Volume 113A, number 4 PHYSICS LETTERS 16 December 1985

S T A B I L I T Y T H E O R Y F O R N O N L I N E A R K L E I N - G O R D O N K I N K S AND M O R S E ' S I N D E X T H E O R E M

Hiroyuki ITO and Hal T A S A K I

Department of Physics, Faculty of Science, University of Tokyo, Hongo, Bunkyo- ku, Tokyo 113, Japan

Received 19 July 1985; accepted in revised form 16 October 1985

A rigorous stability theory for kink solutions in N-component nonlinear Klein-Gordon equations in 1 + 1 dimensions is developed. We show that the number of independent unstable perturbations is given by the Morse index. As applications we prove the stability and the instability of various kink solutions in one- and two-component systems.

Introduction. The concept o f "soliton", original- ly developed in the study of completely integrable sys- tems, is now being extended to a more general notion which denotes fundamental objects governing the be- haviour of various nonlinear systems. In such attempts, stability theories (both for the phase stability and the structural stability) for solitons play a crucial role.

In the present letter, we study the phase stability (i.e., the dependence on small perturbations o f initial conditions) of kink solutions in N-component linear Klein-Gordon equations in 1 + 1 dimensions. We point out that a classical result o f Morse in the calculus of variations in the large (Morse's index theorem) can be used to give a mathematically clear meaning to the lin- ear stability analysis, and to provide a simple expres- sion (in terms of a topological index) for the number of independent unstable perturbation modes ,1

As applications, we prove the stability and the in- stability o f various kink solutions in arbitrary one- component systems, and in the two-component ani- sotropic I~ol4-system. The results in the latter exam- ple have been beyond the reach of the traditional ap- proaches which treat the eigenvalue problems directly.

.1 The Morse theory has been successfully applied, by Taubes [8], to prove the existence of a saddle point solution in a spontaneously broken gauge theory. Some attempts to ap- ply the Taubes method to other models have been made [9]. We are grateful to the referee for letting us know these references.

General theory. We consider an N-component clas- sical field theory in 1 + 1 dimensions, whose hamilto- nian at time t for a field configuration ~ x , t) E R N is given by

~f(~O) = f d x 1(0t~o)2 + Jfstat

= f d x [l(0ttp)2 + l(0xtp)2 + V(~0)] , (1)

where the potential Visa real valued smooth function on R N. The corresponding equation of motion is

(02 - 02) ~ x , t) = - d V(~o)/d~l~o(x,t) . (2)

By a kink solution ~0K(X, t), we mean a solution o f eq. (2) which can be written as ~0K(X, t) = q(s) with s = x - