |
and for n > -n wit/2 n„ S 0 ani/ constant
(3.34b) |R 2 (e , r i ) | S e Jllb (2 )llMC<n0),
where C(nn) ïö ff constant depending only on n , which can be given inexplicit form.
If, instead of (3.30b), we make the stronger asswrrption
( 3 . 3 5 ) b ( j ) ( k ) = 0 ( | k | " ' ) , k ' ±» , j = 0 , 1 , 2 ,
then Ui.: a1 so have the. representation
(3.36) -^J(e.'i) = b(0)Ai(1)(n) + r, (..,•]), uit':
(3.37a) Ir^E.n)! S ell (kb) ( 2 ) ll |n" 3 / 2, ƒ«• f • 0, n - 0
(3.37b) |r,(e,Ti)| S E II (kb) ( 2 ) ll„C(n0), pi- L- - 0, r, - -nQ,
51
where C(nn) denotes the same constant as in (3.34b).
Remark:
( i) Clearly the condition b (k) =b(-k) can be omitted. We include it
because it simplifies the proof somewhat and b (k) has this
property.
( ii ) Observe that (3.33) corresponds to the second order Taylor expansion
for n fixed of the function e>-+ J(E,n) near E = 0. Let us motivate
this choice.
First of all, in view of (3.9) we are interested in the behaviour
of J(c,ri) as E 4- 0. Thus it is quite natural to look at the Taylor
expansion near c = 0. The problem is of course: which order must
we take? In the light of future estimates the answer is rather
simple: such an order n that the remainder term R (e,n) isn
integrable over (1 < n < +°°). Now, with due perseverence one can
show that R (E,H) behaves as n as n + +« for c > 0 fixed,
provided b has n bounded derivatives. Thus we must take n 5 2.
However, it is easily seen that choosing n > 2 does not improve j
the estimates much (in fact it does so only, when for some m the j
subsequent derivatives b J (0), j = 0,1,2,...,m vanish). Hence the
choice n = 2 is singled out. j
Proof: The relation b (k) = b(-k) enables us to rewrite (3.31) as (
R i(3.38) J(e,n) = - Hm Re J b(cs)e1't'ds, <j> = ns + - . I
71 R^° 0J J I
Hence, it is a direct consequence of lemma 3.1 that for any e N 0 the •
function J(e,n) is a well-defined member of C2 (—<= < n < +«>) satisfying j
(3.32). ;
Let us prove the representation (3.33-34).
To this end we insert into (3.31) the Taylor expansion
(3.39a) b(k) = b(0) + kb(1)(0) + b2(k)
(3.39b) b2(k) = Qf (k - k)bv '(k)dk.
In view of (3.3-29) this yields (3.33) with
52
(3.40a) R,(e,n) = 4~ lim f b,(es)eX<t>ds = - Re L( E,n)2 2vr R-x» -RJ 2 it 2
(3.40b) R (£,n) = lim [R b (es)e1(t>ds.
Next, fix E > 0. Set g(s) = b„(es). Then for s £ K
(3.41a) |g(s)| < is^ e2 JH>C2> l)
(3.41b) |g O )(s)| < ( s ^ l l b ^ n ^
(3.41c) |g(2)(s)| s £2llb(2)llM.
Moreover, (3.30) implies that g satisfies condition (3.10) of lemma 3.1.
To prove (3.34b) let i; > 0 be a constant. Put s. = /1 + ru. Then for
n £-n0, ^(e,!-)) n a s t'ie representation (3.15-16-17). Using (3.41a) we find
from (3.16a)
. I s f °1 ' 0J
Furthermore, applying (3.41a-b) and (3.20) we obtain from (3.16b)
(3.42a) 11. I s f ° is2e2|lb(2)ll ds = £2llb(2)ll | s*
1 ' J •» » 6 0
(3.42b) |I2| < cillb^'ll^ (sQ + is^ + s^).
Next, combining (3.41) and (3.20) we get from (3.16c-17)
(3.42c) |I3| S £2|lbU;Ij(s0) with
Together (3.15), (3.40) and (3.42) imply that for n i -r\Q the estimate
(3.34b) holds with
(3.44) C(nn) = (sn + is' + s' t y(sn))/it.U U U o 0 U
To proceed with, let us prove (3.34a).
Let n > 0. Then integrating by parts twice we arrive at (3.21) with sQ
replaced by -R and g(s) = b,,(es). Taking R -+ «° and exploiting (3.30)
we obtain
(3.45) 2irR2(E,n) = - [ G(n,s)e1<!)ds
53
with G(n,s) given by (3.17). An application of (3.41) now gives us
(3.46) |R2(£,n)| i ~ E2llb(2)|
„ . ( 2 ) . . 7 - 3 / 2lib II -sfl
CO 8
which p roves ( 3 . 3 4 a ) .
Lastly, let us show that under the condition (3.35) the representation
(3.36-37) is valid. To do so, observe that for fixed c > 0 the function
g(s) = b(es) easily meets the requirement (3.13) of lemma 3.1. Combining
(3.14) with (3.38) we therefore have
(3.47a) |^(f-,n) = t:"1J(. ,n)
(3.47b) J(e,n) = -j- lira ƒ* b(f.s)ex4>ds
(3.47c) b(k) = ikb(k).
Evidently b satisfies (3.30). Hence J(e,n) has the representation
(3.33-34) with b(k) replaced by b(k). Substituting back we immediately
find (3.36-37), which completes the proof of this lemma. a
Now is the time to apply the above results to the integral (3.9a), to
translate from J(c,n)-language into S.' (x+y;t) —language and to see what we
have got. Doing so we arrive at
Theorem 3.3. Assume that the right reflection r».;c? •"ƒ'icier;r b (k) aatisfi,?^
(3.48) b is of class C2 (B) and the X'ficvzr.MVt.: b ^ ( k ) , j = 0,1,2
are bounded on E.
Then il is iM r->'Ljl ;i differential)! e in V with }'es^e<yt : j x •.;.' .';'.'.",; :\';"»:*
(x,t), x £ E , t > 0. Let t.hf derivation be denoted iv/ .,"•'.
Furthermore, let y > 0, x £ K, t • 0.
Let \i and v denote arbitrary nonnvgative otinstaiitti.
I'ut
( 3 . 4 9 a ) w = x + y + M, b ( k , u ) = b r ( k ) e " 2 i k l ' , b ( j ) = ( ^ ) j b
(3.49b) T = ( 3 t ) 1 / 3 , Z = w(3t)"1/3.
Then one has the representation
(3.50) syx+y;t) = f1br(0)Ai(Z) - JiT~2b(1 } (0,n)Ai(1) (Z) + R(Z, T,y)
with
(3.51a) |R(Z,T,M)| S ï"3»b ( 2 )« t o^ z"3/2 for T > 0, Z -0
(3.51b) |R(Z,T,u) | S T~3llb(2)lloo j C(v) / o r T > 0, Z * -v
wheve C(v) denotes the constant ('6.44) with n replaced by u.
If we make the additional assumption
(3.52) b^j)(k) = (KM"'), k •* +», j = 0,1,2,
then we also have
(3.53) !^(x+y;t) = T % (0)Aiu-" (Z) + r(Z,T,u) wit^
(3.54a) |r(Z,T,u)| < I"3» (kb) ( 2 ) II m •— z"3/2 /or T - 0, Z > 0
(3.54b) |r(Z,T,n)| S T"3« (kb) ( 2 ) II m j C(v) /or T - 0, Z > -v
C(v) as in (3.51b).
Remark. If (3.52) holds then it follows from (3.51-54) that the functions
R(Z,T,u) and r(Z,T,p) can be estimated simultaneously by
(3.55) max(|R(2,T,y)|, |r(Z,T,p)|) S pT~3(1 + u+Z)~3/'2 for T - 0, Z Ï -v,
where the constant p is given by
(3.56a) p = DN2
(3.56b) D = D(v) = — h + (1+v)(| C(v)) 2 / 3j 3 / 2
(3.56c) N2 = H2(ii,br) = maxfjllb^ll^, II (kb) ( 2 ) II V
Proof: Let p S 0 be arbitrarily fixed. Then (3.48) obviously implies that
b(k) = b(k,p) satisfies condition (3.30) of lemma 3.2.
55
Now, let J(e,n) for n £ E , e > 0 be defined by (3.31). Reasoning as in
the beginning of this section we then obtain for y > 0, x £ E, t > 0
(3.57a) «c(x+y;t) = 2eJ(e,n)
(3.57b) e = 2 T , n = 2
with T and Z as in (3.49).
The assertions of this theorem are now easily proven by combining (3.57)
with lemma 3.2.
Specifically, the representation (3.50-51) is merely a transcription of
(3.33-34). Furthermore, lemma 3.2 tells us that ^c(^;t) belongs to
C2 (-00 < £ < +») with
(3.58) lira (-^-)jBc(5;t) = 0 , j = 0,1,2.
Plainly, this implies that for all a £ E
(3.59) N(a,t) = sup | (-^)2«c(5;t) I < +».
With the help of (3.59) it is not hard to show that fi is strongly x-
differentiable. Indeed, let (x,t), x £ E , t > 0 be an arbitrary but
fixed point. Then one has for h £ E, 0 < |h| < 1
a (x+h+y;t) - V. (x+y;t) 3ft(3.60) sup C C
sup0<y <+o it
h nJ
(h - fi)—-r- (x + h + y;t)dh
S jN(x-1,t)|h| = OC|h|) as h + 0.
Consequently, fl is strongly x-differentiable at (x,t) with derivative
(3.61) ft'(x+y;t) = 4E 2 |^ (e,n).c on
Finally, if b satisfies (3.52), then b fulfills condition (3.35) of
lemma 3.2. Together (3.36-37), (3.61) and (3.57b) yield the desired
representation (3.53-54) and so the proof is done.
56
The results laid down in theorem 3.3 are remarkable for two reasons.
Firstly, they display in a strikingly explicit way the structure as well
as the magnitude of the functions y»-»- Si (x+y;t), H'(x+y;t) in the1/3 C C
parameter region x a - u - v(3t) . This will be extensively used below,
when we start estimating in the norms (3.65). Secondly, the conditions imposed
on the right reflection coefficient b (k) are only very weak. In this
respect, let us note that since any initial function un(x) considered in
this paper is supposed to satisfy at least the conditions stated at the
beginning of subsection 2.1, we already know from that subsection that
b r £ L fi C Q C E ) such that b (k) = o(]k]~ ) for k -* ±». Hence, in view or
an interpolation argument, the only extra condition imposed on b by
(3.48-52) is
(3.62) b r £ C2(K) such that b^ 2 ) (k) = (KIM~ 1) f°r k * ±°°-
In general, initial functions satisfying (1.2) will fulfill condition
(3.62). A mild algebraic decay of u„(x) and a number of its derivatives
is already sufficient. Specifically if un satisfies (1.3) then b belongs
at least to C (R) with b J (k) = 0(|k| ), k -»•+<», j = 0,1,2 6, so
that (3.62) is amply fulfilled.
We now turn to the construction of bounds for fl , fi' in variousc' c
settings. To-avoid any misunderstandings and to have an easy reference we
stress the following:
From now on we shall assume that the reflection coefficient b (k) fulfills i
the requirements of theorem 3.3, i.e.: 1
(3.63) b is of class C' (E) and the derivatives b ( j )(k), j = 0,1,2 j
satisfy i
b^j)(k) = Od k f 1 ) , k •* ±».
With the help of theorem 3.3 it is now an easy matter to show that in the
parameter region
(3.64) T = (3t) 5 1, x ? - u - vT, where \t and v are nonnegative
constants,
57
one has the estimates
(3.65a) llficll = sup | « c ( x + y ; t ) | S Y"T ' ,
.00
(3 .65b) II fi II = \il ( x + y ; t ) | d y % |b (CC L1 O-1 C r
(3 .65c) Ihl'll = sup | f ! ' ( x + y ; t ) | e yT ,
f™ - 1
(3.65d) lla'll = !si ' (x+y;t) |dy > ,-T ' ,
C L1 0J " C
where the constant y i s given by
(3.66a) Y = AN + BN , with N as in (3.56c) and
i | d „ )
sup |Ai(n)|,sup |Ai(1)(n)|, [ |Ai(i) (n) |d-i\(3.66b) A = kM = max( |
(3.66c) B = B(v) = iC(v)(1+u) + g- with C(v) as in (3.51b),
(3.66d) N = N1(y,br) = max(2|br(0)|, |b(1} (O.ti) |) with
b(k,ti) = br(k)e"2lku.
Note that (3.65b) hinges on property (3.3) of the Airy function.
Next, let us apply the above results to investigate the mapping T in
the parameter region (3.64). By theorem 3.3 and (3.7-55) there is in this
region a function c(t) such that
(3.67) |iic(x+y;t)| + |^(x+y;t)| •• c(t) (i+y)"3/2.
As a result, the function
r(3.68) (Tg)(y)= o (x+y+z;t)g(z)dz
C 0J °
is continuous in y, since the integrand is dominated by c(t)(1+z) ~!'s'l.
On the other hand
(3.69) llTcgll <- llgll [ | ï ! c ( x + y ; t ) | d y .
Hence, in view of (3.65b), T is a continuous mapping of V into Y with a
norm that satisfies
h f° \ -1(3.70) llTcll 5 l b
r ( ° ) ( K + |Ai(r,)|dr,J + YT ' .
Finally, reasoning as in Chapter 1, we extract from (3.67), that T
is strongly x-differentiable in V with derivative
r(3.71) (T'g)(y) = s:'(x+y+z;t)g(z)dz.
c 0J °From (3.65d) we find in the region (3.64) the following estimate
(3.72) IIT'II S rT~1.
For future reference, lot us note that, in addition to (3.65),
theorem 3.3 gives us useful bounds containing both x and t. In particular,
fixing 6 € (0,1) in (3.7), we obtain in the region (3.64)
(3.73) maxjV^HI, lloMlj < yoT~2exp[- |o(^v + ' )
T-3/', xn
+ PT V1 + v + —
where the constant f( is given by
(3.74) , = AN., A = A(0,v) = max(an (v), a, (v))
with the constants a, N-, •<. (\>) as in (3.56a-66d-7) respectively.
Of course, in the degenerate exceptional case (2.20) the estimates
(3.65-70-72-73) can be improved. Specifically, then theorem 3.3 tells us
that in the parameter region (3.64)
(3.75a) 11:11 ';: fl~2 IITJI • vT"1
(3.75b) ll:;'ll i YT" 3 IIT'll S VT~2,
c c_9
with i still given by (3.66a). Moreover, (3.73) holds with the factor T
in front of the exponential function on the right replaced by T .It is
easily verified that further simplifications of this type occur when also
b (0) vanishes.
As mentioned in subsection 2.1 it rarely occurs that b (0) = 0. However, in
the usual case b (0) * 0 one can still simplify the discussion somewhat
by working in the specially selected parameter region (3.64) with u
chosen to be u = b (0)/(2ib (0)), in which case the derivative of the
Airy function disappears from (3.50). In the present discussion
however we shall stick to (3.64) with u arbitrary.
4. Solution of the KdV initial value problem in the absence of solitons.
In this section we study the asymptotic behaviour of the solution
u(x,t) of the KdV equation evolving from an initial function u,.(x),
which generates no bound states in the Schrödinger scattering problem.
It is assumed that the condition (3.63) is fulfilled.
We shall work in the coordinate region t U , x ^ -;, ^ = p + «I,1/3 .
T = (3t) , where u, v and t are nonnegative constants, with v and
t = t (n,v,b ) to be specified presently.
Clearly, in the absence of bound states the Gel'fand-Levitan equation
reduces to
(4.1) (I + T )3 = -° •c c
For notational convenience we introduce the number v > 0, uniquely
determined by
(4.2) _v f° |Ai(n)|dn = | .
c
The existence of v is guaranteed by the fact (see [14]) that Ai('i) is not
absolutely integrable over -<» < n < 0. As for the numerical value of v ,
we obtain Crom [3], p. 478
(4.3) vc = 1.39.
Moreover, [3] tells us that
(4.4) Ai(n) •• 0 for M ? -v^
Let us now work out our specification procedure.
Firstly, we select: j such that
60
(4.5) O s v < vc.
As a consequence of (4.2) and (4.4) one then has
(4.6) aQ ,(v) s -j + _J |Ai(n)|dn < 1.
Secondly, we fix u 6 0 independently of v.
Thirdly, bearing in mind that |b (0)| s 1, we select tc such that
(4.7) tc > •
with Y = y(lJ,v,b ) as in (3.66a).
After the above specification it is clear from (3.70) that in the
ameter region t s t , x £ -£, £ = u + vT, T =
T occurring in (4.1) can be estimated as follows
parameter region t s t , x £ -£, £ = u + vT, T = (3t) the operator
(4.8) IIT II i a < 1 with a = |b (0) |aQ (v) + Y ( 3 t c ) ~1 / 3 .
This implies that I + T is invertible on the Banach space V.
As a result, (4.1) has a unique solution S € V satisfying
( 4 . 9 ) llgll S io1llficll, w i t h u 1 = (1 - o ) ~ 1 .
Furthermore, (4.1) implies that 3 is strongly x-differentiable with
derivative
(4.10) S' = -(I + T )"'(T'g + £!').c c c
From (4.3-9-10) and (3.72) we obtain the estimate
(4.11) lie1» S w0T~'lls: II + u> llfl'll, with w- = :.;2y.
Recall that
(4.12) u(x,t) = - -£- 8(0+;x,t).
Since
(4.13) | -^ R(0+;x,t)| ; sup | ö(y;x.t) | = IK-MI,0-;y<+t»
we arrive at the following result
61
(4.14) | u ( x , t ) | s U()T~1IISÏcll + n^hï'J.
In particular, it follows from (3.65a-c) that
(4.15) |u(x,t)| < io2T~2, with u 2 = (uQ + w^-y.
The above results are summarized in
Theorem 4.1. Let u(x,t) he the solution of the Xor;,e,.>ej-de Vries problem
( -co . x • +<», t • 0
(4.16) { t X XXX
u(x,O) = u n(x),
where the. real initial function u.(x) is sufficient-1 y smooth and decays
sufficiently rapidly for |x| -> •» for the uhoie of the inVrrae scait-~'riny
method to work and to guarantee the regularity and decay :;ropi-rf-y (;'•.•!/•)
of the reflection coefficient b (k). Amamc that, aa a poten1 la? in :he
Sohrödinger scattering •.n'obl.cm, u„(x) jenei'ate.:; no bound siates. Lei
\>, v and t be nonnegatioe constant a, uith \> and t sat.isf.iina (•>.!•) wi 1
(4.7) respectively.
Then, in the coordinate region t •'• t_, x -• -r,, r, ~ \i + \>T, T= (3t)
one has the foi I owing cstitn.üt- f 'he solution
(4.17) |u(x,t)| • .^T' 1 sup |;:c(x+y;t)| + ,^ sup ! : ' c
O-.y.+o» o-y-+»
w h e r e u a n d ...i. a r e t h e c o n s t a n t s i n t r o d u c e d i n ( - ' . I I I a n : ,'•;..".'
r e s p e c t i v e l y a n d '.I ( x + y ; t ) (';; j i v e n .•>./ ( / - . I ) .W i t h t h e a m i ; U n i t :><„ a s i n ( - l . l h ) : j e ::ane- Cor t - C
2 " c
( 4 . 1 8 ) sup | u ( x , t ) ! • . . ,T~ 2 .x — C
Let us mention some implications of the preceding theorem.
As a consequence of (4.17) and (3.73) tlie solution u(x,t) of (4.16)1 /3satisfies, in the coordinate region t t , x • -!., r, = u + vT, T = (3t) ,
the following x aud t dependent bound
(4.19a) |u(x,t)| ' al~2ey->\- 0f
(4.19b) a = (u)Q + u^Yg, b = (ui0 + w^,,
with 6 £ (0,1), y , p the constants introduced at the end of section 3.y
Hence, for t £ t
(4.20) j |u(x,t)|dx < aT"1 [ exp -^0s 3 / 2]ds + 2bT~2.
Combining (4.18-20) with the formulae (see [18])CO |..„
(4.21a) u(x,t)dx = - - log(1 - |br(k)|2)dk
(4
we
(4
U
21b)
obtain
22a)
22b)
Remark.
fJ U2
- j
f u
(x,
(x,
2 (x
t)dx =
t)dx =
,t)dx =
_ 2
r
f? or
k2 logd
logd -
k2logd
- |b r (k ) |
|b r(k) ')
- |b r(k)
2)dk,
dk + Q(t
! 2 ) d k + 0
- 1 / 3 ) as t
(t ) as t
( i) Observe that (4.22a) can also be derived from (2.28), since by
(3.65a) and (4.9)
(4.23) [ u(x,t)dx = p(O+;-(;st) = 0(t~'/'3) as t > -.
( ii) If u0 satisfies (1.3) then all of the conditions of theorem 4.1 are
fulfilled.
(iii) In the degenerate exceptional case (2.20) the above estimates can~2 ~3
be improved. For instance, in (4.18-19) we can replace T by T
( iv) For a physical interpretation of (4.21-22-23) we refer to the
discussion in section 7.
5. The operator (I + T ) .
When, as a potential in the Schrodinger scattering problem, un(x)
produces bound states, then the operator I + T, makes his entrance in the
Gel' fand-Levitan equation. The following lemma shows that this operator
has a nice inverse.
Lemma 5.1. For any value of the parameters x £ R, C > 0, the operator
I + T. is invertible on the Banaah apace V with inverse S = (I + T.)d d
given by
N -2K.yJ(5.1a) (Sf)(y) = f(y) - . | 1 A^e J
N / f ™ ~2K.Z v(5.1b) A. = i J 1 Bi-(2 e l f(z)dzj ,
2K.xwhere ( g . . ) i s tóe inverse of the matrix A = ( [ c . ( t ) ] e S. . + (K ,+K . ) ) .
Furthermore, S and i is strong ^.-derivative S' satisfy the bounds
(5.2)
(5.3a)
(5.3b)
IIS» S a Q ,
1 + .i
N . . = 2 ( K , K . )
I I S ' x £ t > 0 ,
, Nj - IT . , a( = 2aQ i ?=
N;;>
i
£=1£*i
Nn
P=1
K . +KJ P
K . —KJ P
Thus, IISII and IISMI a r e uniformly bounded for x £ K, t > 0 and t^e bounds
are explicitly given in terms of the K. only.
Proof: For any fixed x £ E, t > 0 we may solve the equation
(5.A) (I + Td)g = f, f,g £ V
to find
N -2K.y(5.5) g(y) = f(y) - X. A.e J ,
where the A. satisfy
(5.6a) a. .A./•» -ZK . Z
J = 2 0J 6 '
f(z)dz,
(5.6b) a.. = a.6. . +lj j IJ
i = 1,2,....N
2K .x2
We shall show that the matrix A = (a..) has positive determinant
and thus has an inverse A = (6..), so that I + T, is an invertible6 . .
)operator on the Banach space V with inverse S = (I + T.) given by (5.1).
Furthermore, we shall prove that the matrix elements fi.. are uniformly
bounded for x £ E, t > 0, where the bound is explicitly given in terms of
the K• by
N(5.7) |B..| , 2 ^ . ) ^
NII
p=1
K . +K
_J PK . — KJ P
s N. . .IJ
To achieve our goal, let us first introduce some notation. By X we
denote the real Hubert space L2 (O,») with inner product f,g> = _
= nf f(y)g(y)dy. In JC we consider trie ele.nents e. defined by e.(y) = e
^ 1 1 'We write A for the Gram matrix of the vectors e.,e_,...,e , i.e.
aij ' aij < ei' ej' i KjSince the vectors e. ,e->•>•9eN are linearly independent, it is clear [5],
that det A > 0.
Let us write (A) = (g.-).
Next, select f ,f.,..., f € 3C such that <i^,e.> = 0 and <f. ,f .•» = a. 6. .
s
the
for i,j = 1,2,...,N. Put g. = f. + e.. Then a.. = <g-,g->, which shows
that A = (a.-) is the Gram matrix of the vectors g.,g9,...,g . Since
vectors e. ,e,,. . • .e,. are linearly independent, the same holds for
g, ,g„,... ,g,,. Hence we have proven, that det A •> 0 and the existence of
A~' = ((3..) is guaranteed. N
To obtain the estimate (5.7) we introduce the vectors h. = .£ B--e..
Clearly, <h.,e.> = fi.. and <h.,h.-- = ?>...
Now let P be the projection of Jf onto span ( g1 ,g_,. .. ,g,.f. Then
N N(5.3) Ph. = . f B. .-;h.,g >g. = -E, P-.g-9
so that <Ph.,h.> = 8.. and
I i j I i* i"~ j * i " ~ i i j j "
By d i r e c t c a l c u l a t i o n ( see [ 5 ] ) we o b t a i n
(5.10)
det
and so t h e proof of ( 5 . 7 ) i s c o m p l e t e .
Note , t h a t by ( 5 . 1 b - 7 )
(5 .11) | A . | % llfll . £ , ^ N . . .
which implies the bound (5.2-3) for IISII.
It remains to estimate the strong x-derivative of S which by (5.1)
is given by
N -2K.y
(5.12a) (S'f)(y) = - ^ Aje J
N 2K
(5.12b) A! = -2K.A. + . £ . g.. — - 2 - A .J J J i,P=1 iJ <i
+Kp P
Using (5.7-11-12b) one gets
/ N N 2K
(5.13) |A!| < IIfII 2K. . I . - S . . + . .1 . — N» —-r 2- N. .' j' V J 1 = 1 Ki XJ i»£,P=1 <£ £p K^+Kp U
from which the bound (5.2-3) for IIS'II is an immediate consequence.
D
Remark. From the above proof it is clear that lemma 5.1 is still valid
(with the same bounds (5.2-3)) if the time evolution of the normalization
coefficients is not given by (2.24) - as prescribed by the KdV equation -
but is instead completely arbitrary.
Corollary to lemma 5.1. let x £ I and t • 0. Then the equation
(5.14) (I + T,)0 = -a,a a
admits a unique solution B, £ V and we have
(5.15a) Sd(y;x,t) = - 2 , J = ) p..e ^
N f -2< y N -2> .y(5.15b) B'(y;x,t) = 4 £ | < ^ ^e P - ? - ^ - 3 e J'
Remark. Let us recall that B, produces the pure N-soliton solution of the
KdV equation associated with un(x) through the formula
(5.16a) u,(x,t) = - f- 6,(0 ;x,t)
N
£,p=1 p
, N N
1 - . ? , — ! — 3. . I.V i,j = 1 K.+K u/
Since
66
i_ 2K x N
e p ^
we find for u, two additional representations that are useful as well:a
N r -2 2 % V 2 \2
(5.16b) ud(x,t) = -4 g1 K [r (t)] e v { ^ B£ ) ,
N 2 -2K X/ N N2
(5.16c) ud(x,t) = -4 Z K [cr(t)] e P M - i 5 = 1 +>; Bi- 1 .
Consequently
(5.18) 0 Èud(x,t) £-4a 0 . J = ) K.N..,
so that u,(x,t) is uniformly bounded for x € B., t > 0 and the bound does
not involve the c. but depends only on the K. in a simple explicit way.
Combining (5.7-16) and (2.6-24) we obtain
(5.19a) |ud(x,t)|dx = 0(e ) as t -> »
(5.19b) ud(x,üdx = OCe ' ) ast-*».
Recall that (see [10])
=, N(5.20a) ud(x,t)dx = -4 ^ K
(5.20b) _ ƒ u|(x,t)dx-f p^ K».
Starting from (5.16b-c) it is shown in [15] that as t approaches infinity
the pure ïl-soliton solution decomposes into N solitons uniformly with
respect to x on 1R. More precisely one has
N +(5.21a) lira sup |u (x,t) - ^ (-2K * sech2 [< (x-x -k^ t) ]) | = 0 ,
67
6. Solution of the Gel'fand-Levitan equation in the presence of bound
states.
Under the condition (3.63) we now proceed to investigate the full
Gel'fand-Levitan equation
(6.1) (I + T d + Tc)3 = -a
' /3in the parameter region t 5 t ,, x S -<;, c. = M + uT, T = (3t) , whe^-e
u, \> and t , are nonnegative constants, with \> satisfying (4.5) and with
t c d = tcd(n,v,br,K1,K2,...,KN) > j to be specified shortly.
Applying the operator S = (I + T.) we can rewrite (6.1) as
(6.2) (I + STc)p = -Sn.
To ensure the invertibility of the operator I + ST , it suffices to prove
that ST has norm smaller than 1, as was the case in our comovins»c
coordinate analysis [8]. However, in the present situation the operator
ST is less manageable. Let us circumvent this difficulty and consider
the operator T S instead.
From (5.1-11) and the estimate-2v:. z
J(6.3)
we obtain
r«> - 2 v : . z| H ( x + y + z ; t ) e J d z | •; — Ils2
Q J C ^ C
i
( 6 . 4 ) UT Sll < IIT II + II£2 II . ? . —•— N . . .c c c i , j = 1 2K .K . i j
Hence, by (3.65a-70)
(6.5a) 11X Sll a jb (0)|a_ , (v) + yï"1, where an , (v) is given by (4.6)C IT U ) » U y I
and
(6.5b)
We now se lec t t , such thatcd
(6.6) t_A > i i n a x | i , Y 3 / i - |b r (O) |a f t »
For t < t . w e then havecd
(6.7) llTcSH t 5 < 1, with 5 = |br(O)|aQ , (v) + ï O t ^ ) 1/3.
This shows, that the operator I + T S is invertible on the Banach space
V. Consequently, the same holds for I + ST and we have
(6.8a) (I + S T c ) ~1 = I - S(I + T cS)"
1T c,
but also
(6.8b) (I + S T c ) ~ ' = S(I + TcS)~1S~'.
We conclude that, in the parameter region t -• t ,, x " -c,, i, = \j + vT,
1 /3 C
T = (3t) , the equation (6.1) has a unique solution 3 6 V . This
solution can be represented in terms of S, T and V. by means of a Neumann
series:(6.9) B = m|0 (-STc)
m(-SS2) .
Note that, while in general this series converges rather slowly, the
convergence in the degenerate exceptional case (2.20) is rapid for large
t by virtue of (3.75a).
7. Decomposition of the solution of the KdV problem when the initial
data generate solitons.
Let us put
(7.)) (S = Sd + 3 C, with
(7.2) e, = -SS2,.
d d
Introducing the decomposition (7.1) into (6.2), we find
(7.3) (I + ST c)@ c = -S(JJc + T c 3 d ) .
From (6.8b) it is clear that (7.3) has a unique solution p £ V,satisfying
(7.4) lig II s (1 - IIT Sll) II:
By (5.7-15a) and (6.3) one has
N ,(7.5) l lT
cSa" s llfJc" i ?=1 K V* i
so t h a t , i n v iew of ( 5 . 2 - 3 a ) and ( 6 . 7 ) ,
( 7 . 6 a ) IIS II < ü J f i H, w i t h
(7.6b) üj = (1 - 5)"1a20.
Let us keep in mind that the solution of the KdV equation is given by
(7.7) u(x,t) = u,(x,t) - -^- B (O+;x,t),O aX C
where u,(x,t) denotes the pure N-soliton solution introduced in (5.16).
Therefore, we need estimates of the derivative of B with respect to x.
From (6.8) and (7.3) it is clear that (3 is strongly x-differentiable,
the derivative 3' being uniquely determined by
( 7 . 8 ) ( I + S T ) 0 f = - S { T ' ( 6 + 3 , ) + fi' + T R j } - S'{V. + T ( p + S , ) } .c c c c d c c d c c c d
Using (5.3a-7-15b) and (6.3) one gets
N
(7.9) IIT 6'II S 2ajn II . ? . N. ..c d 0 c i,j=1 ij
Furthermore, (3.71) and (5.7-15a) imply j
(7.10) DT'g.ll s llfiMl . ? , — N. . . !cd c i,j = 1 K. i j i
t'
Combining ( 3 . 7 2 ) , ( 5 . 2 - 3 a ) , ( 6 . 7 - 8 ) and ( 7 . 5 - 6 - 9 - 1 0 ) we o b t a i n from ( 7 . 8 ) 1the following estimate
(7.11a) II6Ml S üo l lnc l l + i , IIf!Ml, w i t h
-U~ ~ N \ -1
(7.11b) uQ = M^aQ fuijO + 2aQ i t f r r . If+ a^aQ (aQ + u ^ ) 2 .
Evidently
(7.12) 1 - ^ - 6 ( 0 + ; x , t ) | S IIBMI.oX C C
70
By virtue of (3.65a-c) this yields
(7.13) |- ~ @c(0+;x,t)| < ^ T " 1 , with u 2 = (~iQ + w
Summarizing the above results we obtain
Theorem 7.1. Let u(x,t) be the solution of the Korteweg-de Vries problem
u - 6uu + u = 0 , - o o < x < + » , t •• 0
u(x,0) = UQM
where the real initial function un(x) is sufficiently smooth and decays
sufficiently rapidly for |x| •+ «> for the whole of the inverse scattering
method to work and to guarantee the regularity and decay property (3.6?)
of the reflection coefficient b (k). Assume that, as a potential in the
Schvödinger scattering problem, u~(x) produces N 5 1 bound states.
Let u, v and t , be nonnegative constants, with v and t , satisfying (4.S)
and (6.6) respectively.
Then, in the coordinate region t ? t ,, x 5 -£, r, = y + \>J, T = (3t)
one has the following decomposition of the solution
(7.15a) u(x,t) = ud(x,t) + uc(x,t),
(7.15b) |uc(x,t)| £ ~ Q sup |ïïc(x+y;t)| +.7^ sup | — i^ (x+y; t) | ,
where u , ( x , t ) is the puve Ksot.iton solution (é.16), .o and ~ :rc .'./:,-
constants introduced in (7.11b) and (7.6b) rusvectioelij ?';.,' ;: ( x + y ; t )
Is given by (3.1).
With the constant ,.>,. as in (7.13) we !iaut? <\>r t • t .2 • cd
( 7 . 1 6 ) sup | u c ( x , t ) | -, T^T" 1 .
Evidently, in the degenerate exceptional cuse (2.20) the estimate
(7.16) can be improved, since T can be replaced by T ~. Although similar
remarks apply to the estimates below, they are omittod to avoid
interference with the reasoning.
Let us emphasize that all of the requirements of theorem 7.1 are
fulfilled if uQ satisfies (1.3).
71
Theorem 7.1 has a number of interesting consequences.
Firstly, by combining (7.16) with (5.21) it is found that the
solution u(x,t) of (7.14) splits up into N solitons as t + » in the
following way:
Corollary to theorem 7.1. Let {b (k), K. > * •• ... > r.„, c , c , ..., c }
be the right scattering data associated with un(x). Then the solution of
(7.14) satisfies
N(7.17) lira sup |u(x,t) - | 1 •{-!>
2 sech2 [r. (X-X +-4K 2 t) ]) | = 0t-H» X?-C. ^ P P P
with x as in (5.21b).P
Furthermore, it follows from (7.15) and (3.73) that the nonsoliton
part of the solution satisfies, in the coordinate region t 3 t ,, x s — c;,1 /3
C = U + vT, T = (3t) the x and t dependent bound
(7.13a) |uc(x,t)| S IT"'exp[ - 10 (^-^j J + bT"2(i + ?-LA"
(7.18b) a = (SQ + ü,)Yö, 6 = (5Q + I O ^ P ,
with 0 € (0,1), Y , p the constants introduced at the end of section 3.
The estimate (7.18) is to be compared with the estimate (4.19) obtained in
the absence of solitons. Clearly, by contrast with (4.20), the bound
(7.18) does not permit us to conclude that the L ' -r, > x •- +»)-norm of
u (x,t) tends to zero as t + ».
However, we obtain from (2.28), (3.65a), (7.6)
(7.19) ! uc(x,t)dx = ec(O+;-C,t) =Q(t" 1 / 3) as t - ».
Hence, in view of (5.19a-20a)
(7.20a) j u(x,t)dx = -4 Z, v + 0(t ) as t -• «•.
From the formula (see [18])
r 2 r N
(7.21) I u(x,t)dx = - log(1 - |br(k)|2)dk - 4 ^ K
we find for the complementary integral
f-C 2 f"" -1/1(7.20b) ! u(x,t)dx=-^ I log(1 - |br(k)|
2)dk + 0(t )
as t ->
In particular, in the case of a nonzero reflection coefficient, it follows
from (2.12), (7.20) that there exists a t such that for t • t~
(7,0 r
.22) u(x,t)dx • 0 and u(x,t)dx -• 0.-J oJ
Thus, in the light of (5.18), the reflectionless solutions of the KdV
equation can be characterized as the only nontrivial solutions that do not
assume positive values.
Let us mention, that the time independent quantity (7.21) is usually
referred to (cf. [17]) as the total momentum associated with the
solution u(x,t) of (7.14). For nonzero b , (7.20) suggests that as time
goes on there is a definite positive momentum associated with the disper-
sive wavetrain given by (7.20b), as well as a definite negative
momentum associated with the pure N-soliton solution given by (7.20a).
Incidentally, observe that (7.21) gives an immediate proof of the
following result which is partly known from quantum mechanics [11]:
Let U Q ( X ) be an arbitrary potential in the Schrb'dinger scattering problem,
satisfying the conditions of subsection 2.1.i
If ƒ u_(x)dx 0 thei iin has at least one bound st:itc; if tu # 0 and {
ƒ u„(x)dx = 0, then u„ has at least one bound state and a nonzero ;'U ° „, !
reflection coefficient; if x/ u (x)dx • 0 then n. has a nonzero i
reflection coefficient.
We continue our list of consequences of theorem 7.1 by considering
Lz-estimates. From (7.18) we find
'.23) j u2(x,t)dx = 0(t"'/3) as t > -.(7.
Since by (5.20a) and (7.16)
r -1 N
(7.24) j |uc(x,t)ud(x,t)|dx • 4~,T ^ ,y
we conclude from (5.19b-2Ob) that
(7.25a) | u*(x,t)dx = -y p I , -p + 0(t"1 / 3) a s t
Using the formula (see [18])Ju 2 (x , t )dx = - - k2 log(1 - jb (k ) | 2 )dk + - £ I , • 3 ,
Ti Q J r j p-i pwe o b t a i n as a c o u n t e r p a r t t o ( 7 . 2 5 a )
J - C o f" _ 1 i-,
u 2 ( x , t ) d x = - - k M o g ( l - !b ( k ) ! J ) d k + Q ( t ' J )oJ r
a s t > ••'.
In the literature [10] the time independent quantity (7.2b), sometimes
[17] with a factor J in front of it, is referred to as the energy
associated with the solution u(x,t) of (7.Ü). For nonzero b , (7.25)
suggests that as t •* »• the dispersive wave tra in moving to the left, though
it may decay asymptotically to zero amplitude, still carries a finite
amount of energy given by (7.25b), while on the other side of the line
the N-soliton solution, falling apart into N solitons moving to the right,
carries the energy given by (7.25a). It is interesting to compare the
0(t ) term in (7.25b), due to the interaction between the dispersive
wavetrain and the N solitons, with the Q(t ) term in (4.22b) caused by
the self-interaction of the dispersive wavetrain, in the absence of
solitons.
Finally, let us remark that (7.16) improves Tanaka's result ([16],
Theorem 1.1), which can be reformulated as
(7.27) lim sup |u (x,t)| = 0 for v 0 arbitrarily fixed.
Moreover, it is not difficult to derive more precise versions of (7.27).
As a first example, let us make the additional assumption
74
(7.28) There is an integer tt S 2 such that b Ê c"(K) and all
derivatives b (k), j = 0,1,...,n satisfy
b^j)(k) =0(|kf 3) k -> ±».
Then, it follows from a slight modification of Chapter 1, Appendix B, that,
given the positive constant v, there is a constant yn such that
(7.29) \ncUlt)\ • | |-i! c(ut)| • vo(~n for ' - vt • 0.
Hence, by (7.15b) we can choose t , such that
(7.30) |uc(x,t)| Ï pQx"n, t i t c d, x < vt,
where y is some constant. Thus we arrive at
(7.31) sup |uc(x,t)j = 0(t~") as txjvt
Note that, at the same time, (7.31) improves some of the results obtained
in Chapter 1, since it is easy to show that (7.29-30-31) with slightly
different constants u„, ü-, t , are still valid if the condition (7.28)
is relaxed to that stated in Chapter 1, Appendix B, with n = n.
Incidentally, we can apply the above results to estimate the decay
rate of the solution u(x,t) of (7.14) as x v +•" for fixed t '• t ,. Since,
in view of (7.5-16c), u, (x,t) decays exponentially as x • +•••, we find
from (7.15a) and (7.30) that for t • t ,cd
(7.32) u(x,t) = 0(x~") -is x - +...
If uQ satisfies (1.3), then we obtain from (2.22) that (7.2S) holds with
n = ii;i.i + 2 - (,/2). Hence, for t • t
(7.33) u(x,t) = 0(^('/2)"3Ml1"2) as s - +-.
Herewith, for t t , the estimate
(7.34) u(x,t) = 0(x ' ' ') as x • +•,
which was obtained in [4] for any fixed t 0, is improved.
As a second example, let us suppose that u.-. satisfies (1.3) and
furthermore, that there exists an (.„ • 0 such that u„(x) = 0(exP(~2i:,,x))
as x •+ +<».
Since un satisfies (1.3), we can appLy theorem 7.1 to find a constant
t d such that (7.15) holds for t 2 t d, x > 0. Now, fix v > 0. Then it
follows from combining the last remark of Subsection 2.1 with Chapter 1,
Appendix A, that, given c with 0 < r.. •' min(c ,»: ), there exists a
constant y. such that
(7.35) |flc(f.;t)| + I •£- »c(f.;t)| ' ^,exp(-2i ,r + 8t>t), t • t^, C • v
Hence, by (7.15b)
(7.36) |uc(x,t)| - >1exp(-2L1x + 8» ]t), t - t^, x • vt
where y. is some constant.
Firstly, (7.36) gives us the decay rate of the solution u(x,t) of (7.14)
as x •> +°° for fixed t s t ,. Since, by (2.6) and (5.7-16c), u (x,t) =
= 0(exp(-2r s)) as x -> +<», we conclude from (7.15a) and (7.36) that for
t - t ,cd
(7 .37 ) u ( x , t ; = 0(exp(-2> x ) ) as x •> +«• fo r any • w i th
0 • : , - m i n ( . . 0 > - N ) .
References
t 1] M.J. Ablowitz and 11. Segur, Asymptotic solutions of the Kort owe i;--deVries equation, Stud. Appl. Math. 57 (1977), 13-44.
iS e c o n d l y , c h o o s i n g >. s u c h t l i a t 0 • • > •- rain(t: ,» , J > v ) , we o b t a i n f r o m >
( 7 . 3 & ) |
j
( 7 . 3 8 ) s u p | u ( x , t ) | = 0 ( e x p ( - ( l ) t ) ) a s t ^ - I
with a = 2c (v-4(z) •• 0. '•
And so we have obtained another more precise version of (7.27). •
t 2] M.J. Ablowitz and H. Segur, Solitons and the Inverse ScatteringTransform, Philadelphia, SIAM, 1981.
[ 3] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions,National Bureau of Standards Applied Mathematics Serie1-., No. 55,U.S. Department of Commerce, 1964.
[ 4] A. Cohen, Existence and regularity for solutions of the Korteweg-deVries equation, Arch, for Rat. Mech. and Anal. 71 (1979), 143-175.
[ 5] P.J. Davis, Interpolation and Approximation, Dover, New York, 1963.
[ 6] P. Deift and E. Trubowitz, Inverse scattering on the line, Comm.Pure Appl. Math. 32 (1979), 121-251.
[ 7] W. Eckhaus and A. van Harten, The Inverse Scattering Transformationand the Theory of Solitons, North-Holland Mathematics Studies 50,1981.
[ 3] W. Eckhaus and P. Schuur, The emergence of solitons of the Korteweg—de Vries equation from arbitrary initial conditions, Math. Meth. inthe Appl. Sci. 5 (1983), 97-116.
[ 9] C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura, Method forsolving the Korteweg-de Vries equation, Phys. Rev. Lett. 19 (1967),1095-1097.
[10] C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura, Korteweg-deVries equation and generalizations VI, Comm. Pure Appl. Math. 27(1974), 97- 133.
[11] L. Landau and E. Lifschitz, Quantum Mechanics, Nonrelativistic Theory,Pergamon Press, New York, 1958.
[12] P.D. Lax, Integrals of nonlinear equations of evolution and solitarywaves, Comm. Pure Appl. Math. 21 (1963), 467-490.
[13] J.W. Miles, The asymptotic solution of the Korteweg-de Vries equationin the absence of soiirons, Stud. Appl. Math. 60 (1979), 59-72.
[14] F.W. Olver, Asymptbtics and Special Functions, A'.ademic Press, NewYork, 1974.
[15] S. Tanaka, On the N-tuple wave solutions of the Korteweg-dc Vriesequation, Publ. R.I.M.S. Kyoto Univ. 8 (1972), 419-427.
[16] S. Tanaka, Korteweg-de Vries equation; asymptotic behavior ofsolutions, Publ. R.I.M.S. Kyoto Univ. 10 (1975), 367-379.
[17] N.J. Zabusky, Solitoiis and bound states of the time-independentSchrSdinger equation, Phys. Rev. 168 (1968), 124-128.
[18] V.E. Zakharov and L.D. Faddeev, Korteweg-de Vrics equation, acompletely integrable llamil tonian system, Funct . Anal. Appl. 5 (1971),280-287.
77
CHATTER THREE
MULTISOLITON PHASE SHIFTS FOR THE KORTEWEG-DE VRIES EQUATION IN THE CASE
OF A NONZERO REFLECTION COEFFICIENT
We study multisoliton solutions of the Korteweg-de Vries equation in ]
the case of a nonzero reflection coefficient. An explicit phase shift
formula is derived that clearly displays the nature of the interaction of
each soliton with the other ones and with the dispersive wavetrain. In '
particular, this formula shows that eacli soliton experiences in addition S
to the ordinary N—soliton phase shift an extra phase shift to the left !
caused by fhe collision with the dispersive wavotrain. I
I. Introdurtion.
We consider the Korteweg-de Vries (KdV) equation u - i;ui +
+ u = 0 with arbitrary real initial conditions u(x,0) = U „ ( N ) , whichxxx 3 ' 0
are sufficiently smooth and decay sufficiently rapidly for x • • for the
whole of the inverse scattering method to work and to guarantee certain
regularity and decay properties of the scntterini; data, lo be stated
further on. The long-time behaviour of rlie solution U ( X , L ) of the KdV
78
problem has been discussed by numerous authors. The general picture is,
that as t •* +00 the solution decomposes into N solitons moving to the
right and a dispersive wavetrain moving to the left. As t • - the
arrangement is reversed. The emergence of the N solitons as t • +"> for
rather arbitrary classes of initial conditions was demonstrated
rigorously in [8], Chapter 1 in this thesis (see also the discussion in
[7]). Earlier - but less detailed and not widely known - results in that
direction were given in [11]. Further extensions of the asymptotic
analysis and improvements of results were recently presented ir [10],
Chapter 2 of the present thesis. In the literature many attempts were made
to calculate the phase shifts of the solitons as they interact both with
the other solitons and with the dispersive wavetrain. Many incorrect
results were given (cf. [11] and [12]), until finally the question was
settled by Ablowitz and Kodama [1], who presented a correct phase shift
formula.
In this paper we rederive this phase shift formula, starting from our
asymptotic analysis of the solution given in Chapter 2. l.'o next sliov how ;i
simple substitution produces .1 more transparent, formula th.it cleirly dis-
plays the nature of the interaction of each soliton with the other ones .md
with the dispersive wavetrain. Fron our phase shift formula it is evident,
that each soliton experiences, in addition to the ordinary "-soliton phase
shift, an extra phase shift to the ' ft, the so-called continuous phase
shift, caused by the collision with the dispersive wavetrain. Thus, the
presence of reflection causes a delay in the soliton motion. Furthermore,
our formula shows that the total phase shift is completely determined bv
the bound states and the ri;T,ht reflet'.on coefficient. Hence, tlK-re is no
dependence on the ri;;ht normalization coefficients.
From the original formula the above facts are hard to see. i
The composition of this paper is as follows. In .section 2 we briefly 1
discuss the left and right scattering data associated with u-(x) and show
how the left scattering data can be expressed in terms of the right
scattering data in a convenient way. In section 3 we recall a result known
from Chapter 2, concerning the asymptotic behaviour of u(x,t) as t • +•".
By a symmetry argument we derive from this result the asymptotic behaviour
of u(x,t) as t ->• -». Next, in section 4, the two asymptotic results are
74
combined to give the Ablowitz-Kodama phase shift formula. The
representation of the left normalization coefficients in terms of the
right scattering data, which was obtained in section 2, then enables us
to write the phase shift formula in a more transparent form. Finally,
as an exercise, we calculate in section 5 the continuous phase shifts
arising from a sech2 initial function.
2. Scattering data and their properties.
For Ira k 2 0 we introduce the Jost functions tf/ (x,k) and i(i„(x,k), two
special solutions of the Schrödinger equation
(2.1) ty + (k2 - u (x))i); = 0 , -co < x < +~
determined by
(2.2a) 'i, (x,k) = e~lkxR(x,k), lim R(x,k) = 1, lim R (x,k) = 0
x>
(2.2b) <|<»(x,k) = e L(x,k), lim L(x,k) = 1, lim L (x,k) = 0.X-*-+<» X->-+»
We set j
(2.3a) r (k) = 1 - (2ik)"' [ u (y)R(y,k)dy k 6 C \{0} j
- i(2.3b) r+(k) = (2ik)"' j e~2lkyuQ(y)R(y,k)dy k t E\iO} j
-1 r - i(2.3c) £+(k) = 1 - (2ik) uQ ;L(y,k)dy k £ C+V0} i
jU.3d) £_(k) = (2ik)"' j e2lkyu0(y)T,(y,k)dy k € R\iO(.
Note that r_(k) = £+(k), whereas r^(k) = -C_(-k). It is well known [7],
that r_(k) is analytic on C with at most finitely many zeros, all simple
and on the imaginary axis. Let us denote them by it , m = 1,2,...,N and
order
(2.4) ^ - «, ... > K N •- 0.
80
Bearing in mind that if'„(x,iic ) and \|i (x,i< ) are both real-valued and
square integrable, we introduce
(2.5a) c = ! j ^(x.i/c )dx , the right normalization coefficients,m L_J l m J
(2.5b) c = ty2 (x,i< )dx , the left normalization coefficients.m L_J r m JFurthermore, we introduce the following quantities for k £ E\{0}
(2.6a) a = r , the right transmission coefficient
(2.6b) a.» = L , the left transmission coefficient,
(2.6c) b = r+r_ , the right reflection coefficient
(2.6d) b» = £_•£" , the left reflection coefficient.
Assuming that uQ(x) decays sufficiently rapidly (see [7]) we can extend
a , a,, b , b„ in a natural way to continuous functions on all of K.
We shall call the aggregate of quantities {a (k),b (k),K ,c } the right
scattering data of the potential un. Similarly we refer toI
{a„ (k) ,b»(k) ,K ,c } as the left scattering data associated with u„. In a
different, but equivalent way the ~ight scattering data were already
introduced in Chapter 2, section 2 (see also [7], Ch. 4).I
We claim that a.,b» and c can be expressed in terms of the right
scattering data in the following way
a (k)(2.7a) a£(k) - ar(k) , b£(k) = - & 5_k) br(-k),
r
o - i f fKm t°° log(1-|b_(k) | 2) T-, N IK +K
(2.7b) cl = [cr] 12»c exp -^ x- dk I!, -"—p-
m mJ ml v\ n QJ k2 + ƒƒ p=1 \*^v
Indeed, the relations (2.7a) are obvious. To derive (2.7b) we combine
certain familiar facts from [6], [7]. Firstly, from [7], p. 110 we know
(2.8) i> (x,iK ) = a t|y„(x,iK ), with a £ K M O } ,r m m t_ m m
Hence, by (2.5)
31
(2.9) I a \cZ.1 m' m
Next, by r 7 ] , '(4.3.18) one has
dr_ ,-1k=i.K
m
Eliminating a from (2.9) and (2.10) we findm
(2.11) c c dk lk=i< = 1.
Lastly, from [6], p. 154 we obtain the representation
( f , f°° l o g ( 1 ~ | b (in) | 2 ) -i-j N k-itr( 2 .12 ) r (k) = UyiP\~ : dM I' —^-E . , lm k •• 0 .
Consequently
dr_(2.13) dk k=i:c 2K f
0J
m p
ra p
where we have used that b (k) = b (-k).
Combining (2.11) and (2.13) we arrive at the desired formula (2.7b).
3. Forward and backward asymptotic o
Once the right scattering data of u (x) are known, the solution u(x,t)
of the forward KdV problem
r u - 6uu + u = 0 , t - 0(3.1) f * x xxx
1 u(x,0) = uQ(x)
can in principle be computed by the inverse scattering method [7].
Concerning the asymptotic behaviour of the solution we have obtained the
following result in Chapter 2, section 7.
Lemma 3 .1 • Assume that
(3.2) b (k) is of class C2 (H) and the derivatives b J ( k ) , j = 0 ,1 ,2
satisfy
v
Then one has
( 3 . 3 ) l i m S U P / | u ( x , t ) - t;. ( - 2 K 2 s e c h 2 [»- ( x - x - V 2 t ) ] ) ( = 0 ,t->«> x ï - t '
f [ c ] 2
Xm 2K i o g \ 2K p=1 \K +Km l m ^ N p
f [ c ] 2 m-1 ,K -K \ 2 1
Let us now consider the backward KdV problem, starting from the same
initial function u~(x), i.e.
I- u - 6uu + u = 0 , t O(3 .5 ) C x xxx
L u(x,0) = u Q (x ) .
Clearly , i f u ( x , t ) s a t i s f i e s ( 3 . 5 ) , then w(x , t ) = u ( - x , - t ) s a t i s f i e s
r w - 6ww + w = 0 , t O
(3.6) { l X XXX
<• w(x,0) = u Q ( - x ) ,
so that w(x,t) satisfies the forward KdV problem with initial function
uo(-x). To solve (3.5) it is therefore sufficient to determine the right
scattering data n^-TociaLed with uQ(-x) and apply the inverse scattering
method to (3.6). . >wever, it is readily verified that the right scattering
data associated with u„(-x) are equal to the left scattering data
associated with uQ(x), which were studied in the previous section. Thus,
to find the asymptotic behaviour of the solution u(x,t) of (3.5) for
t •* -oo we merely a^piy lemma 3.1 to problem (3.6) and perform the
transcription u(x,t) = w(-x,-t). This yields
Lemma 3.2. Assume that
(3.7) b„(k) is of class C2 (E) and the derivatives b^ (k), j = 0,1,2
satisfy
odkf1), k
Then one has
(3.8) lira sup |u(x,t) - T, (-2K2sech2[K (X-X -4K2t)])| = 0,• • •• '3 ' m = 1 m m m m '
where
r[c ] 2 t H - 1 /K - K v 2 i
<3-9> x« = - 27- M ^ F - pï ifeV1)}-m l m ^ N p m / J
4. An explicit phase shift formula.
Let us assume that b and b, satisfy the conditions (3.2) and (3.7).
Then the convergence results (3.3) and (3.8) display clearly how the
solution u(x,t) of the KdV equation evolving from u(x,0) = Un(x) splits
up into N solitons as t + ±«>.
In particular, we find for the m-th soliton the following phase shift
r ?c c \2 m-1 /K -< \i
=1 K^rj ĥin in in p
This formula was first derived by Ablowitz and Segur [2] for the N = 1
case and by Ablowitz and Kodama [1] for the N > 1 case (see also the
discussion in [3]).
It is a remarkable fact that the formulae (3.9) and (4.1) become
both more transparent and more meaningful if one inserts the representation
(2.7b). Summarizing, this leads to
r r -i, .
(4.2a) xm - 2K ^ S V 2K I ' K p=1
IT ~
84
0
(4.3a) S = Sd + SC
m m m
( 4- 3 b )
(4.3c)
Tn S we recognize the pure N-soliton phase shift (caused by pairwise
interaction of the m-th soliton with the other ones). The quantity S
(which is negative for nonzero b ) can be seen as the shift caused by
the interaction of the m-th soliton with the dispersive wavetrain. Note
that the phase shift S is completely determined by the bound states <
and the right reflection coefficient b and is thus independent of the
right normalization coefficients, a fact not in the least suggested by
the original formula (4.1). For nonzero b we obviously have
(4.4) 0 > SC. > S^ > ... > S?l.
Thus, the collision with the dispersive wavetrain causes a delay in the
motion of the solitons and the effect is most heavily felt Sy the
smallest one, corresponding to K,,.
Using the formula (see [3])
N
p=1 Kp(4.5) I uQ(x)dx = " f J 1 OS(' " |br(k)|*)dk - 4
cwe obtain for the continuous phase shift S the following estimate in
terms of the initial function un(x) and the bound states t: :0 p
(A-6) 10 s sr (_ƒIn estimating the size of S one has to distinguish two cases, the
"generic case" and the "exceptional case" (see [5], [6], as well as
Chapter 2, subsection 2.1). In the generic case, the Jost functions
ijj (x,0) and i|;„(x,0) are linearly independent, whereas in the exceptional
case they are not. In the exceptional case one has
(4.7) B = sup |b (k)| - 1,. k£ K
whence
85
(4.3) |S^| S ~ 27m
In the generid case there is an a z O with
(4.9) b (k) = -1 + ak + o(|k|) as k >• O,
so that in the integral defining S the contribution of k = 0 becomes
important. In particular, fixing |b |, we find for K + 0
(4.10a) SC ~ - — log(1 - |b (0)|2) in the exceptional casem
c 1(4.10b) S ~ — log K in the generic case,
m < mm
c dClearly, in general the sizes of S and S are incomparable.
On the other hand one can cr>sily construct examples in which one of the
two dominates. For instance, consider a generic case with two bound states
K and K ? = |K.. Then, for fixed |b |, the discrete phase shifts Sc
dominate for K. •* +», whereas the continuous phase shifts S dominate for
K. I 0; in the K. + 0 case the familiar picture of a KdV soliton over-
taking a smaller one, where the smaller one is shifted to the left and the
larger one to the right, changes, since now both are shifted to the left.
5. An example: the continuous phase shifts arising from a sech2 initial
function.
To illustrate the previous discussion let us compute the continuous
phase shifts arising from the initial function
(5.1) uQ(x) = -A(\ + 1)sech2x, A 0.
From [9] we find
,«. , , ,. . r(a)r(b) . ,,, r(c-a-h): (.-i)r(b) .(5.2a) ar(k) = r ( c ) r ( a + b_ c ) , \(k) = y^=^nc.h)r(a+b.cy . «"h
(5.2b) a = 1 + A - ik, b = - \ - ik, c = 1 - ik,
where r denotes the gamma function ([4], p. 253). Clearly, a is analytic
86
on C \ { K ,K ,..,,< } with simple poles at the bound states K ,K„ , ... ,>:,,•
Here N Ï 1 is the unique integer such that N-1 < \ i N and the r are
given by
(5.3) K = 1 + A - p, p = 1,2,...,N.
Note, that ur.(x) is reflectionless (i.e. b a 0) if and only if
A = 1,2,..., in which case N = A. For the other values of .'• we find that
b (0) = —1 so that we are in the generic case.
To compute the continuous phase shifts S we notice that by (5.2)
On the other hand, by (2.12)
r log(1-|b (k)|2) n N v-i-
7 J -FT7 dk}} P"I ' • °-Equating both expressions we obtain, after repeated use of the recurrence
formula r(z+1) = zT(z), the following identity
(5-6) 7 J k^-r^ dk - - 7 losIr(i+v^-N)r(,-;+J
where B refers to the beta function ([4], p. 258).
Finally, combining (5.3) and (5.6), we find that the continuous phase
shifts S are given by
. c 1 ƒ B(2-nH-A,1-m+Q^ m 1+A-m °S{B(2-m+2A-N,1-ni+N
However, to get an idea of the magnitude of S it is much simpler to
employ the estimate (4.6) which gives us immediately
(5.8) Is!I S -
a 7
References
[ 1] M.J. Ablowitz, Y. Kodama: Note on asymptotic solutions of theKorteweg-de Vries equation with solitons. Stud. Appl. Math. 66 (1982)No. 2, 159-170.
f 2] M.J. Ablowitz, H. Segur: Asymptotic Solutions of the K&tteweg-deVries Equation. Stud. Appl. Math. 57 (1977), 13-44.
[ 3] M.J. Ablowitz, H. Segur: Solitons and the Inverse ScatteringTransform, Philadelphia, SIAM, 1981.
[ 4] M. Abramowitz, I.A. Stegun: Handbook of mathematical functions.National Bureau of Standards Applied Mathematics Series, No. 55.U.S. Department of Commerce, 1964.
[ 5] A. Cohen: Existence and Regularity for Solutions of the Korteweg-deVries equation. Arch, for Rat. Mech. and Anal. 71 (1979), 143-175.
[ 6] P. Deift, E. Trubowitz: Inverse scattering on the line. Comm. PureAppl. Math. 32 (1979, 121-251.
[ 7] W. Er'.haus, A. van Harten: The inverse scattering transformation andthe theory of solitons. North-Holland Mathematics Studies 30, 1981.
[ 8] W. Eckhaus, P. Schuur: The Emergence of Solitons of the Korteweg-deVries Equation from Arbitrary Initial Conditions. Math. Meth. in theAppl. Sci. 5 (1983), 97-116.
[ 9] G.L. Lamb Jr.: Elements of soliton theory. Wiley-Interscience, 1930.
[10] P. Schuur: Asymptotic estimates of solutions of the Korteweg-de Vriesequation on right ' Lf lines slowly moving to the left, preprint 330,Mathematical Institute Utrecht (1984).
[11] S. Tanaka: Korteweg-de Vries Equation; Asymptotic Behavior ofSolutions. Publ. R.I.M.S. Kyoto Univ. 10 (1975), 367-379.
[12] V.E. Zakharov: Kinetic equation for solitons. Soviet Phys. JETP 33(1971), 538-541.
83
CHAPTER FOUR
ON THE APPROXIMATION OF A REAL POTENTIAL IN THE ZAKHAROV-SHABAT SYSTEM BY
ITS REFLECTIONLESS PART
In this paper Che inverse scattering algorithm associated with the
Zakharov—Shabat system with real potential is simplified considerably.
Exploiting this simplification we derive an estimate which clearly
displays how well the potential is approximated by its reflectionless
part.
1. Introduction.
The inverse scattering method associated with the Zakharov-Shabat
system with real potential [14] can be used to solve a rich class of
integrable nonlinear evolution equations, counting the modified Korteweg-
de Vries equation and the sine-Gordon equation among its most distinguished
members (cf. [2], [9], [13]). However, the only solutions of these
equations that can be computed in explicit form are the so-called
reflectionless solutions, i.e. solutions whose associated right
reflection coefficient is zero. In a more general setting this situation
leads automatically to the following question: Given an arbitrary real
potential in the Zakharov-Shabat system, in which sense is it
approximated by its reflectionless part?
In this paper we shall give an answer to this question.
To this end we first simplify the inverse scattering algorithm by
showing how the Gel'fand-Levitan equation that appears in the literature
can be simplified to a scalar integral equation containing only a single
integral. The newly found Gel'fand-Levitan operator has, when considered
in the complex Hubert space L2(0,<"), the remarkable structure of the
identity plus an antisymmetric operator. Exploiting this structure we
shall derive a pointwise estimate of the difference between the potential
and its reflectionless part, which is remarkably simple in form and depends
only on the bound states and the right reflection coefficient associated
with the potential.
Let us emphasize that in applications of the inverse scattering
method (cf. [3], [9]) the scattering data are usually known in explicit
form. Therefore this estimate has immediate consequences in a practical
case.
In Chapter 5 and Chapter 7 we shall use our estimate for an asymptotic
analysis of the modified Korteweg-de Vries and the sine-Gordon equation
respectively.
The composition of this paper is as follows.
In section 2 we review the direct scattering problem for the Zakharov-
Shabat system with real potential. In section 3 the inverse problem is
discussed and simplified. Next, in section 4 we state our main result,
which, after the introduction of a convenient notation and the derivation
of a useful lemma in ssction 5, is proven in section 6.
2. Construction and properties of the scattering data.
Let us briefly discuss the direct scattering problem for the
Zakharov-Shabat system
90
(2.D 1) -( r , • = £ ,
where q = q(x) is a real function and z, a complex parameter.
For details and proofs we refer to [1], [2], [6], [12]. Our notation is
similar to that used in [6].
Following [6] we assume that the potential q satisfies the
hypotheses:
(2.2a) q 6 c'(R)
(2.2b) lim q(x) = lim q'(x) = 0| x [ •+» | x | -M°
(2.2c) j (|q(s)| + |q'(s)|)ds <. +».
In addition we shall need some conditions on the zeros of the
Wronskian of the right and left Jost solutions, to be specified
presently in (2.13).
For lm C Ï 0 we define the (right and left) Jost solutions >JJ (x,r,)
and v,(x,r,) as the special solutions of (2.1) uniquely determined by
(2.3a) v (x,Ü = e~1CXR(x,,;), H m R(x,r.) = ! )
irx A(2.3b) v„(x,rj = e ' L(x,t), lim L(x,rJ =
x^+m ^U
The vector functions R and h are continuous in (x,r.) on E^C and analytic
in r, on € + for each x £ K. Furthermore their components satisfy
(2.4) max sup |R.(x,rj[, sup !L.(x,rJ!J- exp{ [ !q(s)jds[,LI.'-C + 1R-C+
X J I~«J J
i = 1,2.
For Im .", ' 0 we set
(2.5a)
(2.5b) h ( ) ( ft
-i|>„ (x,-r.) -I|J„ (x,c )1 1
It is readily verified that i(i and ifi„ are solutions of (2.1). Moreover,
for x, K £ E one has
(2.6a) J
(2.6b) W(J £,^) = JL, (x,C> jz + |L2(x,f;)|
2 = 1,
where W(IJJ,<|>) = ip,c(> — tp cfi denotes the Wronskian of ij) and $. Hence, for C real,
the pairs IJJ ,<\> and iK>, ipp constitute fundamental systems of solutions
of equation (2.1). In particular, we have for x, z, £ E
(2.7a) <l>r(x,<;) = r+(c)^(x,c) + v_(.O^^(xtrJ
(2.7b) r+(c) = W(^,*r)
(2.7c) rJO = Wdi^,^)-
The representation (2.7c) makes it possible to extend r_(c) to a function
analytic on Im (; '• 0 and continuous on Im r, r> 0. The following properties
are easily demonstrated:
(2.8a) |r+(?)|2 + |r_(t)|J = 1 , C 6 R
(2.8b) r*(c) = r+(-c), f, € E
(2.8c) r*(c) = r_(-?*), Im C ~: 0.
Furthermore one can derive the integral representations
(2.9a) r+(c) = - j q(s)e~2lfsR] (s.rjds, r. e E
(2.9b) r_(c) = 1 + | q(s)R2(s,Ods, Im c - 0.
In combination with (2.6a) these yield
(2.10) max sup jr. (<;) I , sup |i - r (c)|j - [ |q(s)|ds.Lc£R r, € m -I -J
92
At £ = 0, the scattering problem (2.1) can be solved in closed form. One
has
(2.11) R1(x,0)=cosj I q(s)dsj, R2(x,O) = -sinf [ q(s)dsj
so that by (2.9)
(2.12) r+(0) = -sinf f q(s)dsL r_(0) = cos[ f q(s)ds|.
In terms of r we make our final assumptions:
(2.13a) r_(c) * 0 for r, £ Ü.
(2.13b) All zeros of r in <C+ are simple.
Let us emphasize that, strictly spoken, condition (2.13b) is not
necessary. In fact, a very elegant direct and inverse scattering
formalism using only (2.2) and (2.13a) has been developed by Tanaka in
[12]. Our motivation for requiring (2.13b) is that it simplifies the
reasoning considerably.
Condition (2.13a), on the other hand, cannot be omitted. It poses an
implicit restriction on the potential q and forms therefore a weak point
in the Zakharov-Shabat scattering theory, as developed so far. An obvious
consequence of (2.13a) is, in view of (2.12),
(2.14) [ q(s)ds * (k + J)*, k f. 7L.JOther explicit consequences for the potential q are still to be found.
Note, however, that for small potentials no problems arise.
Specifically, if
(2.15) |q(s)|ds < 1
then (2.10) shows that (2.13a) is fulfilled.
Moreover, if
(2.16) j |q(s)jds < 0.904
then (2.13a) and (2.13b) are trivially fulfilled since r_(O * 0 for
h ? ? 0 (see [2] for a proof).
93
We now proceed with the construction of the scattering data associated
with q(x). As a result of (2.13a) the function r_(O has at most finitely
many zeros t,. ,{,„,... ,£„, Im t,. > 0. By (2.8c-13b) they are all simple and
distributed symmetrically with respect to the imaginary axis. Let P be the
number of purely imaginary zeros and set M = (N - P)/2. We order the
in such a way that
(2.17) j = 1,2,...,N,
where a denotes the permutation among integers between 1 and N defined by
(2.13) o(j) = j
= j
= j
+ 1
- 1
j
j
odd
even
j
5 2M
•= 2 M
• 2 M
It is a remarkable fact, that the r,. are precisely the eigenvalues of (2.1)
in the upper half plane (the so-called bound states). The associated
L2-eigenspaces are one-dimensional and spanned by the exponentially
decaying vector functions n> „ (x,£,.), j = 1,2,...,N. Note that by (2.7c)
there are nonzero constants a(^.) such that
(2.19) ^ ( x , ^ ) = a(Cj)^(x,?j).
One can now derive the representation
(2.20) ^ = (;.) --2i.*(t,> f *, ,i,-)v,, (s,r )ds.J l2 J
Bearing in mind that the integral on the right does not vanish because of
(2.13b), we define the (right) normalization coefficients by
r ! C°° 1-1(2.21) C. = Ji ,|;„ (s,c.)>Jj. (s,c.)ds .
It is easily seen that they satisfy the same symmetry relation as the r,. ,
i.e.
(? ??} r = —(c >"
Next, we introduce the following functions of c € K
(2.23a) a (J;) = 1/r_(c), the (right) transmission coefficient
94
(2.23b) b (c) = r (c)/r_(O, the (right) reflection coefficient.
By (2.3) one has for C € K
(2.24a) a (4) = a (-5), b (c) = b (-£)
(2.24b) | a r ( c ) | 2 " | b r ( O | 2 = 1.
In [6] i t is shown that b^ i s an element of C fl L1 0 L2 (3Ü, which
behaves as o( |?
of (2.12) that
behaves as o ( | c | ) for £ -> +°°. Furthermore, i t i s an obvious consequence
(2.25) b (0) = - tan { q(s)ds ,r L-coJ -I
which shows that in general the reflection coefficient may assume
arbitrarily large values.
Clearly, by imposing stronger regularity and decay conditions on q(x)
in addition to (2.2-13), one can improve the behaviour of b (f,) . For
instance, if q(x) has rapidly decaying derivatives, then b (r;) has
rapidly decaying derivatives as well. The converse also holds. In
particular, q is in the Schwartz class if and only if b is in the
Schwartz class (see [12]).
Moreover, if, for a potential q satisfying (2.2-13), there exists an
t. • 0 such that q(x) = Q(exp(-2f»x)) as x + +<*, then it follows from
(2.9) that for any s: . •- 0 with i- • i.Q and f • Im r, . , j = 1,2,...,N,
the function b (c) is analytic on 0 •- Im c •• >• . a"d continuous and
bounded on 0 •; Ira r, •" i , .
Uc shall call the aggregate of quantities {b (') ,L. . ,C.} the (right)
scattering data associated with the potential q. Their importance lies
in the fact, that a potential is completely determined by its
scattering data.
In concluding this section, let us point out that, as in the
Scliröd int;er case (see Chapter 2, subsection 2.1), it is usually not
possible to obtain the scattering data in closed form. Also here, there is
an interesting exception: for the potential q(x) = .< sech :•;,
< € K\;k+i;k £ Z] one can solve the scattering problem (2.1) in closed
form. This is shown in Chapter 6, section 5 of this thesis.
r>5
3. Simplification of the inverse scattering algorithm.
Let q be any potential satisfying (2.2-13). Then q can be recovered
from its scattering data {b (<;),£;.,C.} by solving the inverse
scattering problem.
For that purpose one defines the following functions of s £ E
(3.1a) .ft(s) = S2d(s) + <2c(s),
K 2i5.s(3.1b) £2d(s) = -2i .£ C.e J ,
(3.1c) u (s) = - ( b (rje2lcsdr,.c TI r
-co-'
Because of (2.17-22-24a) both ;:,(s) and ;'• (s) are real functions. Sinced c
b is in C n L'(E), the integral in (3.1c) converges absolutely and ',".
belongs to CQ n L2(H).
Next, introduce the 2x2 matrix
, o -n(s(3.2) g(s) = {
\:(s) o
and consider the Gel'fand-Levitan equation (see [1], [2], [6], [12])
(3.3) tó(y;x) + s;(x+y) + g(z;x)L!(x+y+z)dz = 0
with y :• 0, x £ E. In this integral equation the unknown £(y;x) is a
2x2 matrix function of the variable y, whereas x is a parameter.
Observe that some authors use a slightly different version of the
Gel'fand-Levitan equation which can be transformed into (3.3) by a
change of variables (see [6], p. 46).
In [2] it is shown that for each x € E there is a unique solution9 x 0
<j(y;x) to (3.3) in (L2 ) (0 < y • +»). it has the form
(3.4) j.where a(y;x) and p(y;x) are real functions belonging to C 0 L1 fl L2
(0 •' y •' +<*>), which vanish as y •+ +». The inverse scattering problem is
now solved, since the functions a and ;•• are related to the potential q
in the following way
(3.5a) q(x) = 3(0+;x)
(3.5b) j q2(s)ds =-a(O+;x), x £ R.
Using (3.4), the matrix integral equation (3.3) can be reduced to a
scalar integral equation involving only I'
-TO -OD
(3.6) S(y;x) + Q(x+y) + (? (z;x)s;(z+s+x):,:(s+y+x)Jsdz = 0.
This is the form of the Gel'fand-Levitan equation that appears in the
literature (cf. [1], [12]) and is frequently used in the asymptotic
analysis of nonlinear equations solvable via the Zakharov-Shabat inverse
scattering formalism.
It has, in our view, a number of disadvantages. Firstly, the information
about a, which was still present in (3.3), is now lost. Secondly, the
equation contains a double integral which is of course harder to analyse
than a single one. The third objection is of an algebraic nature: the
structure of g, which is simply the matrix representation of a complex
number, is violated.
Let us try to mend these shortcomings, starting with the last one.
Set
(3.7) y(y;x) = m(y;x) + ifi(y;x).
Then it is straightforward to deduce the following integral equation
from (3.3-4)
r(3.3) ,(y;x) + i;:(x+y) + i ::(x+y+z), (z;x)dz = 0.
Clearly, equation (3.3) has none of the disadvantages mentioned
above. In particular, the information about « and 3 is obtained by taking
real and imaginary parts. This yields
(3.9a) q(x) = 1m r(0+;x)
(3.9b) qJ(s)ds = -Re f(0+;x).
x-"
Ivote, that the equations (3.3) and (3.3-4) are equivalent. In fact, let
ij) denote the mapping that identifies a complex number with its matrix
representation in the following way
(3.10) <K£ + in) = ( ) C, n £ K.n f.
Then (3.3-4) is the image of (3.8) under *.
4. Statement of the main result.
At the moment no method is known that produces explicit solutions of
the Gel'fand—Levitan equation (3.8), associated with arbitrary scattering
data. However, if b = 0 , then the equation gets degenerated and reduces
essentially to a system of N linear algebraic equations, which can be
solved in explicit form by standard procedures.
If q is a potential with scattering data {b (c,),t-»C.} then ther
potential q with scattering data {0,£.,C.} is called the reflectionless
part of q. The structure of the reflectionless part has been discussed
by several authors (see [2], [9], [10]). In the next section we shall
derive a rather elegant representation of q, in terms of determinants.
In applications of the inverse scattering method the following
situation is generic: by some procedure the scattering data of a potential
are known in explicit form. The potential itself is predicted to exist by
the general theory, but its explicit form is unknown. The only thing one
can calculate explicitly is its reflectionless part. Thus, a natural
question to ask is the following:
In •jShich sense is l,>ir potent-la', appyccimi4^ i hy f.'r, ;•.•/*',•.-• :.".•'..-.• ••;."•."
Our main result is the next theorem which gives an answer to this question.
T h e o r e m 4 . 1 . L"l q l-c .i i:'t.ci;.ia' in ' •';• .";-•<'.••?(••• v-."-1 ;•>,•• .•,/.•••<••: :".
:jhu;h .;jtijfC.\i (S.'>!/•) .rii h.i.*. > -V .••;:.• -.'fit.j :::i ib (.O,',- ,Cr.}.
L>:<- q dcnove r.h.c jv;7t.v>' i. •;/••.:.; pr.ri .'? q. '."<•."•'. ;".J." •• i "': x £ F
(4.1) jq(x) ~q d(x)! • a.2! ! ;..c(x+y)[»dy + sup : (x+y) '^0 0- y • + ••
• j i : h ."• j i v e ^ b y ( S . l ? ) m d a . :k*- ;"•)• ',j:j:'nj • .- '.'•'• ^'•.-.'•' •
93
bound states z,.
(4-2a) ao = f
(4.2b) * - 2 U . 11k=1
Herewith, q - q, is estimated completely in terms of the scattering Jala.
More oreai.-ely, the bound -jiven by (4. 1-2) depends only on the-
re flection coefficient b (O and the bound states t,. and not on ther
normalization coefficients C.
Corollary to theorem 4.1. Under the conditions of '.heoreti 4.1 -,v hajc :-.'.?
a priori bound
a 2 r°"
(4.3) sup |q(x) - q(j(J<)| - — (|br(t)| + !t>r(rj |
2 )d^.
Let us mention an important application of theorem 4.1. Consider a
family of potentials q(x,t), depending on a parameter t • 0 referred to
as time. Suppose that q(x,t), which is assumed sufficiently smooth and
rapidly decaying for |xj -* ™, satisfies the initial value problem for a
suitable nonlinear evolution equation of AKNS class [2], e.g. the
modified Korteweg-de Vries equation. Then the bound states i,. do not
change with time, whereas the associated normalization coefficients and
reflection coefficient change in a simple way. The estimate (4.1-2) now
tells us how well the solution q(x,t) is approximated by its soliton
part q,(x,t). Since a„ is invariant uith time, only the behaviour of
:. (x+y;t) is of importance. In particular, for those nonlinear
evolution equations, whose linearized version has a negative sroup
velocity associated witli it (cf. [7], or Chapter 1, Appendix C) one can
construct coordinate regions of the form
(4.4) t • tQ, x • ,(t)
with t„ a nonnegative constant and i(t) a function of time characteristic
for the problem, such that (x+y;t) is small in \? fl \T (0 y • +.-••)
(see [11], or Chapter 2, section 3, for an example of such a
construction). In that case, (4.1—2) shows that in the region (4.4) the
solution q(x,t) is given by qd(x,t) plus a small correction term, which
dies out as C + ».
We shall prove theorem 4.1 in section G. Before doing so we introduce
some notation and derive a useful lemma in section 5.
5. Auxiliary results.
From now on we shall assume that the conditions of theorem 4.1 are
fulfilled.
To start with, let us give our reasoning an appropriate abstract setting.
To this end we introduce the Banach space G of all complex-valued,
continuous and bounded functions g on (0,»0, equipped with the supremun
norm
llgll = sup |g(y)|.0<y <+c«
Furthermore, l e t JC^denote the complex Hube r t space L2 (0,m) with inner
product < f , g > = ! f(y)g (y)dy and corresponding norm II II,.0J
From sect ion 3 we know that for each x € E the functions y >-<• :. (x+y),
Sid(x+y) belong to S 0 K.
Next, keeping x 6 E fixed, we formally wri te
(5.1) (Tdg)(y) = ! Hd(x+y+z)g(z)dz
(5.2) (T g)(y) = } s; (x+y+z)g(z)dz.
0 J
Evidently, T, can be considered as a mapping from 3 into S, but
equally well as a mapping from 3f into 3f. On the other hand, T is not
necessarily a mapping from u into B. However, an obvious adaptation of
formula (4.5.10) in [6] shows that T maps ?f into JC with a norm that
satisfies
(5.3) IIT II < sup jb <c)|.
100
In view of (2.25) we do not expect this norm to be particularly small.
Since both ft, and Q are real-valued, the operators T, and T are self-d c ' ' d c
adjoint on Jf. This fact will play a crucial role in our analysis. Actually,
to such an extent that the size of UT IL is irrelevant.
In the above abstract language, the Gel'fand-Levitan equation (3.8) takes
the form
(5.4a) (I + iTc + iTd)y = -in
(5.4b) tt = fi + SJ ,
where I is the identity mapping.
A first advantage of this formulation is readily seen. Since T + T, is
self-adjoint, the operator I + iT + iT is invertible on K and so we
know at oa"e that (5.4) has a unique solution y £ Jf. Note that this fact
was already mentioned in section 3, from which we recall that, moreover,
V £ B n jff.
For the proof of theorem 4.1 the following lerana is basic.
Lemma 5.1. For any value of the parameter x € K, the opct'ator I+iT, is
invertib'le on the Banaah space B with inverse S = (I + iT.) niocn b'j
N(5.5a) (Sf)(y) = f(y) - .]
0 J(5.5b) A. = p I , (Jpj(2 J f(z)e l C p Zdz) ,
( [cf] e J -hnïv (b •) i-s the inverse of the matrix A = ( [cf] e
ufih^rmjre, the operator S satisfies the bound
_.+i(? +c.) j'.
(5.6) USII i a0, x € K,
(3.7a) (Im r,
J, i ü _j> 'S_( 5 . 7 b ) !1 . = 2 ( l m £ . p ) 2 ( I m ? j ) 2 ^
r,:u», I1SII -ïiï un if om !^ oounded for x e
' ; j r-k'
101
Proof: Let x £ 1R he arbitrarily chosen.
We first consider T. as an operator from the Hubert space ')( into itself.
Since T, is sél^-adjoint, I + iT is an invertible operator on Jf. From the
relation
(5.8) II (I + il'd)gll2 = «3«2 + «Tjgll^, 8 E 3f,
we obtain tiie following bounds for the inverse S = (1 + iT,)
(5.9) IISII2 : 1, HTJSII., • 1.
Next, let us consider T, as an operator from 3 into 5 and show that
I + iT is invertible on 3. Suppose that (1 + iT.)i; = 0 for some ", £ S.
Then g = -iT g € Jf and thus g is identically zero by the preceding
argument. This shows that I + iT is one to one on 8. However, T, is an
operator of finite rank and hence compact. It follows that 1 + iT, is an
invertible operator on the Banach space S.
Furthermore, solving the equation
(5.10) (I + iTd)- = f, f,g € D
we find
N 2 L\ .y(5.11) g(y) = f(y) - X} A.e J" ,
where the A. satisfy
N (•»• 2 if. z
(5.12a) .rittp.A. = 2 j f(2)e «* Jjs. p.,,2,....N
(5.12b) u . = n.b . + i(r, +;,.) , .i. = [C.] e J .PJ J PJ P J J J
Since the operator I + iT, is one to one on S, the matrix A = (i -) is
invertible with inverse A = (H . ) . We conclude that the inverse
operator S = (I + iTj) is given in explicit form by (5.5).
We shall now prove that the matrix elements ,- . are bounded is
functions of x € K, where the bound is explicitly given in terms of the
.',. by
(5.13) |Bpj| < 2 (lm Cp
tJ"•lelil M—^j = n ..
To this end we first introduce some notation. In 3f we consider the elements2ic.y
e. defined by e.(y) = e J . Let A denote the Gram matrix of the vectorsJ J * _]
e 1 ' e 2 ' " " V i-e- A = '•"pj^ ^pj = <ep'ej ' = (2i(r'j " S } ) ' S l n C e t h e
vectors e.,e„,...,e are linearly independent, it follows that der A • 0
(see [5]). Let us write (A) = ($ .) and introduce the vectorsN PJ
h = .'L, 3 .e.. Evidently, <h ,e."P J=1 PJ J P J
combination with (5.5) this gives
h . and h ,h.PJ P J
(•; .. InPJ
N(5.1A) Sh , . = h , s - 2 X, 6 .e.,ci(p) n(p) j = 1 pj j'
where o is the permutation defined in (2.13).
Using the identity I - S = iT,S, we get
(5.15) 20 . = <iT.Sh , ,,h.•.PJ d o(p) J
Hence, in view of ( 5 . 9 )
(5.16) 4 If-; .]' pj '
II h , Jl^llh.ll^ = I . . , A...o(p) 2 j 2 o(p)o(p) jj
liy direct calculation (see [5]) we obtain
(5.17)"it
which completes the proof of (5.13).
By (5.5b-13) we have
= B
W( 5 . 1 3 ) :• « f l l (Im 'N p j l
from which the bound (5.6-7) for IISII is obvious.
Corollary to lemma 5 . 1 . Few each x £ IR the. equal i'on
(5.19) (I + iTd)y = ~iild
103
admits a unique solution v , € 8 and ws havea
N 2ic.y(5.20) Yd(y;x) = -2 p ?=1 Bpje
J .
Remark. Let us recall that y, produces the reflectionless part of the
potential q through the formula
+ ?(5.21) q,(x) = Im Yj(0 ;x) = -2 Im ? . B ..
d d p, j = I pj
Clearly, by (5.13) we have the a priori bound
N(5.22) sup |qd(x)| < 2 ?l=] N p j,
which does not involve the C. but depends only on the ?. in a simple
explicit way.
Starting from (5.21) it is easy to obtain a more elegant representation
of q.. Indeed, introducing the matrices
(5.23a) V( , l(^ +^ - )x\ / i£.x >r •/ . \~lv-.r P J \ „ ƒ \ r.
PJ P J J /' 1 \ PJ;
(5.23b) D, =fc^e J 6 .V E = (c . ) , c . = 12 V J PJ/ PJ PJ
we find that
N -1 -1 -1(5.24) 5 ts . = Tr(EA ) = Tr(ED,V D.) = Tr(D.ED„V ),
p,j-1 pj I l 1 I
where we used the fact that V = D AD., as well as the invariance of the
trace under cyclic permutation. Clearly, D.ED = - — V. Therefore,
using the familiar formula (cf. [4], p. 28, (7.17))
(5.25) 4- (det V) = (det V)Tr(^- v"1)dx dx
we arrive at
(5.26) yd(0+;x) = 2 -^ log det V.
An alternative derivation of formula (5.26) is given in [9], pp. 103-105.
Transforming back one gets
i04
(5.27) Yd(O+;x) = 2 -^ logfdetCD^det A]
N= 4i I, ? + 2 -f- log det A.
p=1 ^p dx ö
NNote that ^i ? i-s Purely imaginary. We conclude that
(5.28a) q,(x) = 2lm -~ log det A
or equivalently
(5.23b) qd(x) = 2 Im T r ( ^ A~1) =
N _1= Im £ (-4U a )g with a = [Cr] e
p=1 p P PP P P
Observe how this simplifies (5.21). For time-dependent potentials the
representations (5.28) form a good starting point for the asymptotic
analysis of q,(x,t) as t + t« (cf. [6], 2nd ed., Appendix A1).
6. Proof of theorem 4.1.
After the preparatory work done in section 5 the proof of theorem
4.1, which we present in this section, is comparatively easy.
Let x £ E be arbitrarily fixed. As a first step, let us write the
solution y of (5.4) in the following form
(6.1) Y = Yd + Y » with
(6.2) Yd = -iS«d.
By (3.9a) and (5.21) we plainly have
(Ó.3) q(x) - qd(x) = Im Yc(0+;x).
From the previous section it is clear that both y and y, belong to B 0 3f.
Hence, we already know that y £ 8 Cl ÏC. It remains to find a concrete
estimate of y in the supremum norm. For that purpose we insert the
decomposition (6.1) into (5.4), thereby obtaining
105
(6.4) (I + iT + iTjy = -iT y, - is; .c d c c d c
Consider (6.4) as an equation in the Hubert space 3f. Since T + T. is
self-adjoint, the operator I + iT + iT, is invertible on JC.
Furthermore, the relation (5.8) holds with T^ replaced by T c + T .
Consequently, (6.4) has a unique solution y E 3( satisfying
(6.5) h II, •-• IIT y.ll, + IIi: II,.c J. c d I c /
An application of the generalized Minkowski inequality (see [8], p. 148]
gives us
(6.6) 111
Hence
(6.7) 111
where by (5 .
(6.8) II y
We conclude
( 6 . 9 ) II Y
W!2
c>d»2
13-20)
d ' , =
that
II -
0
£ l l
ro J
|,SJ
with a. given by (4.2a). The trick is now to rewrite equation (6.4) as
(6.10) (I • iT d), c = -iTcvc - i l V , d - i,:c
and to realize that the a priori estimate (6.9) paves the way to
estimate the right hand side of (6.10) in the supremum norm. In fact,
we have by Schwarz' inequality
(6.11) IIT f :: sup f f |\i (x + y + Z)|2dzW f |r (Z;x)[
2dZ)i
"'c 2 * c 2 'c 2 ^ A 1
Moreover, invoking again the generalized Minkowski inequality, one «e
(6.12) llTcïd
11 ' ll!.lcllllvd»r
iOó
Together, (6.11) and (6.12) lead to the estimate
(6-13) ll-i V c - i V d - iu ( 2ucl
Applying lemma 5.1 we obtain from (6.10-13) the following estimate for
r in the supremum norm
( 6 . 14 ) II y II £ a ^ f l l n II * lift l l ^ \c 0 \ c c 2 /
The desired estimate (4.1-2) is now a direct consequence of (6.3-14) and
the obvious fact
(6.15) |lm7c(0+;x)| <: sup | >c<y;x) | = IIY II.
Herewith the proof of theorem 4.1 is completed.
Remark. Note that we have proven more, since by (6.1-14) and (3.9b) the
estimate (4.1-2) is still valid if one replaces the left hand side of
(4.1) by
(6.16) maxhq(x) - qd(x)|, | j (q2(s) - qjj
References
[ 1] M.J. Ablowitz, Lectures on the inverse scattering transform, Stud.Appl. Math 58 (1978), 17-94.
t 2] M.J. Ablowitz, D.J. Kaup, A.C. Newell and :i. Segur, The inversescattering transform-Fourier analysis for nonlinear problems,Stud. Appl. Math. 53 (1974), 249-315.
[ 3] M.J. Ablowitz and H. Segur, Solitons and the Inverse ScatteringTransform, Philadelphia, SIAM, 1981.
[ 4] E.A. Coddington and N. Levinson, Theory of Ordinary DifferentialEquations, Me Graw-Hill, 1955.
[ 5] P.J. Davis, Interpolation and Approximation, Dover, New York, 1963.
[ 6] W. Eckhaus and A. van Harten, The Inverse Scattering Transformationand the Theory of Solitons, North-Holland Mathematics Studies 50,1981 (2nd ed. 1983).
[ 7] W. Eckhaus and P. Schuur, The emergence of solitons of the Korteweg-de Vries equation from arbitrary initial conditions, Math. Meth. inthe Appl. Sci. 5 (1933), 97-116.
107
[ 8] G.H. Hardy, J.E. Littlewood and G. Pólya, Inequalities, 2nd. ed.,Cambridge, 1952.
[ 9] G.L. Lamb Jr., Elements of Soliton Theory, Wiley-Interscience, 1980.
[10] M. Ohmiya, On the generalized soliton solutions of the modifiedKorteweg-de Vries equation, Osaka J. Math. 11 (1974), 61-71.
[11] P. Schuur, Asymptotic estimates of solutions of the Korteweg-de Vriesequation on right half lines slowly moving to the left, Preprint 330Mathematical Institute Utrecht (1984).
[12] S. Tanaka, Non-linear Schrödinger equation and modified Korteweg-deVries equation; construction of solutions in terms of scattering data,Publ. R.I.M.S. Kyoto Univ. 10 (1975), 329-357.
[13] S. Tanaka, Some remarks on the modified Korteweg-de Vries equations,Publ. R.I.M.S. Kyoto Univ. 8 (1972/73), 429-437.
[14] V.E. Zakharov and A.B. Shabat, Exact theory of two-dimensional self-focusing and one—dimensional self-modulation of waves in non-linearmedia, Soviet Phys. JETP (1972), 62-69.
108
CHAPTER FI1/E
DECOMPOSITION AND ESTIMATES OF SOLUTIONS OF THE
MODIFIED KORTEWEG-DE VRIES EQUATION
ON RIGHT HALF LINES SLOWLY MOVING LEFTWARD
We consider the modified Korteweg-de Vries equation q +6q2q +q =C X XXX
= 0 with arbitrary real initial conditions q(x,0) = q_(x), sufficiently
smooth and rapidly decaying as |x] -*• °°, such that q„ is a bona fide
potential in the Zakharov-Shabat scattering problem. Using the method of
the inverse scattering transformation we analyse the behaviour of the
solution q(x,t) in all coordinate regions of the form T = (3t) > 0,
x a - u - vT, where u and v are arbitrary nonnegative constants. We
derive explicit x and t dependent bounds for the non-reflectionless part
of q(x,t). If all bound states are purely imaginary these bounds make it
possible to derive a convergence result, clearly displaying the
emergence of solitons. Furthermore, if the reflectionless part of the
solution is confined to x s 0 as t -> », then the bounds help us to
establish some interesting energy decomposition formulae.
109
1. Introduction.
We focus our attention on the modified Korteweg-de Vries (tnKdV)
problem
d . i a ) q t + 6q 2 q x + q x x x = o, -••• • x • +«", t •• o
(1.1b) q(x,O) = qQ(x),
where the initial function qn(x) is an arbitrary real function on R,
such that
(1.2a) Intx) satisfies the hypotheses (2.2-13) in [8] (i.e. Chapter 4)
and is therefore a bona fide potential in the Zakharov-Shabat
scattering problem.
(1.2b) ^n^x^ *"S su^ficiently smooth and (along with a number of its
derivatives) decays sufficiently rapidly for Jx| -> m:
( i) for the whole of the Zakharov-Shabat inverse scattering
method [1] to work,
(ii) to guarantee certain regularity and decay properties of
the right reflection coefficient to be stated further on.
Note that the method of L2-energy estimates yields uniqueness of
solutions of (1.1) within the class of functions which, together with a
sufficient number of derivatives vanish for |xj * •<> (cf. [3]). Let us
refer to this class as the "modified Lax-class" (after [4]).
In [10] it is shown by an inverse scattering analysis that condition
(1.2a) and a special case of condition (1.2b) (namely q„ in Schwartz
space) guarantee the existence of a real function q(x,t), continuous on
K'[0,"0 , which satisfies (1.1) in the classical sense. Whenever, in this
i.aper, we speak of "the solution" of (1.1) we shall refer to the solution
obtained by inverse scattering (unique within the modified Lax-class).
Let us recall that by the inverse scattering method the solution
q(x,t) of (1.1) is obtained in the following way. First one computes the
(right) scattering data {b (r),r.,C.J associated with qQ(x). For their
definition and properties we refer to Chapter 4. Next one puts (see [1],
[10])
1 10
(1.3a) Cr(t) = cTexp{3i(;?t}, j = 1,2,...,N
(1.3b) br(5,t) = br(c)exp!8ic,3t}, — -- r, •• +«-.
Then by the solvability of the inverse scattering problem, there exists
for each t > 0 a smooth potential q(x,t) satisfying the hypotheses
(2.2-13) in Chapter 4 and having {b (t,, t) ,r,. ,C. (t) } as its scattering
data ([10]). The function q(x,t) is the unique solution of the mKdV
initial value problem (1.1).
Explicit solutions by the above procedure have only been obtained for
b 3 0. The solution q,(x,t) of the mKdV equation with scattering data
(0,r,. ,C. (t)} is called the reflect ionless part of q(x,t).
The long-time behaviour of the solution q(x,t) of (1.1) in the
absence of solitons is discussed in [2], [9].
In [2] the existence is claimed of three distinct asymptotic regions
I. x > 0(t) II. |x| ' 0(t1/3)
III. -x ,' 0(t).
Here Q denotes positive proportionality. Within each region, the
solution q(x,t) has an asymptotic expansion, characteristic for that
region. However, as stated in [2], p. 68 proofs are yet to be given.
Apparently, the asymptotic structure of the solitonless solution is
simpler for mKdV than for KdV, where four asymptotic regions were found.
When solitons are present the degrees of complexity are reversed. The
reason is the asymptotic structure of the reflectionless part q (x,t).
First of all, depending on the location of the bound states r., it may
happen that solitons do not separate out as t • +«>. Furthermore, if they
do separate out, one can expect both sech-shaped solitons as well as
breathers (see the discussion in section 2). The sech-shaped solitons
move nicely to the right, but the breathers are unpredictable: depending
on the r,. , the breather envelope may move to the right, to the left, or
even be at rest. Moreover, if one relaxes condition (1.2a) to Chapter 4,
(2.2-13a) and uses the inverse scattering formalism developed by Tanaka
in 110], the asymptotic structure of the associated reflectionless part
of q(x,t) is even more complicated, since now multiple-pole solutions can
occur, as calculated in [11]. In the sequel, however, we shall stick to
1 1 1
(2.2-13). In summary, only under additional restrictions on the location
of the bound states can we expect q, (x,t) to decompose into a rinite
number of solitons moving to the right.
However, we prefer to consider the general situation. In that case,
the only thing about q(x,t) one can expect (in view of the negative
group velocity associated with the linearized version of (1.1)) is the
leftward motion of the dispersive wavetrain.
Thus we are confronted with the question: Confining ourselves to the
regions I and II how well is the solution q(x,t) approximated by its
reflectionless part q,(x,t)?
In this paper we give a complete answer to this question by analysing
behaviou
of the form
the behaviour of the function q(x,t) - q,(x,t) in all coordinate regions
(1.4) T = (3t)1//3 > 0, x a -a, a = u + vT,
where y and v are arbitrary nonnegative constants. Plainly, (1.4) covers
all of the regions I and II. It is proven that
(1.5) sup |q(x,t) - qd(x,t)| = 0(t"'/3) as t ->• ».xS—a
If all bound states are purely imaginary, then (1.5) leads to a con-
vergence result (see (5.5)) which clearly displays the emergence of
solitons. Moreover, we construct several explicit x and t dependent
bounds for q(x,t) - q,(x,t) valid in the region (1.4). In the special
case that the reflect ionless part is confined to x -- 0 as t -> ™, we
derive the energy decomposition formulae
r N -i/i(1.6a) q2 (x.t)dx = 4 Z. Im c, + 0(t ) as t • •
_aJ P P
(1.6b) | q2(x,t)dx=| j iog(i + |br(i;)|J)dC + 0(t"1/3) as t * ».
The results obtained in this paper are of a similar nature as those
obtained for KdV in [7], i.e. Chapter 2 in this thesis. However, there
are differences. For instance, in the KdV case we could improve (1.5) in
the absence of solitons (i.e. q, = 0). The analysis in this paper
indicates that no such improvement is likely for mUdV. Furthermore, the
112
KdV estimates in Chapter 2 are only valid for 0 :; v v where v is
some fixed number connected with properties of the Airy function, and
for t 2 t , where t , is some critical time. The mKdV estimates obtainedcd cd
here are valid for any value of \> • 0 and for all t • 0. A. third
difference is found in the structure of the energy decomposition formulae
(1.6) versus Chapter 2, (7.25). Finally, we refer to the remarks about
the asymptotic structure of the reflectionless part made above.
The organization of this paper is as follows.
In section 2 we isolate certain properties of the reflectionless part
q,(x,t). In section 3 we recall two essential results obtained previously
in Chapter 4 and Chapter 2. The first is a general theorem in which
q(x,t) - q,(x,t) is estimated in terms of V, (x+y;t). The second is a
lemma revealing the structure of '.1 (x+y;t). In section 4 we apply this
lemma to estimate Q. (x+y;t). Then in section 5 the estimates of
ii (x+y;t) and the theorem are combined to give estimates of
q(x,t) - qd(x,t).
2. Some comments on the asymptotic structure of the reflectionless part.
The reflectionlest: part of q(x,t) is ™iven in explicit form (see
Chapter 4, (5.28)) by
(2.1) qd(x,t) = 2 Im — log det A
where A = (u .) denotes the N*N matrix with elementsPJ
_ -2ir, .x
(2.2) ,tp. = [ j - 1 J r '
If all the bound states are purely imaginary, say c. = in.,
0 ii ... n - n., then the asymptotic structure of q,(x,t) is
relatively simple. In that case, corresponding to M = 0 in Chapter 4,
(2.13), the normalization coefficients are purely imaginary as well, say
C- = iu-t u• £ E\IO}. It is shown in [6], that as t approaches infinity
the reflectionless part of q(x.t) decomposes into N solitons uniformly
1 1)
with respect to x on E. Specifically
i + ^1( 2 . 3 a ) lira sup q^ ( x , t ) - ^Z1 j — 2ri s g n ( u r i ) s e c h [ 2 n r i ( x - x ~ 4 r i 2 t ) ] j [ = 0 ,
f*» x£R
( 2 . 3 b ) x + = -^-
Note how closely this resembles the corresponding formula Chapter 2,
(5.21) for the KdV case.
If M > 0 in Chapter 4, (2.13), then the structure of q,(x,t) is more
complicated. Let us consider the simplest case: M = 2, {,, = r, + i;i,* r r
[,2 = ~C + in = "(;,> with f=, n > 0. Then C = • + ip and C. = - A + in,
where A and JJ are real constants that do not vanish simultaneously.
Using (2.1) one gets (see [5], [11])rn , % / „i / .... [sin <f + (ii/>')cos <> tanh S'l(2.4) q.U.t) = 4ri seen V , , . ' ' —• -,—- ,
d L (n/v cos2 * sech2 fj
with
(2.5a) * = 2f,x + 8f,(-"2 - 3n2)t + *
(2.5b) f = 2nx + 8,i(3fz - n2)t + v,
where the constants $ and ^ satisfy
(2.6a) exp(-v) = J (>-./n) ( \2 + |.2)^(f2 + n2 ) ~ J
(2.6b) sin $= (-\n+ nO(X2 + u 2 ) " * ^ 2 + n2 ) " *
(2.6c) cos ,), = (<<•+ i„^(.^ + ^n~hr* + n 2)"-.
Thus q (x,t) has the structure of an oscillating function that is
modulated by an envelope having the shape of a hyperbolic secant. The
envelope and phase velocities are found from (2.5) to be v = 4(M2 - 3 "2)
and v , = 4(3')2 - ' , 2 ) , respectively. Because of the undulations in its
profile, this solution is usually referred to as a breather. Note that
the sign of v is undetermined. Hence, the breather envelope may
propagate to the right, to the left or be at rest.
If M •• 0 in Chapter 4, (2.13) and M 2, then, generically, q,(x,t)
will decompose into M breathers and (N - 2M) sech-shaped solitons.
114
However, it is easy to construct examples (e.g. N = 3, r,. = 5 + in,1
r,2 > 3f,2 , ?_ = i(n2 ~ 3C2) ) , in which no complete decomposition into
breathers and sech-shaped solitons takes place.
Looking at the envelope velocity v , one would expect that for large t
the function q, (x,t) will be concentrated on the positive x-axis,
provided that the breather bound states c = f, + in satisfyP P P y
(2.7) r,p - 3e» > 0, P = 1,2,....2M.
Let us confirm this mathematically. By Chapter 4, (5.28b) we have
M / -2i£
(2.3) qd(x,t) -H.pl, (-4iS[cJ(t)]-1e * J3pp,
where B = (g .) is the inverse of the matrix A given by (2.2). Hence,
using Chapter 4, (5.13), one gets
(2.9a) |qd(x,t)|
(2.9b) iji (x,t) = 8t(Im r, )((Im c ) 2 - 3(Re t )2) - 2(lm z, )x,VP P P 'P ^P
where the constants N ., introduced in Chapter 4, (5.7b) reappear in
this paper in (3.4b). Now, suppose (2.7) holds, then by (2.9)
(2.10a) sup |q (x,t)| = 0(e '>t:) as t + «
r° •> t-(2.10b) qd(x,t)dx = 0(e ) as t -. »,
-co-'
where .; is the positive constant defined by
(2.11) . = mini8(Im c. )((lm r,^)1 - 3(Re j, ) 2); p = 1,2,...,M|.
Thus, indeed, (2.7) implies that, as c ime goes on, qf)(x,t) is confined
to the positive x-axis.With regard to (2.10b), recall that (see [2])
(2.12) J qd(x,t)dx = 4 pS, In. r,p.
1 15
3. Two useful results obtained previously.
To start with let us benefit by the work we have done in Chapter 4.
Since for each t -• 0 the solution q(x,t) of (1.1) satisfies the
hypotheses (2.2-13) in Chapter 4, required for a bona fide potential in
the Zakharov-Shabat scattering problem, we immediately have the
following result, which is established in Chapter 4 in the form of theorem
4.1 and its corollary.
Theorem 3 . 1 . Let q ( x , t ) be the solution of the '•luJtfied Kor: e^ej-de Vries
problem
(3.1)q + 6q2q + q = 0 , —" • x • +»', t • 0
q(x,0) = qQ(x),
where the initial function q „ ( x ) ia an arbitrary real '"iviciijn ''i E ,
isatisfyinrj (l.P,a-b(U). hcl {b ( O ,S • . c f } be the a.ia:.le.>'in-j .tati
associated with q n ( x ) . Then fjr each x £ E .m.i t • 0 •;>:<• ::ar,
( 3 . 2 ) | q ( x , t ) - q d ( x , t ) | • a ^ [ |- (x+y; t ) ! = dy + sup | :.• (x+y; t ) I ) ,
y»
where q , ( x , t ) is the rofLc.ation't-us part >ƒ q ( x , t ) ;::'i\~n by
(3.3) ac(stt) - J- ,-., s €
( 3 - 4 a ) ao
( 3 . 4 b ) N . = 2 (lm c, )PJ P
r..)N N
c, - r J k=1j 'k
Furthermore?, the folloüing a priori, Ixmn.i in ihi'i\'
a 2 (•"• /
(3.5) sup | q ( x , t ) - q d ( x , t ) | • - ^ | M b r ( r . ) | + ! b r ( ü | 2 W ,
Note that a is invariant with time. Hence, to get nn idea of the
magnitude of q(x,t) - qj(x,t) in the region (1.4), only the behaviour of
. (x+y;t) is important. Fortunately, we can now once again benefit from
1 lo
our previous work. In fact, a direct quotation of Chapter 2, theorem 3.3
gives us
Lemma 3.2. In the situation of theorem i. 1, assume that the r'i-yhi
reflection coefficient b (;) satisfies
(3.6) b is of class C2 (») and the derivative:-, b J (rj, j = 0,1,2
are bounded on E.
Let y •• 0, x E R, t > 0. Furthermore, Let \i, v Jrnoti' jrbilv.T.r-j non-
n&jative constants. Put
(3.7a) w = x+y+y, b(c,p> = b r ( r , ) e~ 2 i r ' J , b ( j ) = (~^) jb
(3.7b) T = ( 3 t ) 1 / 3 , Z = w ( 3 t ) " 1 / 3 .
Then one has the representation
(3.3) iic(x+y;t) = T"'br (O)Ai(Z) - Jiï"2b(1 } (0,;,)Ai( ' } (Z) + R(Z,T,n),
(3.9a) |R(Z,T,u)| '1 ï"3llb(2) 11 ~ Z~3/2 ."•..- T • 0, Z • 0,
(3.9b) |R(Z,T,u)| :' T"3llb(2)llto C(v) ;'.,. T • 0, Z - -v,
with C(\) M: in ::haptcr 'i, ('A.hibS and with Ai(n) the Airy function
Chapter i', (ö.j).
Note that the first term in the representation (3.8) is directly connected
with the initial function qQ(x), since by Chapter 4, (2.25)
(3.10) br(0) = -tan j qQ(s)ds .
4. Estimates of i (x+y;t).
Let us assume that the requirements of lemma 3.2 are fulfilled. Then
it is readily verified that in the parameter region
1 17
(4.1) T = (3t) ï 1 , x i: -u, u = u + oT, where u and v arenonnegative constants,
the following estimates hold
(4.2a) sup |£lc(x+y;t)| s YT~'0<y<+co
(4.2b) f |j2 (x+y;t)|dyï |b CO) | f'-l + f | Ai(n) ]dr,) + 7T~'
where the constant >• is given by Chapter 2, (3.66a), with N? replaced by
ilb(2)l..
In addition to the bounds (4.2), which depend only on t, lemma 3.2 also
gives us useful bounds containing both x and t. In particular, fixing
6 E (0,1) in Chapter 2, (3.7), we obtain in the region (4.1)
(4.3) sup |n (x+y;t)| s V W - 1 *&r>0<y<+oo ••
where p denotes the constant Chapter 2, (3.56a) with No replaced by(2)
Jllb II and y is given by Chapter 2, (3 .74) .
If q,,(x) enjoys the specia l property
(4.4) j qQ(s)ds = kn, k € 2 ,
then, in view of (3.10), the estimates (4.2-3) can be improved. For instance,
by lemma 3.2 one has in the parameter region (4.1)
(4.5) sup \<l (x+y;t)| < >ï"2,
with i as in (4.2). Moreover, (4.3) hoLds with the factor T in front of-2
the exponential function on the right replaced by T .Of course, further
simplifications of this type occur when also b (0) = 0. If, as usual,
(4.4) is not fulfilled, one may simplify the representation (3.3) by
working with u = b (0)/(2ib (0)), thereby removing the derivative of
the Airy function.
Though in the present discussion we keep v > 0 arbitrary, our choice of
the bound (4.3) is motivated by this property.
Let us point out that (3.6) is only a weak condition on the
reflection coefficient, which can easily be fulfilled. Actually, as noted
in Chapter 4, if qn(x) 'las rapidly decreasing derivatives then the same
1 18
is true for b (Ü.
In particular, if q„ is in the Schwartz class, then so is b . In that
case, there is not a shadow of a doubt that (3.6) holds.
5. Estimates of q(x,t) - q,(x,t).
Combining theorem 3.1 with the estimates of '.} (x+y;t) obtained in
section 4 we arrive at the main result of this paper, which can be
stated as follows
Theorem 5.1. Let q(x,t) be the solution of Ike modified Kortoweg-i<? Vries
problem
- q + 6 q 2 q + q = 0 , -«• •; x < +« t • 0
(5.1) J t x xxx1 q(x,0) = qQ(x),
bihe.re the initial function qQ(x) is an arbitrary rea' function on E,
satisfying (1.2) in ouch a way thai (S.ii) ic, fulfilled. Let (b (r.),r,.,C.l
be the serattaring data associateJ uiih q_(x). Then for each x € K and
t :> 0 one har,
(5.2) |q(x,t) - q d(x,t)j •= a2J j |s:c (x+y; t) 1
2dy + sup |:Jc (x+y; t) | ) ,
J-ilh q , ( x , t ) the reflectiovlese, part C'..i) of q ( x , t ) , a_ .'/;,- con^ia'U
rive'! :>./ (i'.-l) and a 'he ranmion ini ro.huie-J in l^.,:).- c
:VV'.r; , • ' ; J , y be. arbitrary nemnei.i: ivo i-ove'm: c. i'u'. .i = u + v T ,
T = ( 3 t ) ' / 3 . -hen the followr:j e..-' CH-.I:,, holdü
( 5 . 3 a ) sup j q ( x , t ) - q d ( x , t ) | : A for t • 0X •"-;l
( 5 . 3 b ) s u p | q ( x , t ) - q d ( x , t ) | • > T ~ ' ; ' .•• t • ~x —-i
(5.4a) A = —
1 19
f°(5.4b) Y = a*Yj '
+ Y + |b(O)|(-i + |Ai(„)|dn
with the constant y os in (4.2).
Clearly, if in addition qn satisfies (4.4), then we can improve the-1 -2
estimate (5.3-4) since T can be replaced by T . Although similar
remarks apply to the estimates below, they are omitted to avoid inter-
ference with the reasoning.
Note the similarity between (5.3-4) and the corresponding estimate
Chapter 2, (7.16) in the KdV case. There are, however, two important
differences. Firstly, the estimate (5.3-4) holds for all t 0, whereas
Chapter 2, (7.16) is only valid for t : t ,, where t , is a certain
critical time. Secondly, the results of Chapter 2, theorem 7.1,
including (7.16), hold only for restricted v-values, whereas theorem 5.1
is valid for any value of v "' 0.
Let us discuss some consequences of theorem 5.1.
Firstly, by combining (5.3-4) with (2.3) it is found that, if all
bound states are purely imaginary, the solution q(x,t) of (5.1) splits
up into N solitons as t + « in the following way
Corollary to theorem 5.1. Support' that q„(x) •'«.•.• / •>.' .h.hi; wri'ij .hi;
{b (c),C-,cT} with c. = ii|., 0 * i) < ... • n • w 3<ui cT = i ;i. , u . £ 1R\' 01.
f'nen the. solution of (h.l) r,.U. Cafics
( 5 . 5 ) l i m s u p | q ( x , t ) - ^ ( - 2 n s g n ( u ) s e c h [ 2 : i ( x - x - 4 n 2 t ) ] ) ! = 0
t-*» x > - v T P P P P P Pijith x an in (",ji>).
P
Furthermore, besides the bound (5.3b), which depends only on t, we
obtain from (4.2-3) and theorem 5.1 in the coordinate region
T = (3t) '•-• 1 , x ' -a, a = u + \>T, the x and t dependent bound
(5.6a) | q ( x > t> - ,d(x,t)| • aT-'exp[ § •l
(5.6b) a = vY i(l» b = n ,
120
with ü £ (0,1), Y„, P the constants introduced in section 4.
u
It is a direct consequence of the last remark of Chapter 4, section 6,
that the estimates (5.2-3-6) remain val id if on the left one replaces
|q(x,t) - qd(x,t)| by
(5.7) J (q2(s,t) - qd(s,t))ds .XJ \ /
In particular, this yields
(5.8) q2(x,t)dx = q2(x,t)dx
_aJ _aJ
Note that by (5.6)
(5.9) ! |q(x,t) - q,(x,t)|2dx = Q(t"1/3) ast
ast
! |q(x,t) - qd(x,t)|2dx = 0(t"1/3)
—a
This gives us another way to derive (5.8) (see Chapter 2, pp. 73, 74).
In the special case that the breather bound states satisfy (2.7) we
find from (2.10b-12) and (5.3)
(5.10a) q2(x,t)dx = 4 I, lm c + 0(t~1/3) as t -> -.—(r
Using the formula (see [2])
(5.11) j q 2 (x , t )dx = | j logM + |b (r.)|2]dr, + 4 ï Ia f
we obtain for the complementary integral
(5.10b) j q2(x,t)dx = | J logM + |br(f,)|2 jdf. +0(t~'/3) as t ->• -.
It is interesting to compare (5.10) with the formulae Chapter 2, (7.20-25)
obtained in the KdV case.
Finally, let us remark that, as in Chapter 2, we can apply theorem 5.1
to obtain estimates in subregions of (1.4), e.g. x ' vt • 0, with v • 0
arbitrarily fixed.
Specifically, if we make the additional assumption
I21
(5.12) There is an integer n £ 2 such that b £ Cn(E) and a! 1(' \ ^
derivatives b (5), j = 0,1,...,n satisfy
then, reasoning as in Chapter 2, p. 75, we obtain
(5.13) sup |q(x,t) - q (x,t)| = 0(t ") as txgvt
Likewise, one can make the extra assumption
(5.14) There is an £» ~> 0 such that 1n(x) - (Kexp(-2t;,.x)) as x >
and combine the remark at the end of Chapter 4, section 2, with the
reasoning in Chapter 2, pp. 75, 76. Choosing c > 0 to be strictly
less than t;_, b/v and Im c., j = 1,2,...,N one then arrives at
(5.15) sup |q(x,t) - q (x,t)| = 0(exp(-u t)) as t •> »xïvt
with a. = 2c (v - 4c2) > 0.
References
L 1] M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, The inversescattering transform - Fourier analysis for nonlinear problems, Stud.Appl. Math. 53 (1974), 249-315.
[ 2] M.J. Ablowitz and H. Segur, Solitons and the Inverse ScatteringTransform, Philadelphia, SIAM 1981.
[ 3] Y. Kametaka, Korteweg-de Vries equation IV. Simplest generalization,Proc. Japan Acad., 45 (1969), 661-665.
[ 4] P.D. Lax, Integrals of nonlinear equations of evolution and solitarywaves, Comm. Pure Appl. Math. 21 (1968), 467-490.
[ 5] G.L. Lamb Jr., Elements of Soliton Theory, Wiley-Interscience, 1980.
[ 6] M. Ohmiya, On the generalized soliton solutions of the modifiediiorteweg-de Vries equation, Osaka J. Math. 11 (1974), 61-71.
[ 7] P. Schuur, Asymptotic estimates of solutions of the Korteweg-de Vriesequation on right half lines slowly moving to rho left, preprint 330,Mathematical Institute Utrecht (1984).
122
[ 8] P. Schuur, On the approximation of a real potential in theZakharov-Shabat system by its reflectionless part, preprint 341,Mathematical Institute Utrecht (1984).
[ 9] il. Segur and M.J. Ablowitz, Asymptotic solutions of nonlinearevolution equations and a Painleve transcendent, Proc. Joint US—USSRSymposium on Soliton Theory, Kiev 1979, V.E. Zakharov andS.V. Manakov eds., North-Holland, Amsterdam, 165-134.
[10] S. Tanaka, Non-linear Schrödinger equation and modified Kortewejj-deVries equation; construction of solutions in terms of scatteringdata, Publ. R.I.M.S. Kyoto Univ. 10 (1975), 329-357.
[11] M. Wadati and K. Ohkuma, Multiple-pole solutions of the modifiedKorteweg-de Vries equation, J. Phys. Soc. Japan, 51 (6) (1982),2029-2035.
CHAPTER SJK
MULTISOLITON PHASE SHIFTS FOR THE MODIFIED KORTEWEG-
DE VRIES EQUATION IN THE CASE OF A NONZERO REFLECTION
COEFFICIENT
We study multisoliton solutions of the modified Korteweg-de Vries
equation in the case of a nonzero reflection coefficient. Confining
ourselves to the case that all bound states are purely imaginary, we
derive an explicit phase shift formula that clearly displays the nature
of the interaction of each soliton with the other ones and with the
dispersive wavetrain. In particular, this formula shows that each soliton
experiences, in addition to the ordinary N-soliton phase shift, an extra
phase shift to the right caused by the interaction with the dispersive
wave train.
1. Introduction.
We consider the modified Korteweg-de Vries (mKdV) equation
q + ó<j2<j + q = 0 with arbitrary real in i t i a l conditions q(x,0) =t X XXX
= q„(x), such that q_ is a bona fide potential in the Zakharov-Shabat
scattering problem, sufficiently smooth and rapidly decaying for
|xj ->• <" for the whole of the Zakharov-Shabat inverse scattering method
124
[1] to work and to guarantee certain regularity and decay properties of
the scattering data, to be stated further on.
The long-time behaviour of the solution q(x,t) of the mKdV problem
has been treated by several authors. For details we refer to [9], i.e.
Chapter 5, where some of the idiosyncrasies are discussed.
In the asymptotic part of this paper (sections 3, A and 5) we
confine ourselves to the case that all bound states are purely imaginary.
Then it is found that as t -> +•» the solution decomposes into N sech-shaped
solitons moving to the right and a dispersive wavetrain moving to the left.
As t + -« the arrangement is reversed. One can now ask for the phase
shifts of the solitons as they interact both with the other solitons and
with the dispersive wavetrain.
In this paper, starting from our asymptotic analysis of the solution
given in Chapter 5, we derive a phase shift formula that closely resembles
that found by Ablowitz and Kodama [2] for the KdV case. We next show that
- as in the KdV case - a simple substitution produces a more transparent
formula. From the latter formula it is evident, that each soliton
experiences, in addition to the ordinary N-soliton phase shift, an extra
phase shift to the right, the so-called continuous phase shift, caused by
the interaction with the dispersive wavetrain. Thus, the presence of
reflection causes an advancement in the soliton motion. Note that in our
KdV analysis [8], i.e. Chapter 3, we found the opposite. There the inter-
action with the dispersive wavetrain causes a delay in the soliton
motion, since the continuous phase shifts are to the left.
The composition of this paper closely resembles that of Chapter 3. In
section 2 we briefly discuss the left and right scattering data associated
with q„(x) and show how, in the general case that the bound states are
distributed symmetrically with respect to the imaginary axis, the left
scattering data can be expressed in terms of the right scattering data in
a convenient way. In the rest of the paper we assume that all bound states
are purely imaginary. In section 3 we quote a result from Chapter 5,
describing the asymptotic behaviour of q(x,t) as f v +1. By the same
symmetry argument as in Chapter 3, we derive from this result the
asymptotic behaviour of q(x,t) as t >• —". Next, in section 4, the two
asymptotic results are combined to give a phase shift formula of Ablowitz-
125
Kodama type. The representation of the left normalization coefficients
in terms of the right scattering data, which was obtained in section 2,
then enables us to write the phase shift formula in a more transparent
form. To illustrate our results, we calculate in section 5 the continuous
phase shifts arising from a sech initial function.
2. Left and right scattering data and their relationship.
For Im z, ~~ 0 we introduce the Jost functions g> (x,0 and ï>„(x,r.),
two special solutions of the Zakharov-Shabat system
(2.0 ( ) = ( )( ). ' = 4-> -~ -* • +->V / \ • J\ J dxZ U i
uniquely determined by
(2.2a) Yr(x,r.) = e~1CXR(x,f;), H m R(x,r.) = ^
(2.2b) i|< (x,<-) = e 1 C XL(x,O, H m I x,-.) = ( )-
Here R and L .irt> vector functions with two components. Ue set
(2.3a) r_(r.) = 1 + | qo(x)R?(x,f.)dx, Im r, 0
(2.3b) r+(rj = - q()(x)e~2l'-XR1 (x,,.)dx, : £ E
(2.3c) C+(r.) = 1 + qo(x)L](x,r,)dx, lm : O
(2.3d) t_(O =- f qQ(x)e2lrxL2(x,f )dx, r. € R.
Note that r_(r) = Z+(.r.) and r+(r.) = ('_(-,-). It is si own in [5] thai
r_(O is analytic on Im r. 0 and continuous on Im c. ' 0.
Following [5] and Chapter 5, we shall assume throughout that r lias nc
zeros on the real axis and that all zeros of r in C are simpK'. As a
result the function r (,-) lias at most finitelv many zeros,» T '
I2o
lm r, •» O, distributed symmetrically with respect to the imaginary axis.
We shall call them the bound states associated with q„. From [5], p. 156
we know that there are nonzero constants n such thatm
(2. A) g»r(x.Cm) = a A ( x , r , m ) .
Furthermore, by [5], (5.3.2) one has
Observe that the integral does not vanish since the r. are simple ze-os of
r_. This enables us to introduce the right normalization coefficients
(2.6a) <-li[J ^^(x.^^
as well as the left normalization coefficients
(2.6b) c f -
Next, we introduce the following functions of r. £ E
(2.7a) a = r_ , the right transmission coefficient,
(2.7b) a,, = £ , the left transmission coefficient,
(2.7c) b = r r_ , the right reflection coefficient,
(2.7d) b,, = H_Z , the left reflection coefficient.
We shall call the aggregate of quantities (a (r,),b (c),t ,C } the righc
scattering data associated with the potential q_. Similarly we refer to
(a„ ((,) ,b , (r) ,r, ,C I as the left scattering data associated with q„.
Note that the right scattering data were already introduced in Chapter 4,
where some of their properties were discussed.
We claim that a,,, b,, and C can be expressed in terms of the right
scattering data in the following way
a (r,)(2.3a) af (O = a^c) bf(O = a\_o br(-r.),
(2.8b) c£ -[«ft"1
p=1pffin
-2c f«> log(i + |b (c) |2 )mi r
•n 0 J ? Cm
Clearly, only formula (2.8b) deserves a proof.
To provide it, note first that by (2.4-6)
Cr = ctJC .m m m
( 2 . 9 )
Next, it follows from (2.5-6b) that
(2.10) d r"
Eliminating a from (2.9) and (2.10) we find
Lastly, from [3], p. 57 we obtain the representation
(2.12) r_(c) = jexpj Im c, - 0.
Differentiating and using the symmetry relation b (c) = b (~O, we find
(2.13)dr-
* 1
p=m ra p
T o g e t h e r , ( 2 . 1 1 ) and ( 2 . 1 3 ) y i e l d the d e s i r e d formula ( 2 . 8 b ) .
3 . Forward and backward a s y m p t o t i c s .
Once t h e r i g h t s c a t t e r i n g d a t a of q (x) a r e known, t he s o l u t i o n q ( x , t )
of t h e forward mKdV problem
(3.1) j qt
qxxx = tO
q(x,0) = qo(x)
123
can in principle be computed by the inverse scattering method [1]. An
asymptotic analysis of the solution was presented in Chapter 5. It was
found there that the asymptotic structure of the reflectionless part of
q(x,t) is rather complicated when the location of the bound states r, is
not suitably restricted.
To avoid any unpleasantness of this kind we shall assume from now on
that all the bound states are purely imaginary. Let us denote them by
(3-2> ?m = inm' With ° *] \ < ••• < n2 '; V
Since ui ( x , i n ) and iJjp(x,in ) a r e r e a l we then have t h a t t h er m cm
normalization coefficients are purely imaginary as well, say
(3.3) Cr = ip r , C = i y \ with u r, \il € KMOhm m m m tn m
The asymptotic behaviour of the solution q(x,t) of (3.1) is now
enlightened by the following lemma, which was established in Chapter 5,
section 5.
Lemma 3.1. Assume that
(3.4) b (c) is of jlass C2 ( E ) and the dpiv.vativus b ^ ( c ) , j = 0 , 1 , 2
jP'j bounded on iR.
Then one hao
(3.5) lim sup1
q(x,t)- z. \-2n sgn(ur)secht2-i (x-x+-4n21) ] f| = 0,m=1 I m & m m m m J
P=1 l^TTm v s p ra
Next, let us consider the backward mKdV problem, starting from the
same initial function q_(x), i.e.
f q,. + 6q2q + q = 0 , — • x • +'••, t 0(3.7) \ t X X X X
L q(x,0) = qQ(x).
120
Plainly, if q(x,t) satisfies (3.7), then w(x,t) = q(-x,-t) satisfies
(3.3)r w. + 6w2 w + w = 0,J t x xxx
*• w(x,O) = qp(-x),
so that w(x,t) satisfies the forward mKdV problem with initial function
q.(-x). To solve (3.7) it is therefore sufficient to determine the right
scattering data associated with q..(-x) and apply the inverse scattering
method to (3.8). However, an easy calculation reveals that the right
scattering data associated with q_(-x) are identical with the left
scattering data associated with q_(x). The latter were examined in the
previous section. Thus, to find the asymptotic behaviour of the solution
q(x,t) of (3.7) for t -* -°° we merely apply lemma 3.1 to problem (3.3) and
perform the transcription q(x,t) = w(-x,-t). This yields
Lemma 3 .2 . Ai'tinni^ tint-
( 3 . 9 ) b f ( r . ) /:.-; >>f ,-l.ics C* ( R ) .v.J
Ü;V boundc\i on E.
) , j = 0,1,2
then onr. nar,
(3.10) lim supt ~ » x: | t | 1/3
• " f f - 11q ( x , t ) - X , ^ - 2 ' s m i ( i : ) s c c h f 2 ' . ( x - x - U - : ? t ) ] = 0 ,
ra=l 1 in ° m m m m f
(3.11) X,X,nV.,\2
ra v ra p m
S i n c e •<• i s r e a l i t f o l l o w s from ( 2 . 9 ) t h a t s n n ( . . ) = s n n ( . . ) . I ' l i e r e fo r im m r.i
i n ( 3 . 5 ) a n d ( 3 . 1 0 ) t h e s a m e s o l i t o n s e m e r g e , a p a r t f r o m a p h a s e s h i f t .
4 . An e x p l i c i t p h a s e s h i f t f o r m u l a .
L e t u s a s s u m e t h a t b a n d b , , s a t i s f y t i n ' c o n d i t i o n s ( i . i ) a n d ( 3 . " ) .
T h e n t h e c o n v e r g e n c e r e s u l t s ( 3 . 3 ) a n d ( 3 . 1 0 ) d i s p l a y c l e . i r l v h o w t l i e
s o l u t i o n q ( x , t ) o f t h e raKdV e q u a t i o n e v o l v i n g f r o m q ( . \ , 0 ) = q . . l x ) S ' - l i l s
up into N solitons as t -> ±«. In particular, we find for the m-th
soliton the following phase shift
;J I m-J /'I ~ i
(4.1) S = x + - x =m ra mm ra m 2r,m
b\ 4,^ P=1m
Note how closely this resembles the phase shift formula derived by
Ablowitz and Kodama [2] for the KdV case (see Chapter 3, (4.1)).
Similarly, the formulae (3.11) and (4.1) become more transparent if
one inserts the representation (2.8b), taking into account (3.2-3). In
summary, this leads to
(4.2a)
(4.2b)
1 r i / P m\+ — p i , '°gL' + ,, )
..2m
(4.3a) Sm = Sd + Si
(4.3b) S ^ f j 'm
(4.3c) S C = - .
m
In S we recognize the pure N-soliton phase shift (caused by pairwise
interaction of the m-th soliton with, the other ones). The quantity Sm
(which is positive for nonzero b ) can be seen as the shift caused by
the interaction of the m-th soliton with the dispersive wavetrain. For
nonzero b we obviously have
(4.4) S^ ... SC2 • S^ > 0.
Thus, surprisingly enough, the interaction with the dispersive wavetrain
advances the solitons in their motion and the effect is most heavily felt
by the smallest one, corresponding to •; . Recall that in our KdV analysis
(Chapter 3) we found the opposite situation. There the interaction with
the dispersive wavetrain causes a delay in the motion of the solitons.
Let us examine where this difference comes from. Apparently, formula
131
(4.3b) for the mKdV pure N-soliton phase shift S coincides with the
corresponding KdV formula Chapter 3, (4.3b), provided we identify n with
K , m = 1,2,...,N. But after this identification it is clear that,
remarkably enough, also the continuous phase shifts S are given by the
same formula, namely
p log|ar(k)|2
where a (k) denotes the right transmission coefficient. Now, suppose b is
not identically zero. Then, in the KdV case we have by Chapter 2, (2.12)
(4.6a) |ar(k)|2 = 1 - |br(k)|
2,
cso that S is negative.By contrast, in the mKdV case, Chapter 4, (2.24b) tells us
(4.6b) |ar(c)|2 = 1 + |br(c)|
2,
c c
leading to a positive sign of S . Thus the difference in sign of S stems
from the difference in sign of |b (k)|2 in the formulae (4.6).
Usin^ the formula (see [3])
r» , p N(4.7) q2j(x)dx = log(l + jb (r.)|2)d? + 4 ± Im r
we obtain for the continuous phase shift S the following estimate in termsm •'
of the initial function q,- (x) and the bound states t = i-i :0 p P
(4.8) 0 a S^S -±( { ^ W d x - 4
5. An example: the continuous phase shifts arising from a sech initial
funct ion.
To make the previous discussion less abstract let us compute the
continuous phase shifts arising from the initial function
132
:5.1) qQ(x) = a sech x, a £ K\{ k+ i; k e Z}.
[t is a remarkable fact that for the potential (5.1) one can solve the
scattering problem (2.1) in closed form. In fact by an obvious modification
}f the calculation performed in [7], section 3, we obtain for x £ ÜR,
lm C S 0
(5.2a) Rt(x,C) = F(a,-a; J-H; z)
(5.2b) R2(x,e) = o T ^ d - z ) * A.F(a,-a; t-it; z)
with
(5.3) z = JO + tanh x) .
Here F denotes the hypergeometric function in the notation of [A], p. 556.
The same argument as in [7] now gives us
Note that the assumptions about r_ made in sections 2 and 3 are fulfilled
since all zeros of r (;) in Im c, 5 0 are simple and lie on the positive
imaginary axis.
Specifically, one has the following situation.
If |a| < i, then r_(q) has no zeros at all. Note that in this case
(5.5) J |qo(x)|dx < \,
which is in agreement with the sufficient condition [1]
(5.6) |qo(x)|dx < 0.904-co-'
for the absence of solitons.
For j ex | > \, let N £ 1 denote the unique integer such that
N - J < | a. | < N + i. Then r_(^) has precisely N zeros given by
(5.7) C p = i(i + \A - P), P = 1,2 K.
Clearly, ^Q(X) is reflectionless (i.e. b a 0) if and only if • £ 72..
133
Moreover, if a £ Z\{0}, then qn(x) is reflectionless with N = |a| bound
states.
As is easily seen, q_ belongs to the Schwartz space. Hence, the
same holds for b , which equals b„, since q is even. We conclude that
(3.4) and (3.9) hold, so that (3.5) and (3.10) are valid.
Now, let us compute the continuous phase shifts S .
By (5.4a) one has
On the other hand, by (2.12), (3.2)
r« log(1+f f r«
(5.9) r (iv) = \exp{- - .: ü v . o.o=1 v+n
P
Equating both expressions and making repeated use of the recurrence formula
r(z + 1) = zln(z), we obtain
| , H 1 )1B(Uv, l + v) ƒ '
where Ë refers to the beta function ([4], p. 258). Combining (5.7) and
(5.10), we arrive at the following expression for the continuous phasec
1 , JB(1 + 2|.»i 1 1 J-O21—7-7- 1i+|'i|-m &\ B(1 + |a
shifts Sin
(5.1D s1- - ' , . J Q M ^ M M - u - N , i - u t :i){m - m, 1 + |.( | - m) ƒ'
To estimate the magnitude of S we may benefit from (4.8), which yields
(5.12) 0 ' SC -in
The lower bound in (5.12) can be improved by means of the inequality
/r i->\ B(c — 2b, c) . b 2 rt , i
(5.1 3) -57 r '—r4 " 1 + — , c • 0, b • <c,B(c - b,c - b) c
due to Gurland [6l. Together with the simple estimate lo.';(l+x) TTX for
O x - I, this tells us
3 (1 - m + N)(J + |a| - m)
References
[ 1] M.J. Ablowitz, D.J. Kaup, A.C. Newell and 11. Segur, The inversescattering transform. Fourier analysis for nonlinear problems, Stud.Appl. Math. 53 (1974), 249-315.
[ 2] M.J. Ablowitz and Y. Kodaraa, Note on asymptotic solutions of theKorteweg-de Vries equation with solitons, Stud. Appl. Math. b6 (1982),No. 2, 159-170.
[ 3] M.J. Ablowitz and H. Segur, Solitons and the Inverse Scattering;Transform, Philadelphia, SIAM, 1931.
[ 4] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions,National Bureau of Standards Applied Mathematics Series, Ho. 55,U.S. Department of Commerce, 1964.
[ 5] W. Eckhaus and A. van Harten, The Inverse Scattering Transformationand the Theory of Solitons, North-Holland Mathematics Studies 50,1981 .
[ 6] J. Gurland, An inequality satisfied by the gamma function, Skand.Aktuarietidskr. 39 (1956), 171-172.
[ 7] J.W. Jliles, An envelope soliton problem, SIAM J. Appl. Math. 41(1931), No. 2, 227-230.
[ 3] P. Schuur, Multisoliton phase shifts in the case of a nonzeroreflection coefficient, Phys. Lett. 102A (1984), No. 9, 337-392.
[ 9] P. Schuur, Decomposition and estimates of solutions of the modifiedKorteweg-de Vries equation on right half lines slowly moving leftward,preprint 342, Mathematical Institute Utrecht (1984).
CHAPTER SEl/EW
ASYMPTOTIC ESTIMATES OF SOLUTIONS OF TIE SINE-GORDON
EQUATION ON RIGHT HALF LINES ALMOST LINEARLY MOVING LEFTWARD
We consider the sine-Gordon equation q = 4 sin[2 _cJ' q(x',t)dx']
with arbitrary real initial conditions q(x,0) = q„(x), sufficiently smooth
and rapidly decaying as |x| -* °=, such that q_ is a bona fide potential in
the Zakharov-Shabat scattering problem. Using the method of the inverse
scattering transformation we analyse the behaviour of the solution q(x,t)
in coordinate regions of the form t > 0, x ? -ii - vt , where u, -.• and r
are nonnegative constants with 6 •• 1.
We derive explicit x and t dependent bounds for the non-reflectionless part
of q(x,t). Owing to the rather explicit structure of the reflectionless
part it is then a small step to obtain estimates of q(x,t) as well as some
interesting energy formulae.
1. Introduction.
iVe study the sine-Gordon problem
X
(1.1a) qt = i sin 2 q(x',t)dx'|, —• • x +-, t • O
(1. 1b) q(x,O) = qo(x),
136
where the initial function qn(x) is an arbitrary real function on K,j
such that
(1.2a) qo(x) satisfies the hypotheses (2.2-13) in [9] (i.e. Chapter 4)
and is therefore a bona fide potential in the Zakharov-Shabat
scattering problem.
(1.2b) There is an integer k_ such that
rI qo(x)dx = kQ!i.
(1.2c) 1n^x^ *s sufficiently smooth and (along with a number of its
derivatives) decays sufficiently rapidly for | :•: | > '•••:
( i) for the whole of the Zakharov-Shabat inverse scattering
method [1], [2] to work,
(ii) to guarantee certain regularity and decay properties of
the right reflection coefficient to be stated further on.
Uniqueness of solutions of (1.1) can be proven within the class of
functions q(x,t) vanishing sufficiently rapidly for |x| • *° and satisfying
(1.3) q(x,t)dx = k0.-.
Suitably adapting the procedure outlined in [11] for the raKdV problem one
can establish by an inverse scattering analysis that condition (1.2)
guarantees the existence of a real function q(x,t), continuous on Rx[0,~),
such that
(1.4a) For each t • 0 th° function q(x,t) satisfies the hypotheses
(2.2-13) in Chapter 4.
(1.4b) q(x,t) has the property (1.3).
(1.4c) q(x,t) satisfies (1.1) in the classical sense.
(1.4d) q(x,t) falls in the class of functions for which uniqueness of
solutions of (1.1) can be proven.
Whenever, in this paper, we speak of "the solution" of (1.1) we shall
refer to the solution obtained by inverse scattering. Let us add that,
157
despite the multitude of sine-Gordon papers appeared so far, we know of
no reference in which the above is spelled out in full detail.
Given the solution q(x,t) of (1.1) we put
rx rx(1.5) a(x,t) = -2 q(x\t)dx\ o Q ( x ) = - 2 q^xMdx'.
— CO* —CO-*
Plainly, a(x,t) satisfies
(I.óa) a = sin a, -°° < x -; +°°, t •• 0
(1.6b) a(x,0) - oQ(x)
and for fixed t S 0 one has
(1.7) lim o(x,t) = 0 , lim o(x,t) = -2k -i.
In discussions based on the Zakharov-Shabat inverse scattering method the
version (1.6) of the sine-Gordon problem is most frequently used (see
Li], [2], [5], [7]), the only cases allowing an easy treatment being
those in which a has the boundary behaviour displayed in (1.7).
Let us recall that by the inverse scattering method the solution q(x,t)
of (1.1) is obtained in the following way.
First one computes the (right) scattering data fb (r,),c.,C.} associated
with q„(x). For their definition and properties we refer to Chapter 4.
Next une puts (see [1], [2])
(1.3a) cT(t) = cTexp{-it/(2c.)}, j = 1,2,...,N
(1.3b) br(c,t) = br(t)exp{-it/(2c)}, -» •' c - +«•.
Then by the solvability of the inverse scattering problem, there exists
for each t • 0 a real potential q(x,t), satisfying the hypotheses (2.2-13)
in Chapter 4 and having {b (r;, t),C-,C. (t)} as its scattering data. Note
that by Chapter 4, (2.25)
r r(1.9) br(O,t) = -tan q(
(x,t)dx ,
so that property (1.3) follows from (1 . 1b-2b-3b-9) and a continuity
argument. The function q(x,t) is the unique solution of the sine-Gordon
138
initial value problem (1.1).
Explicit solutions by the above procedure have only been obtained for
b = 0. The solution q,(x,t) of the sine-Gordon equation (I. la) with
scattering data fO,C.,C.(t)} is called the reflectionless part of q(x,t).
By contrast with KdV and mKdV, the long-time behaviour of the
solution q(x,t) of the sine-Gordon problem (1.1) is not treated
extensively in the literature.
In the absence of solitons, it is suggested in [3], pp. 30, 91, that
there are three distinct asymptotic regions
I. x ; 0(t) II. |x| • 0(t)
III. -x •: Ö ( t ) ,
where 0 denotes positive proportionality, within each of which the
solution q(x,t) of (1.1) has a different asymptotic expansion. Mote that,
in view of the negative group velocity associated with the linearized
version of (1.1), one may expect that the solution will evolve into a
dispersive wavetrain moving to the left.
If solitons are present, then, generic-ally, as :. ime goes on, q.(s,t)
will desintegrate into breathers and sech-shapod solitons (see the
discussion in section 2 ) . But the sech-shapcd solitons as well as the
breather envelopes propagate to the left. Consequently, a decomposition of
q(x,t) into a dispersive and a soliton part will not easily be
demonstrated in this case. The only thing beyond doubt is that for large
t the solution q(x,t) will be confined to the negative x-axis. This raises
the question: How small is q(x,t) in the regions I and 11?
In this paper we give a definite answer to this question by analysing the
behaviour of q(x,t) in coordinate regions of the form
(1. 10) t - 0 , x • -,, , = u + . t'
where ,., and •• are nonnegative constants with ' • 1. Here ;, and , are
arbitrary but •• depends on properties of q .
More precisely, let q_ and a number of its derivatives decay fast enough
to guarantee that b has n . 2 derivatives decaying sufficiently rapidly
(see (4.1)).
139
Choose
(1 .11 ) 0 • X ' n - | , 0 • ö •• 1 - (^~
Then we prove t h a t
(1 .12 ) sup | q C x , t ) | = Q ( t ~ " ) as t > ~.
Moreover , we d e r i v e t he ene rgy formulae
(1 .13a ) j q2 ( x , t ) d x = 0 ( t ~ ' ) as t >••
ra ? r J
(1.13b) q 2 ( x , t ) d x = ^ - log(1 + ! b r ( . . ) | 2 )d . ' + 4 ^ Ira r. + Q(t )
a s t ->• •».
In particular, if q_ is in the Schwartz class then (1.12-13) hold far all
\ • 0 a n d a l l 0 •- 'i • 1 .
The paper runs as follows.
In section 2 we isolate certain properties of the rel'lect ionless part
q,(x,t). In section 3 we recall a theorem estahlished previously in
Chapter 4, in which q(x,t) - q.(x,t) is estimated in terms of (x+y;t).
Section 4 is devoted to the construction of some simple explicit bounds of
i! (x+y;t). Then in section 5 these bounds and the theorem are combined to
yield estimates of q(x,t) - q,(x,t). Since q,(x,t) was already explored in
section 2, we have reached our goal and found estimates of q(x,t).
2. The asymptotic structure of the reflectionless part.
From Chapter 4, (5.28) we obtain for the reflectionless part of q(x,t)
the explicit expression
(2.1) qd(x,t) = 2 Im -^ log det A,
where A = (:i .) denotes the N'N matrix with elementsPJ
-2U.x( 2 . 2 ) * . = [ C r ( t ) ] e J •• . + i ( c + ,',.) ' .
PJ J PJ P J
140
If :i = 0 in Chapter 4, (2.10), then the asymptotic structure of
q, (x,t) is relatively simple. In that case all the bound states and
normalization coefficients are purely imaginary, say r,. = in-,
0 • n.. ••• ... < i|. •• n, and C. = iu.> v • £ R\{0}. An obvious adaptation2 1 j j j
of the reasoning given in [3] shows that as t approaches infinity then
the reflectionless part of q(x,t) decomposes into f! solitons uniformly
with respect to x on R. Specifically
N(2.3a) lira sup
t-*» x£R
q,(x,t)- I. -2M sgnd. )sech[2', (x-x -v t)] = 0Md ' P=1 \ P P P P P "
Note that v has negative sign. Thus, all sech-shaped solitons propagate
to the left.
In M • 0 in Chapter 4, (2.18), then the structure of q (x,t) is more
complicated. Let us consider the simplest case: S = 2, <"., = •' + it,,* r r
C7 = -•" + in = ~C.» with r, 'i • 0. Then C. = • + in and C? = -\ + iu,
where '• and |i are real constants not vanishing simultaneously.
Using (2.1) one gets (cf. [10] (or Chapter 5), (2.4))fsin !• + (:i/: )cos •:• tanh 1'h 1']
! J( 2 . 4 ) q , ( x , t ) = 4-1 sech r —; 7—rrv; rr T5Hd I 1 + ( M / \ ) 2 C O S Z J > sech 2
w i t h
( 2 . 5 a ) v = 2^x - r,(2(.-2 + r , 2 ) ) ~ 1 t + ji
( 2 . 5 b ) ï' = 2:,x + n ( 2 ( ^ + ' , 2 ) ) ~ ' t + i>,
where the constants i and <;> satisfy
(2.6a) ^ ~ {
(2.6b)
(2.6c) cos Q = (A»; + un)(\2 + M2)~*(c2 + -i 2)" 2.
Tlius q , ( x , t ) has t he form of a b r e a t h e r ( s ee [ 6 ] ) w i t h enve lope and phasev e l o c i t i e s v = ( - 4 ( r 2 + - i 2 ) ) " ' and v . = (4(f2 + - ! 2 ) ) ~ 1 = - v ,
e ph e
141
respectively. Observe that the sign of v is negative.
If M > 0 in Chapter 4, (2.18) and N :• 2, then, generically, q,(x,t)
will decompose into M breathers and (N - 2.M) sech-shaped solitons. However,
it is easy to construct examples (e.g. N = 3, r,. = r + in, z = i(r;2 + i2)'),
in which no complete decomposition into breathers and sech—shaped solitons
takes place.
Since sech-shaped solitons as well as breather envelopes propagate to
the left one expects that, regardless of the location of the bound states,
for large t the function q,(x,t) will be concentrated on the negative
x-axis. Let us verify this. By Chapter 4, (5.21) we have
N(2.7) qd(x,t) = -2 lm ? = ] B ,,
where B = (J5 .) is the inverse of the matrix A given by (2.2).
Since
N . -2ir, x N
(2.8) 1 - „ ? = ] — ~ — 8£. = [Cr(t)]~ e P -I, .- .,
we can rewrite (2.7) as
(2.9) qd(x,t) = -2 Im Z^ Cr(t)e P ^1 - £ = ) *f i-. .V
Hence, using Chapter 4, (5.13), one gets
(2.10a) |qd(x,t)| s 2 ^ |C
(2.10b) IJ, (x,t) = 2(lm c ) (x + |2c l~2t),
where the constants N,., introduced in Chapter 4, (5.7b), reappear in this
paper in (3.4b). In particular, this shows that
(2.11a) sup |q (x,t)| =3(e"';t) as tx>0
(2.11b) | q2(x,t)dx = a(e" 2 t) as t , -,
where „ is the positive constant
(2.12) .. = min|2(lm C )|2- \~2; p = 1,2 N|.
142
Thus, indeed, we conclude that, as time goes on, q.(x,t) is confined to
the negative x-axis.
) satisf
r0 N
Since q,(x,t) satisfies Chapter 4, (2.12), we obtain from (2.11b)
(2.13) | q|(x,t)dx = h Im C + 0(e~2wt) as t •» -,
3. A useful result obtained earlier.
The fact, that for each t > 0 the solution q(x,t) of (1.1) satisfies
the hypotheses (2.2-13) in Chapter 4, required for a bona fide potential
in the Zakharov—Shabat scattering problem, immediately yields the
following result, which is established in Chapter 4 in the form of theorem
4.1 and its corollary.
Theorem 3.1. Let q(x,t) be the solution of the sine-Gordon problem
(3.1)q = { s i n 2 q ( x ' , t ) d x ' | , - » < x < +°°, t *
q(x,0) = qQ(x),
•jhere the initial function q o ( x ) is an arbiivar'j i'cal ^unction jti E ,
satisfying {1.2a-b-o(i)). Let {b ( c ) , ^ . , C . } be the cectticrin-j l i t ;
associated with q . ( x ) . Then for each x £ E and t "• 0 ori^: 'as
( 3 . 2 ) | q ( x , t ) - q ( x , t ) l S a^f f |s: (x+y; t ) | 2 dy + sup ! ( x + y ; t ) | ) ,
jlure q , ( x , t ) is the reflect to nlesa part of q ( x , t ) jivci b'j (''..!) .rid
(3.3) r:c(s;t) - 1 f ^ i ü ^ 5 ^ 1 ^ ^ , , s € R,
(3.4a) a o = 1 + p J = , < I n . C p ) - 1 N p j ,
( 3 . 4 b ) N p j = 2 ( ! m C p ) * ( l B c . ) * ^ J _ ^ | k U 1
P t
~"i(i'J','."<'."!Ji'i'', t h e f j i loui'i^ a v y i j r i b o u n d ••:•.*• df
•j"V
143
ao f" / \(3.5) sup | q ( x , t ) - q , ( x , t ) | •: — |b (O | +|b (c) I2 WC
(x,t)£]Rx[0,») — •' K ;
4. Estimates of '' (x+y; t) .
Since in (3.2) the quantity a is invariant with tine, the magnitude
of q(x,t) - q,(x,t) in the region (1.10) depends only on the behaviour of
;,: (x+y;t). This function in turn can be estimated in a simple and explicit
way, as is shown in the next lemma.
Lemma 4 . 1 . In the situation .•;ƒ i i v i v i «•;./, asnifu? ;':.it
There is an integer n : ' 2 such th.n b is . • ƒ i/ • \ r
a l l d e r i v a t i v e s b J ( c , ) , j = 0 , 1 , . . . , n s t ! : ' i \ - y
(4.1) There is an Integer n ;- 2 such ih.n b iü . • ƒ -.•' !;<•• c"(]R) nJr *
Lev t - 0, y > 0 and j
( 4 . 2 ) x •£ - M - v t , 'Sih'i'c p , v j f l . i is . f j ' i 1 ' I . I V H . - J .-.•.•; '. ' .'..'•;*! : ; • ! • , : .
Ii
Pu L I
( 4 . 3 ) i = 1 + ii + v t ' , w = x + y + i , B d \ , t ) = b ( - , ) e ~ l l ! , D = ( — ) m l i .
rhiin, for' t*l.r.t:d n In (4.1) o'ic has
(4.4a) |«. (x+y; t)| • c T V( 2 n " 3 ) / V 3 / 2 ,
( 4 . 4 b ) c = d m a x d l ^ b ( j ) l l ; j = 0 , 1 , . . . . n ) ,n n n r ™ J ' ' '
Jhere d is an
la defined bij
here d i s a c-ontitant, Lnd<*pcnd<?nt of b , ;>, •.• md • .•••;' ..••'.•;-.• • G C ( E )n r n
(4.4c) «,n(t) = max(1,C2n 2 ) , f, € E.
i ' r o o f : F o r f u n c t i o n s f ( x , y , t , r . ) p a r t i a l d e r i v a t i v e s w i t h n - s p e r t t o ." w i l l
be denoted by
144
(4.5) f(m)
Let us write
(4.6) >> =
(4.3-6) and the symmetry relation b^iO - b^i-O (see Chapter 4,
(2.24a)), we can rewrite J. asJ
(4.3) s7c(x+y;;t) = | Re [ el*Bdc.
Evidently, ü belongs to C (-•» • f; -- +•») and by Leibniz' formula
(4.9a) |v' U ) B (C,t)| 'I N i , with
(4.9b) N = 2'"mmax{(!>)j b J 11^; j = 0,1,. ..,m}.
Mow, let 0 < r < R -1" +•». Integrating by parts n times we find
R
(4.10)
r+ j ei?(iT)nBdr;,
where the operator T is defined by
(4.11) If - (sf) ( 1 ) = s ( 1 ) f + sf ( 1 ).
Induction reveals that the £-th iterate of T has the structure
(4.12a) ICf = s^ I u„ f(C"p) with t„ . = t, whereas for pp=U L,p L,U
(4.12b) ••,,, = I aL'p oï£,,.e2,..,e ez
f
where a, « n » are nonne'jative integers, independent of s and f.
Applying Leibniz' formula to the identity (4C2w+t)s = 2f.2 , we find
(4.13) (4^w+t)s(j) + Sjf.ws^"0 + 4j(j-i)ws(J"2) = (2c.2)(J),
from which it is easily seen that
145
(4.14),(j) M,
IJKMO}, j = 1,2,...,
where M. is a constant, independent of p, v and ö.
Thus, in view of (4.12) there are constants A» , independent of u, v and
6, such that for every f of class C (-» < c. •- +«•)
(4.15) |/f| 5 s£ p I Q '-£*£ \f
U~p)[, K € E\{0}, £ = 0,1 n.
Taking r + 0 and R •+ •» in (4.10) we obtain from (4.0-9-15)
(4.16) i^(x+y;t) = £ Re
where the integral is absolutely convergent,
ifete that by (4.7-9-15)
( 4 . 1 7 ) | T " B | S . V - 2 np 0 n t p n
^ a ,n L A .)n n'n p=0 n'P
V 3 / 2 with
Consequently
(4.13a) |ïïc(x+y;t)| s c^ V( 2 n - 3 ) / V 3 / 2 , with
(4.18b) c = d 2~2nN , d = 1 23n~hd,n-h Z A ,n n n,n n r? 2 2 p=0 n,p
where B refers to the beta function ([4], p. 258).
Herewith, the proof of the lemma is completed.
Let the conditions of the preceding lemma bo fulfilled.
Having fixed n in (4.1) we select * such that
(4.19) 0 * \ % n - |.
Next, we choose 5 such that
<4.20) 0 , 6 . ,
Dy virtue of lemma 4.1 we then have in the parameter region
146
i
(4.21) t S 1, x S -p - vt , where y and v are nonnegative constants,
the following estimates
—A — 3/2(4.22a) sup |Oc(x+y;t)| S Pnt (x + T) '
0+
.22b) f |fl (x+y;t)|2dy 5 |p*t 2A(x + i) 2, with-J C
(4O'
(4.23) Pn = (1 + u + v)"cn.
Here i = 1 + y + ut , while c denotes the constant introduced in (4.4).
5. Estimates of q(x,t) - q,(x,t) and q(x,t).
It is now merely a matter of combining theorem 3.1 with the
Lmates of S2 (x+y;t) derived in sect
this paper, which we state as follows
estimates of S2 (x+y;t) derived in section 4 to obtain the main result of
Theorem 5.1. Let q(x,t) be the. solution of the inc-uopdoi problem
r rx isinj2 q(x ,t)dx , -«• •' x < +», t > 0
(5.1) C:q(x,O) = qo(x),
jhfi'c the initial function qQ(x) is an arbitvacit w j i function on K,
oatisfjing (1.2) in such a mij that (1.1) is fulfilled. Let {b (r,), j ; . , C T}
;•'.,' tho <3iiatlevinj daia associated with q n (x ) . Then foi' each x £ E -i".d
t - 0 one hats
( 5 . 2 ) ] q ( x , t ) - q d ( x , t ) | 'i a d h c ( x + y ; t ) | - f l y + s u p i ^ C x + y ; t ) ] j ,
[Jith q d ( x , t ) ',he i't:fit:ct.Lorii<iss part- c:..V . ; ' q ( x , t ) , a Q :.-".• sjnuurr
(Z.4) and "
(5.3) j. • 0, v • 0, 0 • •. • n - | , 0 ^ • 1 -
147
Put a = u + vt . Then one has the estimates
(5.4a) | q ( x , t ) - q d ( x , t ) | 5 A for t > 0, x S -a
_\ — 1/9(5.4b) | q ( x , t ) - q d ( x , t ) | £ p^t (x + 1 + a) ' for t Ï 1 , x g -a
with
a 2 (••»
(5.5a) A = - ^ | ( | b r ( c ) | + | b r ( r j | 2 ) d c
(5.5b) Pn = a^pn(1 + i p n ) ,
ufaira the. constant p in -liven b:i (4.CA).n
To find the behaviour of q(x,t) in the coordinate region t • 0,
x S -a, it clearly suffices to combine the preceding theorem with the
results of section 2.
It follows from the last remark of Chapter 4, section 6, that the
estimates (5.2-4) still hold if one replaces the left hand side by
(5.6) (q2(x\t) - q,(x\t))dx'
In particular, this yields
(5.7) q2(x,t)dx = qd(x,t)dx + 0(t"X) as t - ~.
_ aJ _aJ
However, by (2.10) one has
fw
(5.3) qd(x,t)dx = 0(e ) as t •• ~
-a*
where ~i i s any c o n s t a n t s a t i s f y i n g 0 • ~i • ..i w i t h ..i as in ( 2 . 1 2 ) .
Consequen t ly
(5.9a) { q2(x,t)dx =0(t A) as t •» ~.-a'
formula (see [3])
['" 2 r :i
q2(x,t)dx = - logO + (b (?,) \!)d;. + 4 r Im .'. ,-J " 0J
r P • P
-U'
Using the formula (see [3])
(5.10)
we obtain for the complementary integral
148
jas t ->
,t)dx = - I-N
(5.9b) j qMx,t)dx = - I log(1 + |br(rJ|2)dc + 4 p|1 lm
i'tote that in addition to (5.3) it ai.-o follows from (2.10) that
(5.11) sup |qd(x,t)| = Ove"u)t) as t ->• •».
xi-a
Thus, in view of (5.4), we arrive at
(5.12) sup |q(x,t)| = 0(t"A) a s t * » ,xi-ct
Let us mention some consequences of the above results for the
solution a(x,t), given by (1.5), of the sine-Gordon problem (1.6). By
(1.3) and (5.4b) one has for t 2 1 , x è -a.
(5.13) | o ( x , t ) - o d ( x , t ) + 2 (k o -k 1 )n | g 4f>nt~A (x+1+u) , with
rX
( 5 . 1 4 a ) o d ( x , t ) = - 2 I q ( x ' . t ) d x ' and
(5 .14b) k = 7 q d ( x , t ) d x £ Z.-co - '
Using (2 .10) , we obtain
(5.15) sup | o ( x , t ) + 2k0iT | = Q(t~' ) as t -> «.
xï-a
To conclude with, let us observe that if the initial function qn(x)
in (1.1b) is in the Schwartz class, then so is b .
This implies that (4.1) holds for all n. Hence (5.7-9-12-15) are in this
case valid for all A •> 0 and all 0 i 6 < 1.
References
[ 1] M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, Method forsolving the sine-Gordon equation, Phys. Rev. Lett. 30 (1973),1262-1264.
149
[ 2] M.J. Ablowitz, D.J. Kaup, A.C. Hewell and '1. Segur, The invers-scattering transform - Fourier analysis for nonlinear problems,Stud. Appl. Math. 53 (1974), 249-315.
[ 3] M.J. Ablowitz and U. Segur, Solitons and the Inverse ScatteringTransform, Philadelphia, SIAM, 1981.
[ 4] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions,National Bureau of Standards Applied Mathematics Series, No. 55,U.S. Department of Commerce, 1964.
[ 5] D.J. Kaup and A.C. Newell, The Goursat and Cauchy problems for thesine-Gordon equation, SIAM J. Appl. Math. 34 (1978), 37-54.
[ 6] G.L. Lamb, Jr., Elements of Soliton Theory, Wiley-Interscience, 1380.
[ 7] G.L. Lamb, Jr. and D.W. HcLaughlin, Aspects of soliton physics, in:Solitons (Ed. R.K. Bullough and P.J. Caudrey) Topics in CurrentPhysics 17, Springer-Verlag, Mew York, 1980.
[ 8] M. Ohmiya, On the generalized soliton solutions of the modifiedICorteweg-de Vries equation, Osaka J. Math. 11 (1974), 61-71.
[ 9] P. Schuur, On the approximation of a real potential in theZakharov-Shabat system by its reflectionlcss part, preprint 341,Mathematical Institute Utrecht (1984).
[10] P. Schuur, Decomposition and estimates of solutions of the modifiedtCorteweg-de Vries equation on right half lines slowly moving left-ward, preprint 342, Mathematical Institute Utrecht (1934).
[11] S. Tanaka, Non-linear Schrödinger equation and modified Korteweg-de Vries equation; construction of solutions in terms of scatteringdata, Publ. K.I.M.S. Kyoto Univ. 10 (1975), 329-357.
CHAPTER EIGHT
CU THE APPROXIMATION OF A COMPLEX POTENTIAL IN THE
ZAKHAROV-SHABAT SYSTEM BY ITS REFLECTIONLESS PART
We consider the Zakharov-Shabat system with complex potential and
derive a pointwise estimate of the error made in approximating the
potential by its reflectionless part. As an illustration we apply this
estimate to investigate the lon^-time behaviour of the solution oc the
complex modified Korteweg-de Vries initial value problem.
1. Introduction.
In Lo], i.e. Chapter 4 of this thesis, studying the Zakharov-Shabat
system with real potential, we derived a pointwise estimate of the error
made in approximating the potential by its ref lectionless part. For t'iat
purpose we reduced the matrix Gel'fand-Levitan equation appearing in the
literature to a scalar intejral equation containing only a single
integral. Furthermore, we exploited the fact that the corresponding
scalar Gel'fand-Levitan operator, when considered in the complex Hubert
space Lz(0,™), has the structure of the identity plus an antisymmetric
operator.
lil
In this paper we take a more general standpoint by considering the
Zakharov-Shabat system with a potential that may assume complex values.
This confronts us with a matrix Gel'fand-Levitan equation that can no
longer be reduced in the above way. It has, however, one important
property in common with the scalar case: The corresponding matrix
Gel'fand-Levitan operator has, when considered in the complex Hubert2x2
space (L2(0,<*0) , the structure of the identity plus an antisymmetric
operator.
Motivated by this resemblance we present an analysis of the matrix
Gel'fand-Levitan equation that - as far as its abstract setting is
concerned - parallels the analysis of the scalar Gel'fand-Levitan equation
given in Chapter 4. To increase the similarity matrix norms and notation
are selected with due care. Although the technicalities are different (cf.
the proof of lemma 5.1 with that of Chapter 4, lemma 5.1) this analysis
leads to an estimate of the difference of the potential and its
reflectionless part, which, surprisingly enough, is almost identical
with that obtained in Chapter 4. Since the only alteration consists in some
numerical front factors due to the particular matrix norms involved, we may
truly speak of a generalization of the main result of Chapter 4.
This generalization is of practical importance, since working with a
complex instead of a real potential considerably enlarges the class of
nonlinear evolution equations solvable via the associated inverse scattering
method. For example, the complex modified Korteweg-de Vries equation, as
well as the nonlinear Schrödinger equation can be solved by the complex
but not by the real Zakharov-Shabat inverse scattering method.
The composition of this paper is as follows.
In section 2 we briefly discuss the direct scattering problem for the
Zakharov-Shabat system with complex potential. The inverse scattering
formalism is outlined in section 3. Next, in section 4 we state our main
result, which after the introduction of some convenient notation and the
derivation of a useful lemma in section 5, is proven in section 6. Finally
we apply the aforementioned result to investigate the long-time behaviour
of the solution of the complex modified Korteweg-de Vries initial value
problem.
152
2. Direct scattering.
To begin with let us tersely survey the direct scattering problem for
the Zakharov-Shabat system [11]
(2.1)
where q = q(x) is a complex function and x, a complex parameter. For
details and proofs we refer to [1], [2], [5], [10]. Our notation is
similar to that used in [5].
Following [5] we assume throughout that the potential q has the
regularity and decay properties stated below:
(2.2a) q 6 C1(E)
(2.2b) lira q(x) = lim q'(x) = 0jx | "*» Jx j -wo
(2.2c) j (|q(s)| + |q'(s)|)ds - +».
In addition we shall require some conditions on the zeros of the
Uronskian of the right and left Jost solutions, to be specified presently
in (2.11).
It is interesting to compare the results of this section to those of
Chapter 4, section 2 and to single out the symmetry relations in
Chapter 4 caused by the realness of the potential.
For Ira c ï 0 we define the (right and left) Jost solutions -j; (x,t,)
and o, (x,t) as the special solutions of (2.1) uniquely determined by
(2.3a) ,: (x,e) = e~1CXR(x,<;), lim R(x,c) = (,1)r o
(2.3b) v£(x,f,) = e1 C XL(x,O, lim L(x,O = (°).
*+
The vector functions R and L are continuous in (x,r) on R<C+ and analytic
in I, on C for each x £ "R.
Furthermore, their components satisfy
(2.4) maxi sup iR.(x,c)|, sup JL.(x,:.)' •. exp] | |q(.s)lds[,LE-<C K>-C J l— J '
i = 1,2.
153
For Im t 5 0 we set
(2.f-<!> ( x . f , ) \ /•••„ ( x , ;
*J ". (x,c y*• i L '
It is readily verified that ij* and ip„ are solutions of (2.1).
Moreover, for x, <; € R one has
(2.6a) W0|.r,<l>r) = |rM (x,c) I2 + |R2(x,r,)|
2 = 1
where W(y,cj>) = t'.'K ~ J'T 'I denotes tiie Wronskian of , and ;•. Hence, f
c, real, the pairs if/ ,y and gi„ ,g/„ const itute fundamental systems of
solutions of equation (2.1). In pa~ticular, we have for x, r € E
(2.7a) Y r(x,O = r+(rji,^(x,r,) + r_(r,),i(,(x,r.)
(2.7b) r.(c) = M(f,,, )+ -c r
(2.7c) r (fj) = W(i)/ .v'1?)-
It is not hard to show that
(2.3) |r (c)|2 + |r (O\2 = 1, ;. £ E.
Let us use (2.7c) to extend r_(r,) to a function analytic on Im r. • 0
continuous on Im c, ' 0.
Then the following integral representations hold
(2.9a) r+(r,) =- | q*(s)e~2ir'sR] (s,,;)ds, r, £ K
(2.9b) r_(O = 1 + q(s)R2(s,<;)ds, Ira C - 0.
In combination with (2.6a) these yield
(2.10) inax[ sup |r+(r,)|, sup l i - r _ ( r , ) | ] - [ |q(s)!ds.r.cK
In terms of r_ we make our f inal assumptions:
(2.11a) r (O z 0 for r. £ E
(2.11b) All zeros of r_ in C+ are simple.
Let us point out that condition (2.11b) can be circumvented by using
TanaLca's direct and inverse scattering formalism [10]. We only include
it for reasons of simplicity.
Incidentally, if
(2.12) | |q(s)|ds ,- 1
then (2.10) shows that (2.11a) is f u l f i l l e d .
Moreover, if
(2.13) f IqCs) Ids < 0.904
then (2.11a) and (2.11b) are trivially fulfilled since r_(-) * 0 for
Im ; -i 0 (see [2]).
We now turn to the construction of the scattering data associated
with q(x). As a result of (2.11a) the function r_(r,) has at most finitely
many zeros C, j^o. • • • »C»,> Im (,. • 0. They are all simple by virtue of
(2.11b). It is a remarkable fact, that the t;. are precisely the eigen-
values of (2.1) in the upper half plane (the so-called bound states).
The associated L2-eigenspaces are one-dimensional and spanned by the
exponentially decaying vector functions y»(x,c;.), j = 1,2,...,N. MoteJ
that by (2.7c) there are nonzero constants i(t-) such that
(2.14) <v (x,£.) = u(t.).jv(x,C,).
(2.15) % ( £ • > = -2ia(c,) f i|V (s,c.)^„ (s,r,.)ds.
One can derive the following representation
f i|V ( s , c . ) ^2
iiearing in mind that the integral on the right does not vanish because of
(2.11b) we define the (right) normalization coefficients by
C. = ii[ [ *„ (s,c,)». (s,c,)dsl(2.16) CT. = i i [ [ *„ ( s , c , )» . (s ,c , )ds l .l| J 2
itext, we introduce the following functions of £; £ E
(2.17a) a (c) = 1/r_(c), the (right) transmission coefficient
155
(2.17b) b (O = r+(;)/r_(c), the (right) reflection coefficient.
By (2.3) one has for c É I
(2.13) |ar(c)|* " |br(O|2 = 1.
In [5] it is shown that b is an element of C II L1 (1 L2 (K) , which
behaves as o(|c| ) for t, -* ±=>. Of course, by imposing stronger
regularity and decay conditions on q(x) in addition to (2.2-11), one can
improve the behaviour of b (5). For instance, if q(x) has rapidly
decaying derivatives, then so has b (5).
We shall call the aggregate of quantities {b (f,),t-»C'} t n e (right)
scattering data associated with the potential q. Remarkably enough, a
potential is completely determined by its scattering data.
3. Inverse scattering.
Let q be any potential satisfying (2.2-11). Then q can be recovered
from its scattering data {b (O.C-.C.l by solving the inverse scattering
problem.
For that purpose one defines the following functions of s € E
(3.1a) f;(a) = s.' (s) + :• (s),d cN 2ii;.3
(3.1b) :;,(s) = -2i .1. C^e : ,
(3.1c) ;.c(s) = | j br(r,)e2lCSd:..
Since b is in C D 1- (K), the integral in (3.1c) converges absolutely
and .: belongs to C n L Z(K).
Next, introduce the 2*2 matrix
/ 0 -."(s)^(3.2) .us; = ( I
V.(s) 0 '
and consider the Ge]'fand-Levitan equation (see [1], [2], [5], [10])
(3.3) g(y;x) + iü(x+y) + B(z;x);j(x+y+z)dz = O
0J
with y > O, x £ E. In this integral equation the unknown 3(y;x) is a
2x2 matrix function of the variable y, whereas x is a parameter. Observe
that some authors use a slightly different version of the Gel'fand-
Levitan equation which can be transformed into (3.3) by a change of
variables (see [5], p. 46).
In [2] it is shown that for each x 6 * there is a unique solution 3(y;x)
to (3.3) in (L 2) 2* 2 (0 < y •- +»). It has the form
(3.4)
*a -b
where a(y;x) and b(y;x) are complex functions belonging to C fl L1 f! V
(0 '- y < +°°) , which vanish as y •> +«>. The inverse scattering problem is
now solved, since the functions a and b are related to the potential q
in the following way
(3.5a) q(x) = b(0+;x)
(3.5b) [ |q(s) |2ds = -a(0+;x), x C R.X-"
Using (3.4), tiie matrix integral equation (3.3) can be reduced to a
scalar integral equation involving only b
(3.6) b(y;x) + ;. (x+y) + b(z;x)L (z+s+x) •.; (s+y+x)dsdz = 0.
In this form the Gel'fand-Levitan equation frequently appears in the
literature (cf. [1], [10]). However, for our present analysis the matrix
form (3.3) proves to be more convenient.
4. Statement of the main result.
If q is a potential with scattering data !b (r.),i.,C.! then the
potential q, with scattering data iO,u.,C.l is called the ref lectionless
part of q. The function qr>(x) can be o tained in explicit form (see (5.30))
by solving the Gel'fand-Levitan equation (3.3), which in that case reduces
157
to a system of Ï1 linear algebraic equations. The main result of this
paper is the next theorem which tells us in which sense the potential is
approximated by its reflectionless part.
h i c h :'••'!! CcJ'i-..: (::.;:-!]) -inJ hi.' -..!••- ;:•*/• •-• •ƒ'.'>._.• ;';;.• •. b ( r ) , ' . . , C . }.
T h e o r e m 4 . 1 . /.<•;' q .'•'•' •/ : J'i'
Qhich :'••'!! CcJ'i-..: (::.;:-!]) -inJ
q dcnoii'. ;•'.'>• f.•;~\\?t.L»ii• ::.; .•'./-"; .•;' q. ."•':• •. ;' ;• ,•_?•'. >; c 18
(4.1) iq(x) - q (x) | -,,::(, 1 f '.. (x+y) 'J dy + sup ' (x+y);),u\ 0J L 0 y + ' y
c ' 0 'n^und .sr.ii! n: i,.
J» _,
(4.2a) .,. = 1 + ? , (lm .; ) M .,0 P,J = 1 P PJ
(4.2b) M - 3(lm ,/a* ,^ j, j 72^; J, | 7^i .
;';.. t - . \ j L : . h , q - q j . ' . ! • • . ; . ' . • . ' « ; . • • • , / • • • • « • ' • • : . ;• '•. • • . " - . . • v ' " •'•• . • • . • • • • • • • . • • . . - • • • • • • .
. - - • ^ . • / / i ' - i , . ' . ' ! / b r ( r j „ • : , . ; ' ••/;,• / - - ; / . - - ; • , . • _ .- • •. • • • •" • • . • • - , ; • . • . • : . • • . ' •.
C o r o l l a r y t o t h e o r e m 4 . 1 . .'..••/' ' •. • ••.- .' '. ' .' '' '•.• .'• ' :'. .' ' • .''
2 pt' i.Jl'i. '••• •' .1 'hi
(4.3) sup |q(x) - qd(x)| -~ [ C-h^,-)' + ,1' br (• ) = )d • .
We shall prove theorem 4.1 in section h. Before doiiv; so we introducé
some notation and derive a useful lemma in section =>.
5. Tirst steps to the proof.
In the remainder of this paper it is understood that the conditions
of theorem 4.1 are fulfilled.
We begin by introducing some useful concents and notation.
Kor 2<2 matrices
133
(5.0 A - P " 3 ) . B - p "3
2 4 2 "4
with complex entries we set
( 5 . 2 ) <A,J3>E = i | 1 u . 0 . , IIAIIE = ( A , A E ) * .
By B we denote the linear space of all 2v-2 matrix functions
/&, (y) S-,(y)\(5.3) g(y) =
V,2(y) s4(>,such that each g. is a complex-valued, continuous and bounded function on
(0,a)). We turn S into a complex Banach space by equipping it with the norm
(5.4) llgll = sup ll3(y)llE.
?>:2
Furthermore, we write 3f to indicate tlie complex Hilbert space (L2(0,«>))~
with inner product
(5.5) -.f,g> = f <f(y),g(y)-.dy
and corresponding norm II II2 .
The choice of the above spaces is motivated by our wisli to exploit
analogies with the scalar Gel' fand-Levitan equation (3.8) in Chapter 4.
Returning to (3.2), let us put
(5.6a) ..,(s) = •.,d(s) + -.c(s),
/ 0 -!-\(s)(5.6b) ,.d(s) = (
(5.6c) .;c(s) = I C
From section 3 we know that for each x 6 B the functions y ••* •..< (x+y),
;.',(x+y) belong to 3 D 3C.
Next, keeping x € 1R fixed, we formally write
15')
(5.7a) (Tdg)(y) = j g(z)ü>d(x+y+z)dz
f™(5.7b) (T g)(y) = g(z)u (x+y+z)dzc
0J c
with g as in (5,3). Plainly, T, can be considered as a mapping from S in
B. but equally well as a mapping from JC into 3C. On the other hand, T is
not necessarily a mapping from B into 8. However, suitably modifying
formula (4.5.10) in [5] one can easily show that T reaps Jf into K with a
norm that satisfies
(5.8) IIT II2 S sup |b (c)|.c (;€E
It is straightforward to verify that the operators T and T are both* * " c
antisymmetric on JC, i.e. Td = ~TJ» T = -T . This fact plays a dominatin
role in our analysis.
In the above abstract language, the Gel'fand-Levitan equation (3.3) take
t he fo rm
(5.9a) (I + T + T,)3 = -•»c d
(5.9b) u = ui + ;..,c d
where I i s the iden t i ty mapping. A f i r s t advantage of t h i s formulation i
read i ly seen. Since T + T is antisymmetric, the operator I + T + T, i
i n v e r t i b l e on JC and so we know at once that (5.9) has a unique so lu t ion
3 £ JC. Hote tha t t h i s fact was already mentioned in sect ion 3 , from whic
we r e c a l l t h a t , moreover, fl £ Ó fl JC.
For the proof of theorem 4.1 the following lemma is bas i c .
Lemma 5 . 1 . FOP :m i vaiutf of ih>/ :'tiparm': et' x £ R , r .V ••:<• :•.>: JP I + T , :j" ' _ i ^
inVCPlibic on ihf. B.inaah a:>usc 8 .Jiih l<rO'i:%: S = ( I+T , ) .-.'.V' /'•'d
(5 .10a ) ( S f ) ( y ) = f ( y ) - \ . \ ( P ' ) , • (y) = e ' p ' ,p ~ ] p \ . p ( y ) o / p
(5 .10b) A
1Ó0
P J =
' j K*J U ' P . j + ^ ' ''p+N,j+N
where (g ) is the inverse of the 21I*2N matrixrs
Z(5.11a) A = f !, wi t^
V-D Z t y
( 5 . 1 1 c ) D = ( Ö . 6 . ) , 6 . = ( - 2 i C . ) e J , D = ( 6 . 6 . ) .J r J J J J rJ
Furthermore, the operator S satisfies the bound
( 5 . 1 2 ) IISU & aQ, x £ E
II( 5 . 1 3 a ) a n = 1 + Z , (lm t, ) M .
U P j j = ' P PJ
*
(5.13b) Mp. = 8(lm t/Vm K-^ & | ^ - | | ,»,
Thus, llsll is uniformly bounded for x £ E, where the bound is an explicit
function of the 5..
Proof: Let x £ E be arbitrarily fixed.
Recall that, when considered as an operator from the Hubert space X into
itself, T, is antisymmetric. Hence I+T, is invertible on X and one has
( 5 . 1 4 ) II ( I + T d ) g l l ! = llgll| + H T d g l l | , g £ 3C,
so that the inverse S = (I+T,) satisfies the boundsd
(5.15) IISII2 i 1, HTdSU2 < 1.
Shifting our gaze, let us consider T, as an operator from 3 into 3 and
show that I+Td is invertible on 3. Suppose that (I+Td)g = 0 for some g £ B.
Then g = ~Tjg £ JC and thus g is identically zero by the preceding
argument. This tells us that I+T, is one to one on S. However, Tj is ofa a
finite rank and therefore compact. It follows that 1+T, is invertible on
the Banach space S.
Next, consider in a f) iff the elements e. ,e0,.. . ,e.„ defined by
161
(5.1óa)O
,0 0 s / 0 0(5.16b) e^0 M(y) = f *
Solving the equation
(5.17) (I + Td)g = f, f,g 6 3
we find
N , o >-*<y>(5.13) g (y) = f(y) - p S , Ap( ^
where the A satisfy
" U
(5.19a) Ap )
" Up+3N
(5.19b) s S ] „ r 8 ( S ) = >{ r )
"s+2H l'r+2U
r™( 5 . 1 9 c ) 3 q = j < f ( y ) , e q ( y ) ^ d y j q = 1 , 2 , . . . . 4 N
with A = (n ) the matrix given by (5.11).
Since the operator I+T, is one to one on B, tlie matrix A = (.\ ) isv d rs
invertible. More directly, one can verify the invertibility of A by
writing it as a positive definite matrix plus an antisymmetric one.
Denoting the inverse by A = (,' ), we obtain from (5.19)r s
( 5 . 20) A =P
2N('
P
p , s
I'.,s
s
s+2N
sP+N
W,s"sbs+2N
= i - i v r J , A , P > J , P+"'J ;•
j + 2N ' j + 3M ' ' p , j + N p + N . j + N
l'ogether, (5.16), (5.13) and (5.20) imply tliat the inverse operator
S = (I+T ) is given in explicit form by (5.10).
Iö2
We shall now prove that the matrix elements R are bounded asrs
functions of x E E. In fact, we shall estimate the x dependent matrix
that occurs in the right hand side of (5.10b), in the following explicit
way
(5.21)3p+N,j+Il
I-I= M
PJ
A
lror this purpose we develop some further notation.
By A we denote the Gram matrix of the vectors e, ,e_,...,e. , introduced
in (5.16a), i.e. A = (3 r g), 5r <er,eg>.
Since the vectors e.,e„,...,e„ are linearly independent, it follows that
det A > 0 (see [3]).
Evidently
„t
(5.22) A =^0 V
(A)
2NLet us write (A) = (3 ) and introduce the vectors h = £, 3 e .
rs r s=1 rs sNote that <h ,e > = 5 and <h ,h > = 8
r' s rs r' s rsIn combination with (5.10-20) this gives
2N
(5.23) (I - S)hs = rJi 6rser.
Using the identity I - S = T,S, we get
(5 .24) S r s = <T d Sh s , h r >.
Hence, in view of (5 .15)
(5.25) | B r s | ' S llhrll|llhsll^ = B r r 8 s s , r , s = 1,2 211.From (5 .22) i t i s c l e a r t h a t for Z = 1 , 2 , . . . , N
(5.2Ó) ~ - - - 1
Jlt - ) N .PJ P»J=1
4(lmN
p=1
163
The desired estimate (5.21) is, of course, an immediate consequence of
( 5 . 2 5 - 2 6 ) .
By (5.1Ob-21) we have
N( 5 . 2 7 ) IIA IIE S IIfII •5 1 (> 2 Im {,.) H . ,
y i e l d i n g the bound ( 5 . 1 2 - 1 3 ) for IISll.
Corol lary to lemma 5 . 1 . For each x € E the eqwttion
(5.23) (I + Td)3 = -u>d
admits a unique solution f> £ 8 and 'j>j haos
(5.29) ,d(y;x) - - J = , (^ ^p , j P+-V
-Vb
, a ,d d
Remark. Let us recall that 3. produces the reflectionless part of the
potential q through the formula
(5.30) qd(x) = bd(0+;x) = ?=1 i3 . + r
Clearly, by (5.21) we have the a priori bound
N(5.31) sup |q,(x) I < ? , M . ,
x G ^ lqd P,J=1 P J'
which does not involve the C. but depends only on the ",. in a simple
explicit way.
j. Proof of theorem 4.1.
The nature of the results obtained in the previous section enables us
to provide a proof of theorem 4.1 which is remarkably similar in form to
that given in Chapter 4 for the corresponding scalar case.
164
Let x £ E be arbitrarily fixed.
To start with, let us write the solution 6 of (5.9) in the form
(6.1) ö = Sd + Bc, with
(6.2) Sd = -Sud.
By (3.4-5) and (5.29-30) we plainly have
b ac c
w i t h a = a - a , and b = b — b . , such t h a tc d c d
(6.4a) q(x) - qd<x) = bc(0+;x)
(6.4b) [ (|q(s)|* - |qd(s)|2)ds =-ac(0
+;x).
From section 5 it is clear that both fi and S, belong to B H X. Hence, we
already know that •/, € 8 0 5f. It remains to find a concrete estimate of
3 in the norm (5.4). For that purpose we insert the decomposition (6.1)
into (5.9), thereby obtaining
(6.5) (I + T + T )3 = -T Ë - ,-. .c d c c d c
Consider (6.5) as an equation in the Hubert space X. Since T + T, isc d
antisynimetric, the operator I + T + T is invertible on X. Furthermore,
the relation (5.14) holds with T d replaced by T + Tj. Thus (6.5) has a
unique solution r. € 3f satisfying
(6.6) «.•: II, • IIT .-; II + il. II, .c 2 c d 2 c 2
Using the generalized Minkowski inequality (see [6], p. 148) we obtain
(6.7) , V d l , • ƒ ( J ' i ! , d ( Z ; x ) . c ( x + y + z ) i U d y ) ^ƒ (Jo
Hence
(6.3) IT ,•.„!!, •
where by (5.21-29)
r N -1
( 6 . 9 ) «ÉSd«i = j l lR d (z ;x) l lEdz < p ? = 1 (/2 Im C p ) ' M ? J .
We conc lude t h a t
( 6 . 1 0 ) II3CU2 'i ilu. llj(1 + Il6dlli) < a o l lu c « 2
w i t h a g i v e n by ( 5 . 1 3 a ) .
The trick is now to rewrite equation (6.5) as(6.11) (I + T,)3 = -T 6 - T 0 - u
d c c c c d c
and to realize that the a priori estimate (6.10) paves the way to estimate
the right hand side of (6.11) in the norm (5.A). In fact, since
(6.12) II (T 3 )(y;x)ll„É [ «3 (z;x)IIJu> (x+y+z)ll dz,
we have by Schwarz' inequality
(6.13) IIT 3 II = sup ( f II3 (z;x)IIJ,dz) { [ ILo (x+y+z) l|2,dz )c c 0<y<+~ V C / \QJ C '
s He « J U I L < iiu i i | ( i + O R , i i , ) .c 2 c 2 c 2 d
Moreover, invoking again the generalized Minkowski inequality, one gets
(6.14) llT 0,ll £ «io 1111(4,Hi.c d c d
T o g e t h e r , ( 6 . 1 3 ) a n d ( 6 . 1 4 ) p r o v i d e t h e e s t i m a t e
( 6 . 1 5 ) II-T 8 - I B , - u I S (II io II + II us II | ) < 1 + l l 3 . H i ) .c c c d c c c 2 d
Applying lemma 5.1 we obtain from (6.11-15) the following estimate for ;i
in the norm (5.4)
(6.16) «3 II i a*(IU II + llu I In .c 0 c c.
Insert ion of (5.6c) and (6.3) then leads to
(6.17) sup ( | a ( y ; x ) | ' + |b (y ;x ) | * ) ï a ' f / I [ | ;• (x+y) | *dy +0<y<+=o ^ C / U \ 0J C
sup |ft (x+y) | ).0<y<+oo C '
166
Combining (6.17) with (6.4a) we arrive at the desired estimate (4.1-2),
wherewith the proof of theorem 4.1 is completed.
Remark. Actually, we have proven more, since by (6.4b) and (6.17) the
estimate (4.12) still holds if one replaces the left hand side of (4.1)
by
(6.18) max[|q(x) - qd(x)|,
7. An application: cmKdV asymptoties.
ds j
The importance of theorem 4.1 stems from the fact that it is a useful
tool to investigate the asymptotic behaviour of solutions of certain non-
linear evolution equations solvable by the Zakharov-Shabat inverse
scattering method (see the discussion in Chapter 4, section 4).
As an illustration let us consider the complex modified Korteweg-de
Vries (cmKdV) problem
(7.1a) qt + 6|q|2qx + q ^ = 0, -» < x < +», t > 0
(7.1b) q(x,0) = qQ(x),
where the initial function qQ(x) is an arbitrary complex-valued function
on E, such that
(7.2a) 1r/x^ satisfies the hypotheses (2.2-11) and is therefore a bona
fide potential in the Zakharov-Shabat scattering problem (2.1).
(7.2b) 1o^x^ ^s su^^iciently smooth and (along with a number of its
derivatives) decays sufficiently rapidly for |x| -+ =>:
( i) for the whole of the Zakharov-Shabat inverse scattering
method [2] to work,
(ii) to guarantee certain regularity and decay properties of the
ri-rht reflection coefficient to be stated further on.
167
Uniqueness of solutions of (7.1) can be established within the class
of functions which, together with a sufficient number of derivatives vanish
for |x| -> « (of. [7]).
Suitably adapting the procedure outlined in [10] for the real raKdV
problem one can prove by an inverse scattering analysis that condition
(7.2) guarantees the existence of a complex-valued function q(x,t),
continuous on Kx[0,«>), such that
(7.3a) For any value of the time t :? 0 the function q(x,t) satisfies
the hypot-heses (2.2-11).
(7.3b) q(x,t) satisfies (7.1) in the classical sense.
(7.3c) q(x,t) falls in the class of functions for which uniqueness of
solutions of (7.1) can be proven.
Whenever, in the sequel, we speak of "the solution" of (7.1) we refer to
the solution obtained by inverse scattering.
Uniqueness implies that if q~(x) is a real-valued function then so is
q(x,t) for all t • 0. Thus, in that case, the cmKdV problem (7.1) reduces
to the mKdV problem (1.1) in [9], i.e. Chapter 5 of this thesis.
Let us point out that by the inverse scattering method the solution
q(x,t) of (7.1) is obtained as follows.
Having computed the (right) scattering data {b (t),r_.,C.t associated with
q.,(x), one puts (see [4], p. 307)
(7.4a) cT(c) = cjexp{8ir,?t}, j=1,2,...,H
(7.4b) br(r,,t) = b (r.)expl 3 L ;3 1 } , — • -. •. +....
Then by the solvability of the inverse scattering problem, there exists '"or
each t -• 0 a smooth potential q(x,t) satisfying the hypotheses (2.2-11)
and having \b (r., t ) , r,. ,C. (t) ) as its scattering data. The function q(x,t)
is the unique solution of the cmKdV initial value problem (7.1).
Plainly, the reflectionless part q (x,t) of q(x,t) can be obtained in
explicit form by substituting (7.4a) into (5.30).
!f>3
r i <hIn the N = 1 case, setting t,. = f, + in, C = |ie ', with ii, •,> • O,
?, J) c K, this yields
(7.5) q d(x,t) = 2'ie"1*sech
with
(7.óa) •;• = 2.-,x + 8 .•-,(•-.* - 3n2)t + .> + j
(7.6b) i' = 2nx + 8n(3f;2 - M2)t + v
(7.6c) v ^
Thus the one-soliton solution is a single wave packet modulated by an
envelope having the shape of a hyperbolic secant. The envelope and phase
velocities are found from (7.6) to be v = UO 2 - 3''2) ande
v , = <4(3n2 - t,2 ), respectively. According to the sit>n of v the envelope
may propagate to the right, to Ihe left or be at rest.
If Ï1 ' 1, then, generically, for large time q (x,t) will decompose into
N distinct solitons of the structure (7.5). However, it is easy to
construct examples (e.g. N = 2, c, = •. + i-i, v2 • 3'2, c,n = K M 2 - 3r2)*)
in which no such decomposition takes place, but a more complicated
structure is found instead.
ïSecause of the negative group velocity associated witli the linearized
version of (7.1) we expect that for large t, when considered on the
positive x—axis, the solution q(x,t) of (7.1) is approximated with
reasonable accuracy by its reflectionless part q.(x,t).
We shall use theorem A.1 to verify this.
Note first that, if qQ(x) satisfies (7.2a-b(i)), then by (7.3a) the
function q(x,t) satisfies the conditions of theorem 4.1.
Consequently, for each x € R and t • 0 one has
(7.7) !q(x,t) -q d(x,t)| ' ajlti j | (x+y ;t) |2 dy +
+ sup ]:t (x+y;t)|),0- y<+« '
wi th
169
Jt)-l | \ r(7.3) fic(8Jt)-l | \ r ( ü e 2 i C S + 8 U I t d ; , s £ K
and et, the constant given by (4.2).
Next, suppose that q«(x) satisfies (7.2) in such a way that
(7.9) b is of class C2 (E) and the derivatives b '((;), j = 0,1,2
are bounded on E.
Then, reasoning as in Chapter 2, but taking into account that the symmetry
relation b (?) = b (~c) is no longer guaranteed, it is easily seen that
in the parameter region
(7.10) T = (3t) 2 1, x £ -u - \)T, where p and v are nonnegative
constants
one has the estimates
(7.11a) sup |fic(x+y;t)| < YT~'
0J c x+y>t y = r \1 + _ J x n y +
where y is some constant.
Combining (7.7) with the estimates (7.11) we arrive at
Theorem 7.1. Let q(x,t) be the solution of the complex modified Korte-jeg-
de Vries problem
<- q -•- 6|qj2q + q = 0 , -<» < x < +», t * 0(7.12) ' x xxx
L q(x,0) = q Q(x),
where the initial function qn(x) is an arbitvapu aomplejr-valujJ function
on E, satisfying (7.2) in such a way that (7.9) is fulfilled. Let
{b (5),c.fc5} be the scattering data associated with qQ(x). Then fj?
each x 6 E and t y 0 one has
(7.13) |q(x,t) - qd(x,t)| S ad/2 J |r>c(x+y; t) \^ dy +
(x+y;t)|),
0J
1 _ » » i \
+ sup c
170
with q, (x,t) the reflectionless part (5.30), (7.4a) of q(x,t), a~ the
constant given by (4.2) and Q the function introduced in (7.8).
Next, let v and \> be arbitrary nonnegative constants. Put a = u + vT,1 /3
T = (3t) . Then the following estimate holds
(7.14a) sup |q(x,t) - qd(x,t)| i A, for t > 0xS-a
(7.14b) sup |q(x,t) - qd(x,t)| S yT~', for t ^
with
a2 f»(7.15a) ^
(7.15b) Y = afrh + fiy + v^"|br(0)|^|+ j
where y denotes the constant appearing in (7.11).
Clearly, theorem 7.1 generalizes theorem 5.1 in Chapter 5 obtained for
the real case. Generalizations of other results can be derived in a similar
way, but they fall outside the scope of this paper and are therefore left
to the interested reader.
References
[ 1] M.J. Ablowitz, Lectures on the inverse scattering transform, Stud.Appl. Math. 53 (1978), 17-94.
t 2J M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, The inversescattering transform - Fourier analysis for nonlinear problems, Stud.Appl. Math, 53 (1974), 249-315.
[ 3] P.J. Davis, Interpolation and Approximation, Dover, New York, 1963.
[ 4] R.K. Dodd, J.C. Eilbeck, J.D. Gibbon and H.C. Morris, Solitons andNonlinear Wave Equations, Academic Press, 1982.
[ 5] W. Eckhaus and A. van Harten, The Inverse Scattering Transformationand the Theory of Solitons, Worth-Holland Mathematics Studies 50,1981 (2nd ed. 1983).
171
[ 6] G.H. Hardy, J.E. Littlewood and G. Pólya, Inequalities, 2nd ed.,Cambridge 1952.
[ 7] Y. Kametaka, Korteweg-de Vries equation IV. Simplest generalization,Proc. Japan Acad., 45 (1969), 661-665.
[ 8] P. Schuur, On the approximation of a real potential in theZakharov-Shabat system by its reflectionless part, preprint 341,Mathematical Institute Utrecht (1984).
[ 9] P. Schuur, Decomposition and estimates of solutions of the modifiedKorteweg-de Vries equation on right half lines slowly moving leftward,preprint 342, Mathematical Institute Utrecht (1984).
[10] S. Tanaka, Non-linear Schrödinger equation and modified Korteweg-deVries equation; construction of solutions in terms of scatteringdata, Publ. R.I.M.S. Kyoto Univ. 10 (1975), 329-357.
[11] V.E. Zakharov and A.B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in non-linearmedia, Soviet Phys. JETP (1972), 62-69.
172
CHAPTER WINE
INVE11SE SCATTERING FOR THE MATRIX SCI1RODIMGER EQUATION WITH NON-HERJUTIAN
POTENTIAL
We develop an inverse scattering formalism for the nxn matrix
Schrödinger equation with arbitrary, in general non-Hermitian potential
matrix, decaying sufficiently rapidly for |x| •+ •».
1. Introduction.
In 1974, Wadati and ICamijo [5] developed an inverse scattering
formalism for the matrix Schrödinger equation with Hermitian potential
matrix U(x). Assuming si.iilar results in the non-Hermitian case the
authors and later on Calogero and Degasperis [1] found several classes of
solvable nonlinear evolution equations.
To our knowledge this assumption has nowhere been justified in the
literature. Indeed, in 1930 Wadati [4] writes:
"At the moment, it is an open question as to what are the most general
conditions for which the inverse scattering problem can be solved.
I have considered three cases:
173
I. ü(x) is diagonal (and can be complex),
II. U(x) = J'J(x)J; J a constant matrix, J2 = I,
III. Ü (x) = JU(x)J; J a constant matrix, J2 = I.
The first crucial point in the discussions is the definition of the
Wronskian. Although we need extra assumptions, similar arguments seem
to be valid".
The present paper is an attempt to settle this question. In section 2 we
consider the Schrödinger scattering problem for any continuous potential
matrix U(x) decaying sufficiently rapidly for |x| -• °>. Superimposing some
rather natural regularity conditions on the right transmission coefficient
that are partly familiar from the Zakharov-Shabat scattering problem
(see [3]), we show that - apart from the unique solvability of the
Gel'fand-Levitan equation - the inverse scattering problem can be solved
completely. Though initially the approach parallels that of [5], our ways
split up in subsection 2.2. The reason is that the Wronskians employed in
[5], (2.17) to extend the scattering coefficients to the upper half plane
are of no use in the non-Hermitian case. To circumvent this difficulty
we perform the extension by means of integral representations in terms of
the potential and the Jost functions (see (2.20)).
Throughout we shall use the following notation:
If A = (a. .) is an n*n matrix with a.. £ -, then we write:J-J ij
nI A. | = max .£ la. . I .
The determinant of A will be denoted by det A. Furthermore, \»e write
I for the nxn identity matrix.
Our notation for the Jost functions, scattering coefficients, etc. closely
resembles that used in [3]. This notation is different from the one in
[5]. For convenience we specify here the relation bet-ween these different
notations:
F1 = V F2 = V C11 = R+' C12 = R-' C22 = '-'
C2) = L+, K = Br, K^x.y) = ;Ux,y-x).
174
2. The inverse scattering problem for the matrix Schrödinger equation.
We consider the differential equation
(2.1) -i'" + (k2 " U(x))Y = 0 , ' = "fj » -» < x < +»,
where U(x) is a complex n*n matrix and if is a vector function with n
components. Equation (2.1) is called the nxn matrix Schrödinger equation.
Accordingly, the complex parameter k2 and the matrix U(x) are referred to
as energy and potential, respectively.
We shall impose the following conditions on the potential:
(2.2a) U(x) is continuous on -•*> < x < +«,
(2.2b) lira U(x) = lim U(x) = 0,
(2.2c) I (1 + jx|) |U(x)|dx < +», C = 0 or 1.— CO''
We call (2.2) a growth condition of order I and write [[£]] to indicate
which growth condition is meant. For many results it suffices to
a.-3urne [[0]]. However, we need [[1]] to prove existence and regularity of
the Jost functions for k = 0 (see subsection 2.1), a result which is
essential in the derivation of the Gel'fand-Levitan equation in sub-
section 2.4.
Clearly, any system of n solutions of (2.1) can be represented as an
nxn matrix ï satisfying the equation
(2.3) T" + (k2 - U(x))'i = 0, — < x • +••-.
Conversely, the columns of any solution of (2._>> are solutions of (2,1).
Following [5] we shall not study (2.1) directly, but instead focus our
attention on (2.3). This enables us to exploit analogies 'Jith the scalar
Schrödinger equation with real potential, of which the inverse
scattering mechanism is now perfectly understood (see [2] and [3]).
2.1. Jost functions.
In this subsection we generalize some results that are familiar from
the n = 1 case with real potential as described in [2] and [3]. Since
the proofs of these generalizations are analogous, we omit them.
?or l i k H we introduce the Jost functions Ï (x,k) and ^«(x.k), two
special solutions of (2.3) satisfying plane wave boundary conditions at
infinity, i.e.
(2.4) Vr(x,k) = e"l k xR(x,k), H^Cx.k) = e l k xL(x,k),
(2.5) R" = 2ikR' + UR, lim R(x,k) = I, lim R'(x,k) = 0,
(2.6) L" = -2ikL' + UL, lira L(x,k) = I, H m L'(x,k) = 0.
x-«° x-«°
The problems (2.5-6) for R and L can be reformulated as integral
equations solvable by iteration and yielding useful information concerning
regularity, asymptotic behaviour and Fourier representation of the Jost
functions. The essential results are as follows.
Lemma 2.1. If k £ cAfO} and [[0]] oi< if k = 0 and [[1]], : hrv w !,\ti>e
( i) R it; J olasaicai o! it ion of (2.!>) »
R is tjcmr.tnuou.-i in x, bounded for x + -"• nut s,i: l.t^U'r
(2.7a) R(x,k) = I + J G(x,y,k)R(y,k)dy
r(2ik)~ 1ie 2 l k ( x" y ) - 1lU(y), k f C \{0)(2.7b) G(x,y,k) = < +
l(x-y)U(y), k = 0.
(ii) L is a nlausical solution of fl'.f'J «=»
L is continuous in x, howidcd j'o.- x • ™ .;':.:' (i,f• iii'':c.'
(2.8a) L(x,k) = I + J H(x,y,k)L(y,k)dyx^
f(2ik)"1le 2 l k ( y" x ) - i)U(y), k € C \>-0]
(2.3b) iKx.y.k) = j +
l(y-x)U(y), k = 0.
17b
Theorem 2.2. If [[0]] and Im k > 0, k * 0, tWi o«e tea:
( i) The problem (2. 7) for R has a unique solution that i.; continuous in x
and bounded for x -+ -<». ftw solution satisfies (2.i>) in classical
sense and is given by the- Neumann series
(2-9) R = nlo Gm' G0 = l' G m + 1C x ' k ) = _ ƒ C5(x,y,k)Gin(y,k)dy, m - 0.
The G 's satisfy the estimate:
|(2.10) |Gm(x,k)| É (m!)"'{uo(x)/|k|}m, uQ(x) = | |U(y)|dy.
(ii) The problem (2.8) for L has a unique solution that is continuous in x
and bounded for x -»• <». 'phis solution satisfies ('.',.G) in i-'iassical
sense and is given by
( 2 . 1 1 ) L = f 0 H , H o = I, H m + 1 ( x , k ) = [ H ( x , y , k ) H ( y , k ) d y , m > 0 ,xx
(2.12) |Hm(x,k)| i (m!)-'{vo(x)/|k|}m, vQ(x) = [ |ü(y)|dy.
x
Theorem 2.3- Let the potential U satisfy a 'jrowlh condition of order 0.
Then the matrix functions R, R', R", L, L' and L" are
( i) continuous in x and k on R*(IE \{0})
(ii) analytic in k on (C for each x £ E.
Theorem 2.4. Assume [[0]]. Then the limits prescribed in (2.5) and (2.6)
for R, R', L and L' are. unifox-.., in k on compacta t= C \{0}. For the limits
in the non-prescribed directions we find if k € f :
X-*k)"1 f(2.13a) lim R(x,k) « I - (2ik) U(y)R(y,k)dy, lim R f(x,k) = 0,
(2.13b) lim L(x,k) = I - U i k ) " 1 U(y)L(y,k)dy, lim L'(x,k) = 0 .
rte limits in (2.1,') are uniform in k on compaata'zZ .
Theorem 2 . 5 . l,et U satisfy a growth condition oe order 1. Then the problems
(2.7), (2.8) for R, L with k = 0 have a unique solution that is continuous
in x and bounded for x • -•», x -• •» respectively. R, L satisfy (V..b), (".(*>
in classical sense for all ( x , k ) £ R.*£ .
The functions R , R ' , R " , L , L ' -jnd L" arc c-.mt i.nu,<ut' in ( x , k ) ->n 'B^C+. ',';:^
Units ppcjsi'ibi'd in (',',.;>) and (2.6) for R, ! \ ' , L m,l h' are :viiforr> in k ••/
177
Theorem 2 .6 . Suppose [ [ 1 ] ] . Then the matrix L has the Fourier representation
(2.14) L(x,k) = I + e N(x,s)ds, x € R, k £ <C+,
wit/2 H(x,s) £ L"Xn 0 L"X" (0 < s < +») / o r e-rc/z x.
The kernel N i s an element of C™*n(R^) = {W e Cn*n (]Rx[0,»))| Vx € K :
lim W(x,s) = 0 } , which is diffei'entijble with vespt.ot to x as •jell is>
to s and N , N £ c"Xn(E2) .X S U T
r, 'A has the important property
(2.15a) N(x,0+) = J j U(y)dy, ..7u :ftat
(2.15b) U(x) = -2 II ( x , 0 + ) , x 6 H.
2.2. Scattering coefficients.
Let us assume [[0]]. For k e ]R\{0} the pairs '?„ (x,k) .^(x.-k) and
¥ (x,k),ï (x,-k) constitute fundamental systems of solutions of
equation (2.3).
In particular we have
(2.16a) 'i'r(x,k) = H'£(x,k)F+(k) + 'l'£(x,-k)IMk)
(2.16b) V£(x,k) = ?r(x,k)TJ_(k) + '?r(x,-k)L+(k)
where the scattering cot1 cients R+, R_, L_ and L are n«n matrices
depending on k E R\{0}.
Substituting (2.1óa) into (2.16b) and vice versa, we find
( R (k)L (k) + R (-k)L (k) = I, L (k)r. (k) + L (-k)R (k) = 1,(2.17) \
1 R_(k)L_(k) + :l+(-k)L+(k) = 0, L+(k)R+(k) + L_(-k)R_(k) = 0.
From (2.5-6) and (2.16) we obtain the asymptotic behaviour for ':;' * "
of f• ., "/\, i and ;' with k 6 R\fO} fixed:L t. r r
173
(2.18a) f (x,k) « e~ l k xi for x ->• -«
« e ikxR+(k) + e~ikxR_(k) f o r x - +<*
— 1 —i Vv(2.13b) (ik) V ( x , k ) ra -e 1KXI for x * -«
«e l k x R + (k ) - e"lkxR_(k) for x > +"
i if x(2.18c) fe(x,k) « e I for x - +«
« e lkxL+(k) + e"l'<XL_(k) for x - -«
— 1 il/v
(2.18d) ( ik ) ' ^ ( x , k ) « e I for x
p» e 'L (k) - e L (k) for x * -*
Here t h e f» s i g n d e n o t e s t h a t t h e d i f f e r e n c e b e t w e e n l e f t and r i ^ h t hand
s i d e t e n d s t o 0 .
U s i n g ( 2 . 1 3 ) , ( 2 . 5 - 6 ) and ( 2 . 7 - 3 ) we f i n d f o r k C TR\\O}:
( 2 . 1 9 a ) R_(k) = l i m ( 2 i k ) ~ 1 e 1 K X { i k r ( x , k ) - r ' ( x , k ) ;
= l i m R ( x , k ) - j j - k R ' ( x . k ) = 1 im I - - ^ U ( y ) R ( y , k ) J y
(2.1%) R + ( k ) = 7 T k I e~
( 2 . 1 S c ) L + ( k ) = I - - ^ U ( y ) L ( y , k)Jy
( 2 . 1 9 d ) L _ ( k ) = ^ i j r j e 2 l k y l i ( y ) L ( y , k ) J y .
We now o b s e r v e t h a t t l i e r i g h t - h a n d s i d e s of ( 2 . 1 9 a ) and ( 2 . 1 9 c ) a r e w e l l
d e f i n e d f o r a l l k £ C M O J . Thus we e x t e n d t h e domain o f R_ and L from
K. \ i01 t o C \ i O t by d e f i n i n g :
( 2 . 2 0 a ) K_(k) s l - T7jj7 U ( y ) R ( y , k ) d y k C C + V O l
(2 .20K1 L + ( k ) s i - ji-^ ! J ( y ) L ( y , k ) d v !< £ C + ' - . O t .
17')
It follows from theorem 2.3, that ll_ and L are analytic on C+ and
continuous on C \{0}. By (2.13) one has for k £ C+:
(2.21) lim R(x,k) = R_(k), lim L(x,k) = L+(k),
where the limits are uniform in k on compacta c: C .
rurtheriiiore, the determinants of t -i matrices in (2.20) satisfy
/!< (x,k)?„(x,k)(2.22) det R_(k) = det L+(k) = (2ik) det/
x £ 1R, k € £+\{0}.
To prove this we first note that any pair of solutions 'i',,':'o of (2.3)
satisfies
-j— det' ) = 0, i.e. tlie determinant or the 2n-2n njtcixI - VA,KJ TrjVXitw
I ) is constant for x £ K. By continuity it suffices to prove
1 2(2.22) for k £ C+, in which case:
det
v ,tr £ J = det[
¥l ÏÓ ^R'-ikR L'+i
,h R v ,1 R_ xim detl ) = detf ) = (2 ik) n de t R
++- ^-L' -!l'+2ikR ' ^0 2ikR ;
I L= lim det( ) = det( + ) = (2ik)"det 1,
V ^ 2ikL
/I L N
( + ) = (2ik)"det 1,
: 1 imxi
2.3. Bound states.
We now study the bound states of p^uation (2.1) in C , i.e. values
of k with lm k > 0 for which (2.1) possesses a nontrivial quadratically
130
integrable solution ij). They are characterized by:
Lemma 2.1. If k» £ C+ and [[0]], then the following statements are
equivalent
( i) det R_(kQ) = 0
(it) There exists a nontrivial y £ L„(E) such that
(2.23) v" + (k* - 'u(x))g' = 0.
Proof: Suppose det R_(kQ) = 0. Then by (2.22) we can find a,b £ Cn\{0}
with
Putting v(x) s * (x,ko)a = -'1'„(x,kQ)b we obtain from (2.4-5-6) that
Hi decays exponentially for x •* ±°° and therefore belongs to L„(]R) .
Furthermore it is evident that % satisfies (2.23) and Q £ 3.
Conversely, suppose (ii) holds. If det R_(kQ) * 0, then by (2.22) the
columns of "/ (x,k„) , V -.(x,k„) form 2n linearly independent solutions ofr U i- 0
(2.23). Thus there are a,b £ c", (a,b) * (0,0), such that
Using (2.4-5-6) and (2.21-22) we conclude that for x * ±« the function
,' grows exponentially in at least one direction. This contradicts the
fact that *• is in L?(lR)n. a
We nexc present a property of bound states that will play an important
role in subsection 2.4.
T h e o r e m 2 . 3 . . V : !••!•' ; . • _ • < : , " ; ; . i , : L U <; y u ; f j , i , : > ' • : . • : • ' • - . > ; ; ï : .••••: . • ' . • • • ? , • } ' 0 ,
.'•.'• k „ c C + : ) , : ; : . \ o n j . - t r . ? . ' . , • . j f ( S . I ) . A . u ' . i - i , - f < j > - •' • .•••••;.•.•'.• ••',.;• - ; : . - ••;;•.'•.•'
K~ ( k ) ,;.:.: ,1 ,u'W;•••'.? :\>,\? 71 k = k Q -jith iV.^;:.:;-.
(2.24) r>r(k
0) = Hra (k-ko)R~'(k).k-kQ
181
'Thin there in a unique matrix C (k,,) nuc-h thai for x £ K
(2.25) 4-r(x,k0)Dr(k0) = H£(x,k0)C r(k0).
froot: Since for potentials with compact support the proof is immediate,
we shal' apply a truncation procedure.
For U • • I we define u' (x) = i..(x)U(x) with *.,(x) = 1 for 'x' - 'i-1,
a (x) = N-1x| for N-1 • |xj • ;•}, u (x) = 0 for jxj • N.
Plainly, U'* satisfies (2.2) with i' = 1. From (2.4-9-11) and (2.19a-b) itN N '•) T>) -M IJ
is clear that the functions V ,K ,'i'\,V ,TC_ and R associated with V" can
be extended to analytic functions with respect to k on C M O } . Moreover,
the relation (2.16a), which we obtained for k € E•i0', remains valid
for k £ <C\IOl by analytic continuation. Thus for :•: t E and k € C\<0!
one has:
(2.2o) 4'"(x,k) = li'il(x,k)K J(k) + i'!(x,-k)R"(k) .
Concerning the behaviour as N > •" we first note that
(2.27a) j |UH(y) - U(y)idy -'' 'L'(y)'ily • I) for ". • • .
F u r t h e r r a o r e , f o r x € R and lm k 0 t h e f o l l o w i n g e s t i m a t e l i o l d s
( 2 . 2 7 b ) maxi ] R A ' ( x , k ) - R ( x , k ) [ , ! L N ( x , k ) - U x , k ) ! , R ^ ( k ) - K _ ( k ) i
-r~j exp(-^A-) j '"''(yJ-L y) ,dy with A = 'U(y)idy.
Indeed, consider the integral, equation
R""(x5k)-i;(x,k) = j (2ik)~l(e2lk(x~y)-l)(r*(y)-U(y))R(y,k)dy
T -1 -'ik(x-v)+ (2ik) (e~ ' -1)1-' (y)(R (y,k)-R(y,k
liy ( 2 . 9 - 1 J ) we have t he e s t i m a t e j R ( y , k ) | • cxpf-rrr]"}, which y i e l d s
132
(2.28) |RN(x,k) - R(x,k)| < -rij- exp^-yly^ | |UN(y) - U(y)jdy
fIterating (2.28) we find
(2.29) |Ax,k, - R(x,k) | : - e x p ^ j j |u"(y) - U(y) |dy.
Similarly we derive (2.29) with left hand side |LN(x,k) - L(x,k)|. The
proof of (2.27b) is completed by taking x -» «° in (2.29) and using (2.21).
We shall apply the fundamental relation (2.26) for suitably restricted
values of N, x and k.
First we choose ;; = (k||k - k.| '- t }, > •• 0, with 17 <= « and det R_(k) * 0
for k f ÏÏ\{k0}. By (2.27) there is an N , such that for N • N
( i) det R^(k) * 0 for k Ê 3.'
(ii) |LN(x,k) - L(x,k)| • j for x € E and k 6 ÏÏ.
Finally, by virtue of theorem 2.4 we can pick xQ, such that for x - x»
and k £ :L one has |L(x,k) - lj • |. Combination with (ii) yields that
for N "• 11,, x > x^ and k £ '.'. it holi i) that JL' (x,k) - l| •• j, so thatNF.,(x,k) is invertible.
Now let us take N • NQ and x • x . For k t .)•. we can rewrite (2.26) as
follows:
i'(xk)] i'Since t'i'„(x,k)] i',,(x,-k) is continuous on and analytic on , , we obtain
where •)•- is traversed counterclockwise.
133
Next we let N -* °°. By (2.27) the integrand on the left converges uniformly
in k on 3ft, yielding
,nf ^'(x.kH (x,k)R~'(k)dk - Urn ƒ R^(k)tRN(k)]"1dk.9 " l r
N^oo 3 " +
Applying Cauchy's residue theorem we find
3flJ f~1(x)k)>Ci.(x>k)R"
1(k)dk = ~1
Hence
C (kn) = (2Tii)"1 lim ƒ R"(k)[RNCk)]"1dk.
fJ-K»
Thus (2.25) holds for x > x„ and therefore necessarily for all x € E.
2.4. The Gel'fand-Levitan equation.
In this subsection we shall derive a linear integral equation for
N(x,s), the Fourier kernel introduced in (2.14).
Let the potential U satisfy a growth condition of order 1. We shall make
the following extra assumptions:
(2.30a) det R_(k) * 0 for k £ R M O } and for |k| • .-, lm k 0 with
e > 0 small enough.
(2.30b) lira R~'(k) exists.k-i-0lm kïO
(2.30c) All poles of R_ (k) in C+ are simple.
Since det R_(k) is continuous on C M O } and analytic on C , whereas
lim det R_(k) = 1, it follows from (2.30a) that det R_(k) has at most
initely many zeros in C • Let us denote these by k., j = 1,2,...,N.
Because of (2.22) and lemma 2.7 we can characterize k.,k7,...,k, as the
184
bound states of (2.1) in <£+\{0}.
We now introduce
A (k) = R (k), the right transmission coefficient
B (k) = R+(k)R~ (k), the right reflection coefficient.
By (2.30) the matrix A is continuous on C+\{k ,k_,—,k } and analytic
on <C+\{k. ,k„,.. .,kN> with simple poles at k. , k„, •••, k^. Let us
examine B . A priori B is defined on E\{0}. However, theorem 2.5 tells
us that lim L(x,k) = I uniformly in k on C and thus for x„ sufficientlyX-t-oo + 0
large we obtain from (2.16a)
Br(k) = 4'^1(x0,k)4'r(x0,k)Ar(k) - ï"
1 (xo,k)fc(xo>-k), k € K\(0},
where the right hand side is continuous in k on all of E. Therefore, in
a natural way we extend B to ;
and (2.19a-b) it is clear that
a natural way we extend B to a continuous function on R. From (2.9-10)
Br ( k ) =2lk J e~2lkyU(y)dy + 0(p-), k * ±».
Since U £ L" X" fi L" X n(E), Fourier theory implies that kB (k) is an
element of L " X " ( K ) with the property lim kB (k) = 0. Note that both1 *• I k I +» r
— and kB (k) are quadratically integrable on |k| ? 1. Hence by
Cauchy-Schwarz B is absolutely integrable on |k| Ï 1. Thus we have found
Br 6 L"X" n h^n n C n X n(K) and Br(k) = o(-rjU) for k - ±».
As a consequence the function B defined by
cont( 2 . 3 1 ) B
c o n t ( z > = 2 T j e l k Z Br ( k ) d k , z € R
belongs to L n X n n cj"n(ï).
we introduce the right
1,2 N, determined as in theorem 2.3, i.e.
Next we introduce the right normalization coefficients C (k.),
135
(2.32) Vr(x,kjDr(k.) = ï£(x,k.)Cr(k.),
D (k.) = lim (k - k.)A (k).r J J r' J k+k. J *•
In terms of these we form the discrete counterpart of (2.31), namely
N ik. z
(2.33) B,. (z) = -i .£ e J C (k.), z € K.
discr j=1 r j '
Finally, we define
B = B.. + B
discr cont
We shall call the aggregate of quantities IB (k),k.,C (k.)} the (right)
scattering data of the potential U.
Our main result is now given by:
T h e o r e m 2 . 9 . / ƒ t •:,• , :>ni 11: ."-•->.•;<-; ('.'.'. ".' ;jftu (' = 1 ,!>t,; ("..•<:)> a--- fu'j'i' 'r :,
ik.-n the b\».ifU:>' kcrnc * N ( x , s ) i.n t>',jdn:-'i\: '». (D.I!) .i.;r ;.•;,•'/•'.• '•'..•
/..i/ / .'Ui'.nj ffKï.'jr*;/ ,\):i-ii io'i .•".'.•• x 6 E , s : 0 :
(2.34) N(x,s) + B(2x + s) + N(x,t)B(2x + s + t)dt = 0.
0J
//.-•i1.' i.hi' oa>'!ui>l-c x .ii>:n\ii't' •!''• ,i y a i w i - ' ci'.
r'tK' m a t r i x B / / / , ; / •jnOi't'ti:' ; • • ) ; ' ; ' C n ! - \ i > ' . i ' . ' . / ; / . J f i n n ' t t .",*'•;; . ' \", , j ; , - • , ' . > " • . ' . • ' • . ' . • ;
b j t . h i : j ' o ! f o u i n j s ' r ' a ' i c i ' i n j < L I : J : ;•';,• r ' ^ j : ' . : i'< 'j\'< \*! ?'• ".' •'•1-'.•'.''•"'''"' " • ' ^ ( ' O ,
ih,,: bound urai-rn k . . ' ƒ C". /^ ' 'z C + \ ) 0 } .,--,'J .'•'/.• .•','. J-':,- i n-;. .-. ' . ' : : ; • , ' , ' • ;
- - - - - - - ^ cr(k.).
Proof: For a matrix ',s(x,.), depending parametrical ly on x, we shall use
the following notation of the Fourier transform
(F>)(x,k) = j e .:.(x,s)ds, k e TR
(F"'*)(x,s) = - j e l k s:(x,k)dk, s e n .
The proof is based on the fundamental identity (2.16a), which we rewrite
as
186
¥ (x,k)A (k) = Y„(x,k)B (k) + lf (x , -k) , x £ K, k £ E.L ie -C. r -c
H*| lf y 1 If V
S u b s t i t u t i o n of t h e fo rmulas Y = e R, 'K„ = e L ( s e e ( 2 . 4 ) ) y i e l d s
r r + ï,
where L(x,k) a L(x,-k). According to theorem 2.6 we can represent L as
t = I + 2TTF J with J(x,s) = 0 for s < 0 and J(x,s) = N(x,s) for s ? 0.
Inserting this we obtain:
(2.35) RAr - I = e 2 l k X B r + 27<e2xkx(F~1 J)B r + FJ.
Mote that each term in (2.35) belongs to L_ (-«> •• k • +«>). Applying
F to both sides one gets
(2.36) F"'(RAr - I) = F"'(e2lkx13 r) + 2HF"1 (e2 l k x(F"'J)B r> + J .
We shall show that for s ; ' 0
(2.37a) {F"1(e2 l k x i i r)}(x,s) = \Qnt.(2x + s)
( 2 . 3 7 b ) {2TIF ' ( e ^ K X ( F ' j ) B ) } ( x , s ) = N ( x , t ) B ( 2 x + s + t ) d tr QJ cont
(2.37c) fF~1(RAr-I)}(x,s) = -Bdiscr(2x+s) - j N(x,t)Brf.gcr(2x+s+t)dt.
Clearly the integral equation (2. 34) is an immediate consequence ot (2.36-37).
ad (2.37a) Trivial, since the integral in (2.31) converges absolutely.
ad (2.37b) Because the integrals in (2.14-31) are absolutely convergent
we may interchange the order of integration.
ad (2.37c) Observe, that RA - I is continuous on C +\ik ),k 2 >—,k N} and
analytic on C+\(k .t^, • • • ,^) with simple poles at k ,k?,... ,k^,
whereas RA - I = (KTT T ) f° r |k| "* "» lm k 0. Thus, using
standard limit procedures, we find by Cauchy's residue theorem:
N{F~'(RA - I ) J (x , s ) = -j- 2ni £ Urn (k-k. )e l k s(R(x,k)A (k) - I)
r ^ N J"1 k^kj J
= i . ^ e l k J S R(x,k j )D r (k . )
N ik.(2x+s)= i ^ e J L(x,k j)C r(k j)
187
= -B,. (2x + s) - N(x,t)B,. (2x + s + t)dt. •discr 0J ' discr
We shall refer to (2.34) as the Gel'fand-Levitan equation. Though it
remains to be proven, we expect that under mild conditions equation (2.34)
is uniquely solvable for all x € TR. In that case one can reconstruct the
potential from the scattering data by the following procedure:
Given the scattering data {B (k),k.,C (k.)} one calculates the function
B(z) = -i e J ^(kj) + 27 { eikzBr(k)dk, z € E,
and then solves the Gel'fand-Levitan equation (2.34). The potential U is
found from N by the formula U(x) = -2N (x,0 ) as we saw in (2.15b). In
the literature the above procedure is known as inverse scattering.
Finally, let us remark that the notation used in this paper has been
chosen so as to exploit analogies with the reasoning performed in [3].
However, the form (2.34) does not appear in the literature, except for
the reference [3] itself. What does occur in the literature is a more
symmetric version of (2.34), which is readily obtained via the
transcription.
(2 .
(2.
38)
39)
0(y
ÜI(£
;x)
) =
= 2N(x
2B(25)
,2y)
For future reference let us reformulate the aforementioned inverse
scattering procedure in this symmetrized version:
Given the scattering data {B (k),k.,C (k.)}, calculate the function
N 2ik.£ (•«>(2.40) u(c) = -2i X e J Cr(k.) + \ e Z l kS (k)dk, U I.
Next, solve the Gel'fand-Levitan equation
r(2.41) 6(y;x) + u(x+y) + 6(z;x)u(x+y+z)dz = 0
with y > 0, x € E.
The potential U is then found from S by the formula
188
(2.42) U(x) = -Px(O+;x).
References
[1] F. Calogero and A. Degasperis, Nonlinear evolution equations solvableby the inverse spectral transform associated with the multichannelSchroedinger problem, and properties of their solutions, Lett.Nuovo Cimento 15 (1976), 65-69.
[2] P. Deift and E. Trubowitz, Inverse scattering on the line, Comm. PureAppl. xiath. .32 (1979), 121-251.
[3] W. Eckhaus and A. van Harten, The Inverse Scattering Transformationand the Theory of Solitons, North-Holland Mathematics Studies 50,1981 (2nd ed. 1933).
[4] M. Wadati, Generalized matrix form of the inverse scattering method,Topics in Current Physics 17: Solitons (R.K. Bullough and P.J. Caudrey,ed.), Springer, 1930.
[5] M. Wadati and T. Kamijo, On the extension of inverse scattering method,Prog. Theor. Phys. 52 (1974), 397-414.
189
CHAPTER TEN
UNIFICATION OF THE UNDERLYING INVERSE SCATTERING PROBLEMS
1. Introduction.
In this chapter we show that the inverse scattering problems occurring
in the present thesis can all be viewed as special cases of the inverse
scattering problem for the matrix Schrödinger equation treated in
Chapter 9.
For the first three chapters there is nothing to prove since the scalar
Schrödinger problem occurred. In the Chapters 4 to 3 the Zakharov-Shabat
problem was the underlying scattering problem. Hence, it suffices to
concentrate on that.
Let us consider a slightly generalized version of the Zakharov-Shabat
system, as was studied in [i], [2], namely
,-ik
<•••> C M : i nwhere p(x) and q(x) are complex functions and k is a complex parameter.
It is understood that the potentials p and q have the regularity and decay
properties
(1.2a) p,q £ c'(lR)
(1.2b) lira p(x) = lira q(x) = lim p'(x) = lim q'(x) = 0
i X I -x» I X I -*» I X I -••» ' X ] -x»
190
(1.2c) I (|p(x)| + |q(x)|)dx < +0C
(1.2d) j (1 + |x|)(|p(x)q(x)| + [p'CxJ i + |q'(x)|)dx < +».— 00*"
In addition we shall require some conditions on the zeros of the Wronskian
of the right and left Jost solutions, to be specified presently in (4.1).
Clearly, the case p = -q was treated in Chapter 3 under the same
conditions except for a slight relaxation of (1.2d).
The basis of the analysis presented here is the remarkable fact -
already pointed out in [3] - that the system (1.1) is connected with a
special 2x2 matrix Schro'dinger equation. Namely, let , = (i^.^^) satisfy
(1.1). Differentiating with respect to x we obtain
(1.3a) v" + (k2 - U(x))* = 0 with
/PI q'(1.3b) U(x) = (
V pq
By (1.2) the potential U(x) satisfies Ch. 9, (2.2) with ? = 1, so that the
inverse scattering problem for (1.3) was already considered in Chapter 9.
We shall apply the results obtained there to derive an inverse scattering
formalism for (1.1).
This derivation forms an alternative for the one «iven in [2], Ch. 5. In
the sequel all entities introduced in Chapter 9 for general U(x) will -
without any change in notation — be associated with the special
potential U(x) in (1.3) exclusively.
2. Jost functions.
Following [2], we first introduce the Jost functions , (x,k),
j^(x,k) for Im k S 0 and J'r(x,k), ^(x.k) for Im k •: 0 as the special
solutions of (1.1) uniquely determined [2] by
(2.1a) •; (x,k) = e~lkxr(x,k), lim r(x,k) = (hr \j
x->— "•
(2.1b) Cr(x,k) =el k x?(x,k), Jim f(x,k) = (°)
191
(2.1c) ifv(x,k) = elkx£(x,k), H m £(x,k) = (°)
(2.ld) i|^(x,k) = e X£(x,k), H m ü(x,k) = (g).
x-*+°°
liere ^ , r, etc. denote vector functions with two components.
Though easily verified, the following fact is crucial for our analysis:
The Jost functions 1' (x,k) and ll'„(x,k), associated with the 2*2 matrix
Schrödinger problem (1.3) and first introduced in Chapter 9 in a more
general context, as well as the corresponding 2*2 matrix functions R(x,k)
and L(x,k), can all be expressed in terms of the vector functions introduced
in (2.1). Specifically one has for lm k ï 0
(2.2a) H'r(x,k) = (i))r(x,k)Jr(x,-k)).
(2.2b) fc(x,k) = (^(x,-k)i^£(x,k)),
(2.2c) R(x,k) = (r(x,k)r(x,-k)).
(2.2d) L(x,k) = (?(x,-k)£(x,k)),
i.e. ty is the first column of the 2*2 matrix f , etc.
Combining the representations (2.2) with the results derived i'i Chapter 9
we immediately learn new facts about the Zakharov-Shabat Jost functions
that somehow were overlooked in the literature. For instance, Ch. 9,
lemma 2.1 gives us a reformulation of the problems (1.1), (2.1) as
integral equations different from the wellknown form [2] (5.2.1). Further-
more, Ch. 9, theorem 2.2 tells us that r(x,k), r(x,k), C(x,k), f(x,k) can
be represented in the form of a Neumann series in which the n term is
of order |k| as |k] •+ °°. To illustrate why this is pleasant let us
quote from [2], p. 151, where the asymptotic behaviour for |k| -»• « of the
Zakharov-Shabat Jost functions is discussed:
"In the case of the Schrödinger equation the n-th term in the Neumann
series of the solution is of the order |k| , whereas this is no longer
automatically the case here. Though this fact was very pleasant before
(no singularity at k = 0), it works now in the wrong direction: we have
to work harder to get the asymptotics for |k| -*•"•".
This quotation is followed by a formal expansion procedure to find the
192
first few terms in the asymptotic expansion of r(x,k) for |k| ->• <».
Including the correctness-proof the derivation takes two pages whereas
the asymptotics for r, t, t is only mentioned but not proven. It is
therefore surprising that Ch. 9, theorem 2.2 gives us the full asymptotic
expansion of r, r, t, t for |k| -»• •» without any pain. Next, let us
mention some consequences of Chapter 9 and (2.2) that are also occurring
in [2], Ch. 5, albeit after a different reasoning.
By Ch. 9, theorem 2.5, the vector functions r and t are continuous
in (x,k) on RxC and analytic in k on C for each x £ R. Similarly, f
ind I are continuous in (x,k) on K*C_ and analytic in k on C_ for each
x £ K. Another consequence of Ch. 9, theorem 2.5, is that the limits
prescribed in (2.1a-c) and (2.1b-d) for r,£ and r,£ are uniform in k on
C and <C respectively.
Finally, it follows from Ch. 9, theorem 2.6 that the vector functions
t and I have the Fourier representation
(2.3a) £(x,k) = C ) + elkSn(x,s)ds, x C E, lm k -• 00J
(2.3b) £(x,k) = (') + e n(x,s)ds, x € K, Ink •: a,
0j
where the kernels n(x,s) and ?.(x,s) belong to L2 fi \A (0 ••' s ' +»') for
each x. Furthermore they are elements of
C*0(1:p = {w E C' (Kx [0 ,«» ) ) |Vx t K : l i m w(x ,s ) = ( p ) } .
By Ch. 9, (2.15a) and (1.3b) their values at s = 0 are related to the
potentials p and q in the following way
(2.4a) -2n(x,0 ) = ( )-x
fC° p(y)q(y)dy/
,- I™ p(y)q(y)dyv(2.4b) -2n(x,0 ) = (
V p(x) '
3. Scattering coefficients and bound states.
As a next step let us introduce a convenient set of scattering
coefficients for the problem (1.1) and see how these are related to the
Itt
scattering coefficients R+, R_, L_ and L associated with the potential
J(x) in (1.3b).
Mote first that for k G E either of the pairs i/; , ty and ij;., q>„ forms
a basis of solutions of the system (1.1). Thus we have
(3.1a) Yr = r+i|j£ + r j ^ (3.1b) ^ = tji^ + £+^r
where the scattering coefficients r , r etc. depend on k 6 E. They
satisfy a number of relations, of which we mention
(3.2a) r_f+ - f_r+ = 1
r r -1 I I r_ = I , f = l_(3.2b) ( ~\ = ( ~ \ i.e.
The relation (3.2a) is easily derived from the fact that the Hronskian
of ty and ip , i.e. W(ij; fty ) = (ip ) ( ) o — ((|J )O('^ ),» is independent o
x. Further (3.2b) holds by compatibility.
Using (2.2a-b) we can rewrite (3.1) as:
/ 0 r (-k)N ,r (k) 0(3.3a) ¥ (x,k) = ï^(x,k)( ~ ) + V„(x,-k)(
r •" V +(k) 0 ' \ 0 f+(-k)
/ 0 £ (k)x 7l (-k) 0 N
(3.3b) n(x,k) = H' (x,k)( + f (x,-k)(V (-k) 0 J r ^ 0 C (kr
for x,k £ E. From (3.2-3) we conclude that the matrices R+, R_, L_, L+,
defined in Ch. 9, (2.16) for k 6 K\{0), can be extended to k £ 1R and
are given by
(3.A) K (k) = 1 R_(k) = ( "V +(k) 0 ' X 0 i+(-k)'
, 0 -?_(k)N /^(-'O 0 vL (k) = ( ) L (k) = ( ) .
^-r.(-k) 0 ' ^ 0 r (kV
194
The domain of R and L is extended to the upper half plane by Ch. 9,
(2.20). However, owing to special features of the system (1.1) there is
another way to perform the extension. Hereto we note that the Wronskian
W(I)J ,i))/>) = W(r,£) is well-defined for k € C , independent of x and for
k € E. equal to r_. Similarly W<$r ,ijy„) = W(f,•£) is well-defined for
k £ C , independent of x and for k £ TR equal to -r . Hence, we can
extend the domain of R_ and L from R to C. by the following definitions
(3.5) r_ = r ^ 2 - r ^ on C+
On C \{0} the extension given in (3.4-5) coincides with the one in Ch. 9,
(2.20). Indeed, if p and q have compact support this follows from the
principle of analytic continuation. For general p and q it is
demonstrated by applying a truncation procedure.
In the following, when writing R_, t, , we shall mean the extension by
(3.4-5). From Ch. 9, theorem 2.5 it is clear that R_ and L are analytic
on C and continuous on C .
We are now able to determine the limits of r, f, I, C. in the
directions that are not prescribed in (2.1). From Ch. 9, (2.21) and (2.2),
(3.4) we obtain
/ r - ( ! ° \(3.6) lim r(x,k) = I ) uniformly in k on compacta c C
x+co \ 0 J +
/ 0 >lim £(x,k) = i E uniformly in k on compacta cr C+
x->~c» ^ r (!c)'
/ "0 \lini r(x,k) = ( I uniformly in k on compacta c: Cx-«> s f + ( k ) '
/f+(k)\lim C(x,k) = ! ! uniformlv in k on compacta cr C .x-*-» \ 0 '
Evidently we can reformulate (3.5) as follows
(3.7) r_ = det( + ].ij;j,) on ~C+, f+ = detC^J^) on C_.
As an immediate consequence we have
195
Lernna 3 . 1 . If Ira k > O and r_(kg) = 0 then there ia an u (k Q ) £ C\{0}
such that
(3.3a) ijjr(x,ko) = a (k 0 H £ (x ,k 0 ) .
If lm ie, < O and f+(kQ) = O then thar-e is an 5(k0) £ CMO} <JUCT7 r.foi
(3.3b) ^ ( X . Ü Q ) = a(ko)+^(x,ko) .
Furthermore, using (2.1), (3.6-7) and reasoning as in the proof of Ch. 9,
lemma 2.7 we arrive at
Lemma 3.2. If k1 6 C\1R, than the. foilouinj statements ajv equivalent
( i) k0 c (k Ê C+|r_(k) = 0} U {k'e C_|r+(k) = 0}
(ii) k^ is a bound at.ite of (1.1), i.e. t-he.ps wrists a \.»i :!•;''.'•' r'
<p 6 L 2 ( R )2 such thai
4. Inverse scattering.
We shall now show how the results from Ch. 9, subsection 2.4 lead to an
inverse scattering theory for the generalized Zakharov-Shabat system
(1.1). In addition to (1.2) we make the following assumptions:
(4.1a) r_(k) * 0 and f+(k) * 0 for k £ E
(4.1b) All zeros of r_ in C+ and f in C_ are simple.
Plainly, (4.1) guarantees that R_(k) as defined in (3.4) fulfills the
requirements made in Ch. 9, (2.30). Consequently, r_ am) f have at riost
finitely aiany zeros. We shall write k.,k?,...,k, for the zeros of r_ in C+
and k.,k. , ...,k-j for thi zeros of f in C_. Because of (3.7) and lemma 3.2
•jt' can characterize k. ,k_ ,... ,k .,k. ,k9 ,. .. ,k", as the bound states of (1.1)
1'Jf)
in C.
Let us calculate the matrix functions introduced in Ch. '), subsection
2.4. tor the right reflection coefficient we obtain from (3.4)
0 b (-k) r (k) f (k)Br(k) = I ), b (k) 3 — m , hr(k) - -r^r , k H R.
\(k)
Hence, by Ch. ), (2.31)
, 0 b (zKli (.) = { < O n L ), z C K, with
^b (Z) 0 /cont
b (z) * 4~ I elkzb (k)dk, b (z) s -L o~lkZb (k)dk.cont 2'; _ J r com 2 } r
We next compute U,. . Renumbering the hound states !;. , k. if ncrtssarv,discr J J
we may represent the zeros of det R_(k) in (.' as k .,!<.,,. ...k,,, ,k ; ,
*••'kd'kd+l'""'kN' w h e r e k:j-d + i £ ~ ki' ! = l,2,...,d. Thus one has
Dy le inraa 3 . 1 t h e r e a r e . ( k . ) , t ( k . ) £ CWO) s u c h t h a t
( 4 . 2 a ) , r ( x , k . ) = . ( k . ) , f ( x , k . ) , j = 1 , 2
( 4 . 2 b ) , ( x , k . ) = ~ i ( k . ) , . , ( x , k . ) , j = 1 , 2 , . . . , d .
u s i n g C h . i , ( 2 . 3 2 ) a n d ( 2 . 2 ) , ( 3 . 4 ) , ( 4 . 2 ) w e f i n d
for j = l,2,...,N-d
for j = N-d+1,...,d
for j = d+1,...,N
and so by Ch. 9, (2.33)
13.- (z) =discr \b.. (z)discr
b,. (z)discr
discr
ïk.za J
discr
z £ IR, with
- i k .zJ
(k.)
cj ~
r S(k.)
cj ~ f'(E.)J + j
Setting b s D + b.. , b s b + E.. , we finally have° cont discr cont discr
InserLion of these results into Ch. 9, theorem 2.9 gives us:
T h e o r e m 4 . 1 . i f ' ' V ' a o n d L r l o n . - ï { 1 . 2 ) a n d ( 4 . 1 ! < ; ? v ' V : . ' V . ' l e I , .'•'.••";
:•'y.u'Lr'i1 Sijt'ncl N ( x , s ) = ( n ( x , s ) n ( x , s ) ) dcfl'iel in !:\.-'-> .; n /<•;'?• \; •
•'•): '.oiöinj {.ntcjpj; ciua'ion foi» x 6 E , s • 0 :
(A.3) W(x,s) + B(2x + s ) + :- i (x,t)B(2x + s + t )d t = 0.
198
the scattering data associated with the potentials p and q, namely: the
rijhv reflection coefficients b , b , the bound states k., k. of (1.1) in
(D and the right normalization coefficients c , c..
Theorem 4.1 paves the way to an inverse scattering theory:~ ~ r ~ r
Given the scattering data {b ,b ,k.,k.,c.,c. }, one computes B and solves
(4.3). The potentials p and q are then found from !J by (2.4). It should be
noted, however, that the problem of finding sufficient conditions for
unique solvability of (4.3) covering a wide class of potentials is still
open.
It follows from the last remark of Chapter 9 that we can cast the
above inverse scattering procedure in a more symmetric form, namely:
Given the scattering data {b ,b ,k.,k.,c.,c. }, compute the function
(4
(4
(4
.4a) u)
.4b) .2
.4c) :~
Next, solve
(O
(O
the
_ /
= -2
= 2i
Gel
0 ,«,
(O 0 '
d 2ik.
d -2ik.
'fand-Levitan
• ! f™ _11 -J
equation
ik£:br(k)dk
2 l K b (k)dk.
(4.5) ri(y;x) + '..(x+y) + fi(z;x)^(x+y+z)dz = 0
with y • 0, x G E.
The potentials p and q are then found from fi = (b b) by
(4.oa) ;>(x) = -b,(0 ;x)
(4.ób) q(x) = -bj(0 ;x).
References
[I] .-I.J. Ablowitz, Lectures on the inverse scattering transform, Stud.Appl. Math. 53 (1978), 17-94.
[2] W. Kckhaus and A. van Harten, The Inverse Scattering Transformationand the Theory of Solitons, North-'.loLland Mathematics Studies 50,1981 .
[3] M. Hadati and T. Kamijo, On the extension of inverse scatteringmethod, Prog. Theor. Phys. 52 (1074), 397-414.
.'00
CONCLUDING REMARKS
In what follows we have collected a number of additional results,
each with some interest of its own. We present these results without proof.
(i) Let u(x,t) be a real solution of the KdV obtained via 1ST
as described in Chapter 2. Suppose there are x.,x E TR, x. * x~
and t, > t. > 0, such that for all x £ K
uCx^+x,^) = u(xi-x,ti), i = 1,2.
Then one has:
(a) If x. > x2, then u = 0.
(b) If x < x„, then either u = 0 or u(x,t) = -2K2sech2[x(x-x+-4K2t)]
with
K = èV—— and x = — — — .H 2 ll 2 C1
In particular this shows that, apart from the 1-soliton solution,
any nontrivial solution of the KdV obtained via 1ST can have
spatial symmetry for at most one value of the time t.
(ii) By combining the remark made in Chapter 2 after the proof of
lemma 5.1 with the other results from that chapter one easily
arrives at the following theorem:
•••...•midci' JL t-;>ai>am,-.ici' family u(x,t), t s t Q, tQ £ K ,;ƒ JV U ?
;\ t^atiala in the S^hroding^r r.^.ii.teving ;i>jb!cr;, r,at-iüf:i in.j
j'of fi.:-ud t > t the. oMdiiicnc. .::-v;.ed in Ch. 2, hhj.-.jt:i-rn 2.1.
u ( x , t ) . vh'iz-; u d(x,t) f-y th,. >'e f.-;-'.. i •»'..•:.• .-.;,-• •ƒ u ( x , t ) , .'.. .
••':,• : •. lari.;' :^i!': J<J,I: Uyi'-ng .iar.i { 0 , K . ( t ) , c ^ ( t ) } . /... S2_ .•••
•ji.se>! /•;, •••'•. 2 , ( 2 . 2 6 c ) ,:>id .'.. ; V :•• /.• •: :. 4 5 .
A . : : , - , -I, ,•• ••>.'• - . : . • • . : • : • . • ^ . c , , . . . , c N , . • , " . • ; ; . . • •
201
Assume furthermore, that the function y •*• Si (x+y;t) is strong-
l.y diffcpcntiablc in V with respect to x at every point ( x , t ) ,
x 6 E , . t i t... Let there exist a function a:[t_,<») -*-K, uuoh that
in the parameter' region t £ t~., x § a(t),the functions Q and Si',
with ft' the utronij x-dcrivatiw; of SI , satisfy:
(a) max | |s?c(x+y;t) | , |sV (x+y;t) | j S H ( y , t ) , y > 0,
with H(y, t ) a monotonitnil hj decreasing function of y
fur fixed t , such that a ( t ) = sup H(y, t) ->• 0 as t ->• <°.0<y<+«>
(b) 0/°O|ftc(x+y;t) |dy < oQ < 1 and
0/"|i^(x+y;t) |dy £ a, < +», with aQ and a^ constants.
Then we haoc:
sup |u(x,t)-u,(x,t) | = 0(a(t)) as t -»• m.xga(t) a
(iii) The inverse scattering formalism associated with the self-
adjoint Eakharov—Shabat system
where q = q(x) is a real function and C, a complex parameter,
can be simplified. As in Chapter 4 the Gel'fand-Levitan equation
appearing in the literature can be reduced to a scalar integral
equation containing only a single integral. With the help of this
simplification one can analyse in a way similar to Ch. 2, section
the asymptotic behaviour corresponding with the various nonlinear
evolution equations (all solitonless) solvable via the associated
inverse scattering method, such as the solitonless mKdV, sinh-
Gordon, etc.
(iv) The explicit structure of the constants in the niKdV analysis
presented in Chapter 5, shows that related results are valid
in coordinate regions t > 0, x s -u -vt , •• = — + e, with
u,v,e nonnegative constants and e > 0 sufficiently small.
202
AWE SDU
AN OPEN PROBLEM
In this thesis we have succeeded, by a more or less uniform method,
to reveal the asymptotic structure for large time for solutions of a
number of interesting nonlinear evolution equations solvable by the
inverse scattering method. However, any reader familiar with the soliton
field no doubt has noticed the absence of the wellknown nonlinear
Schrödinger equation (NLS)
(1a) iqt = qxx + 2q2qX, -«> < x -• +», t •> 0
(lb) q(x,0) = qQ(x),
where the initial function ^Q(X) is an arbitrary complex-valued function
on R, sufficiently smooth and rapidly decaying for |x[ •* °° and
satisfying conditions similar to Ch. 3, (7.2) so as to make the
Zakharov-Shabat inverse scattering method associated with Ch. 8, (2.1)
work.
Note that the dispersion relation w(O = -i,2 associated with the
linearized version of (la) is an even function, so that the group
velocity dw/di; = -2c, is not of one sign for all real £. This forms a
significant difference with the other problems treated and has direct
consequences for the method employed in this thesis.
Let us nevertheless see what our method produces.
203
To start with let us point out that for the problem (1) the Zakharov-
Shabat inverse scattering method runs as follows (see [1], [2], [4], [6]
for details).
Having computed the (right) scattering data [b^(0 , C • ,CL. } associated
with q o(x), when introduced as a potential into Ch. 8, (2.1), one puts
(2a) c5(t) = cyexp{-4i£2. t }, j = 1,2,...,NJ J J
(2b) b U,t) = b (c)exp{-4i;2t}, -» < ,; • +».
Then by the solvability of the inverse scattering problem, there exists
for each t > 0 a smooth potential q(x,t) having {b (<;,t), c,. ,C. (t) } as
its scattering data. The function q(x,t) is the unique solution to the MLS
initial value problem (1).
Now, let q,(x,t) denote the reflectionless part of the solution
q(x,t) of (1). Then, since the group velocity is not of constant sign,
tnere is no particular region of the x-axis singled out on which
q(x,t) - q,(x,t) might be concentrated as t -+ «. However, we may expect
an overall decay of q(x,t) - q,(x,t) for t -• «>. In fact, some authors
[3], [5] claim the existence of special solutions to (1) such that
q - q, decays in time as t more or less uniformly in x on R.
Let us see what the techniques of Chapter 3 yield in this case. By
theorem 4.t of that chapter we can estimate q — q, in terms of the
scattering data in the following way:
For each x t E and t "> 0 one has
(3) |q(x,t) - qd(x,t)| S u?Q[S2 j |:'c(x+y;t)
+ sup !;:
with
(4) !.c(s;t) = -i
and a 0 the constant given by Ch. 3, (4.2).
Now, a detailed examination of the right hand side of (3) shows that in
all relevant coordinate regions the second term behaves as t " as t •• ™,
but the first term tends to a constant!
Note the difference with the cmKdV problem Ch. 8, (7.1) where theorem 4.1
204
revealed neatly the asymptotic structure (see Ch. 3, (7.7-8) and the sub-
sequent discussion).
As an explicit prototype problem, let us mention the special case
that the initial function qn(x) in (1b) has the scattering data
{b (c) = e ,0,0}. Then the integral (4) can be evaluated in closed
form, yielding
(5) fic(s;t) = it (1 + 4it)~iexp{-s2/(1+4it)}.
Let us write out in full glory the Gel'fand-Levitan equation Ch. 8, (3.2-3)
in the special case (5). For x E R, y > 0 and t > 0 one then has
r(6a) g(y;x,t) + w(x+y;t) + i3(z;x,t)io(x+y+z;t)dz = 0
0J
0 -lr"i(1-4it)~^exp{-s2/(1-4it)}v
(6b) ü>(s;t) = [ i . )V ( 1 + 4it) iexp{-s2/(1+4it)} 0 '
with s £ E.
Here the unknown B(y;x,t) is a 2x2 matrix function of the variable y,
whereas x and t are parameters. Of crucial importance is the behaviour of
8(0 ;x,t) as t ->• +<*> in appropriate regions x S a(t). This gives the
behaviour of q(x,t) through the relations Ch. 3, (3.4-5).
Since the integral equation (6) is an explicit integral equation, no
particular knowledge of inverse scattering is required to attack it.
References
[l] M.J. Ablowitz, Lectures on the inverse scattering transform, Stud.Appl. Math. 58 (1973), 17-94.
[2] M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, The inversescattering transform - Fourier analysis for nonlinear problems, Stud.AppL. Math. 53 (1974), 249-315.
[3] M.J. Ablowitz and U. Segur, Asymptotic solutions and conservationlaws for the nonlinear Schrcidinger equation I, J. Math. Phys., 17(1976), 710-713.
205
[4] W. Eckhaus and A. van Harten, The Inverse Scattering Transformation andthe Theory of Solitons, Uorth-Holland Mathematics Studies 50, 1981(2nd ed. 1983).
[5] H. Segur, Asymptotic solutions and conservation laws for the nonlinearSchrodinger equation II, J. Math. Phys., 17 (1976), 714-716.
[6] S. Tanaka, Non-linear Schrodinger equation and modified Korteweg-deVries equation, construction of solutions in terms of scattering data,Publ. R.I.M.S. Kyoto Univ. 10 (1975), 329-357.
206
SAMENVATTING
In 1967 ontdekten Gardner, Greene, Kruskal en Miura een methodeom het beginwaardeprobleem voor de Korteweg-de Vries vergelijking (KdV)in principe op te lossen. Niet lang daarna bleek dat deze methode nietop zichzelf stond, maar met succes gebruikt kon worden otn andere belang-rijke niet-lineaire partiële differentiaalvergelijkingen te bestuderen.De methode kreeg de naam: inverse scattering technique (1ST).
Aan de hand van de KdV laat zich de 1ST als volgt omschrijven:Het beginwaardeprobleem voor de (niet-lineaire) KdV wordt gereduceerdtot het oplossen van de (lineaire) Gel'fand-Levitan integraalvergelijking.De beginwaarde voorgeschreven voor de oplossing van de KdV wordt alspotentiaal geïntroduceerd in de Schrödinger vergelijking en levert dande zogenaamde scattering data met behulp waarvan de coëfficiënten van deGel'fand-Levitan vergelijking worden gedefinieerd.
Echter, expliciet oplossen van deze integraalvergelijking is slechtsmogelijk als de reflectiecoëfficiënt geassocieerd met de beginwaardenul is. Dan ontstaat de vermaarde "pure N-soliton oplossing" met Nhet aantal discrete eigenwaarden in de Schrödinger vergelijking.Het asymptotisch gedrag van deze oplossing is uitgebreid bestudeerden blijkt te corresponderen met een decompositie in N solitonen, d.w.z.gelokaliseerde golven die na onderlinge interactie hun oorspronkelijkevorm en snelheid behouden en alleen een faseverschuiving (phase shift)aan deze interactie overhouden.
In het algemene geval dat de reflectiecoëfficiënt niet overal nul is,is de asymptotiek aanmerkelijk gecompliceerder. Een eenvoudige heuristi-sche redenering leidt al gauw tot het vermoeden dat naarmate de tijd ver-strijkt de oplossing uiteen zal vallen in twee componenten: een soliton-component bestaande uit N naar rechts lopende solitonen en een nonsoliton-component bestaande uit dispersieve golven die naar links lopen.Het feit dat destijds geen enkel bewijs van dit vermoeden bekend was,vormde de aanleiding tot het onderzoek geboekstaafd in dit proefschrift.
Meer algemeen stellen we ons in dit proefschrift ten doel een volledi-ge, door bewijzen gestaafde beschrijving te geven van de manier waaropsolitonen te voorschijn komen uit verschillende klassen van niet-lineairepartiële differentiaalvergelijkingen oplosbaar via 1ST. Hierbij werken wein die coordinaatgebieden waar de nonsolitoncomponent is op te vatten alseen storing op de solitoncomponent. Een steeds weerkerend element in deanalyse van de verschillende niet-lineaire problemen op de voorafgaandebladzijden is dan ook de aanpak van de Gel'fand-Levitan vergelijkingvia een storingsargument.
De inhoud van dit proefschrift kan als volgt worden geresumeerd:
207
In Hoofdstuk 1 analyseren we de oplossing van het beginwaardeprobleemvoor de KdV in alle naar rechts meelopende coördinaten en geven eenvolledige en nauwkeurige beschrijving van het te voorschijn komen vansolitonen. Deze asymptotische analyse wordt in Hoofdstuk 2 uitgebreidmet zeer expliciete schattingen van de nonsolitoncomponent op rechter-halflijnen die langzaam naar links lopen. In Hoofdstuk 3 berekenen we dephase shifts van de KdV-solitonen wanneer niet alleen de interactie metde andere solitonen maar ook die met de nonsolitoncomponent in aanmerkingwordt genomen. De interactie met de nonsolitoncomponent blijkt te resul-teren in een extra phase shift naar links. In Hoofdstuk 4 beschouwen wede vraag hoe goed een oplossing van een niet-lineaire, via 1ST oplosbaregolfvergelijking wordt benaderd door zijn solitoncomponent in een ruimerkader door schattingen af te leiden voor het verschil van een reëlepotentiaal in het Zakharov-Shabat systeem en zijn reflectieloze component.Verder wordt het bijbehorende inverse scattering formalisme aanzienlijkvereenvoudigd. Gebruikmakend van de schattingen uit Hoofdstuk 4 wordenin Hoofdstuk 5 asymptotische schattingen voor de oplossing van hetmodified KdV (mKdV) beginwaardeprobleem verkregen, welke op hun beurtbenut worden in Hoofdstuk 6 voor de berekening van de phase shifts van demKdV-solitonen. In tegenstelling tot het KdV geval beschouwd in Hoofdstuk3 blijkt bij de mKdV de interactie met de nonsolitoncomponent te resulte-ren in een extra phase shift naar rechts. In Hoofdstuk 7 passen we deresultaten uit Hoofdstuk 4 toe voor een asymptotische analyse van deoplossing van het sine-Gordon beginwaardeprobleem. De schattingen uitHoofdstuk 4 worden in Hoofdstuk 8 gegeneraliseerd tot het geval vancomplexe potentialen. Als toepassing analyseren we het complexe mKdV(cmKdV) beginwaardeprobleem. In Hoofdstuk 9 ontwikkelen we een inversescattering formalisme voor de matrix Schrödinger vergelijking met niet-Hermitische potentiaal. Dit matrix Schrödinger probleem vormt, zoals inHoofdstuk 10 wordt aangetoond, een natuurlijke synthese van de scatteringproblemen optredend in dit proefschrift. In een appendix illustreren weaan de niet-lineaire Schrödinger vergelijking, dat de in dit proefschriftontwikkelde technieken falen voor vergelijkingen waarvan de geassocieerdegroepssnelheid niet tekenvast is en stellen vervolgens een interessantopen probleem.
208
CURRICULUM VITAE
Geboortedatum: 31 januari 1950Geboorteplaats: ZwolleEindexamen gymnasium 3: 1969 (cl.)Doctoraalexamen wiskunde aan de R.U.Groningen: J978 (cl.)Specialisatie: FunctionaalanalyseAfstudeerdocent: Prof. dr. G.E.F. ThomasAfstudeeronderwerp: De Stieltjes integraalvergelijkingPromotieplaats aan het Mathematisch Instituut van deR.U.Utrecht: 1978-1984Huidige functie: medewerker aan de afdeling ToegepasteWiskunde (vakgroep ADAM) van de T.H.Twente
209
SOLITON SOLUTIONS
OTTlti OFFER
TVPING TWISTS
INSUMMABLE INSÖMNTA
STELLINGEN
behorende bij het proefschrift Studies in Soliton Behaviour
van P.C. Sehuur
1. Beschouw in de complexe Hilbertruitnte JC = L2[O,1] met standaard-
inprodukt de twee t-parameter families van operatoren F.(t)
en F„(t), beide met domein C[O,1] , gegeven door
Ft(t)v = v(0), t ï 0,
[F2(t)v](x) =|lév(0)' t > 0
(o t = o.
Dan zijn F,>F2 e n F]F2 s t e r k differentieerbaar in 3C op CfO,1]
met betrekking tot t s 0, terwijl niet geldt
2. Zij R > 0 en f:[O,R)—*C continu met f i 0.
Dan is er een 6 > 0 zodanig dat voor 1 < x < 1 + 6 geldt
( x - 1 ) 20 / R | k | 0 x" k f (yx" k ) | 2 dy < 4 0 ; R | f ( Z ) | 2 dZ.
3. De bewering in W. Rudin, Functional Analysis, p. 341,
dat iedere symmetrische operator in een Hilbertruimte
een gesloten symmetrische voortzetting bezit, kan door
een eenvoudig tegenvoorbeeld weerlegd worden.
4. Zij E de klasse van alle complexwaardige functies g(x) ,
gedefinieerd op (0,»), zodanig dat (1+x) g(x) tot L1(0 < x < +»)
behoort. Zij A 6 C, Re X * 0.
Dan heeft de Stieltjes integraalvergelijking
f(x) = g(x) + Xj £2± dy, x > 0
voor iedere g E E een oplossing f € E.
(P.C. Schuur, De Stieltjes vergelijking, h'.IJ. Gt\niinjcn,
inu-rn rapport, IH?8)
5. Zij S een (niet noodzakelijk dichtgedefinieerde) isometrische
gesloten operator in een complexe Hilbertruimte.
Dan is het spectrum van S een van de volgende verzamelingen:
(i) n e c| |x| i 1} (Ü) {A e c| |A| % 1}
(iii) (C (iv) een deelverzameling van
{x e c| |x| = 1}.
6. In het boek W. Magnus, F. Oberhettinger, R.P. Soni, Formulas and
Theorems for the Special Functions of Mathematical Physics
vertonen twee afzonderlijk vermelde formules bovenaan p. 9
een frappante gelijkenis:
ƒ (cos<)*cos (fit) At — -
Rex> — 1,
{ {cos l)' cos (yt) dt = ~ r(x + J)
» 2 ^(-p+^r-f-'+i)Rex> - I .
7. De buitenkant van het blad Mededelingen van het Wiskundig
Genootschap is voorzien van de zinsnede "verschijnt maandelijks
(negen maal per jaar)". Deze zinsnede is onprecies en inwendig
tegenstrijdig en dus niet passend voor een wiskundig tijdschrift.
8. De rits is een manonvriendelijke uitvinding.
9. Of iemand een baan al dan niet krijgt, wordt te dikwijls bepaald
door het misverstand in psychologische tests, dat een oneindige
rij bepaald zou zijn door specificatie van een eindig aantal
termen.
10. Veel cartoons suggereren dat sokken breien begint bij de teen
en eindigt bij de boord. In werkelijkheid gaat het andersom.
11. Wie blij is met een dode mus is een dierenbeul.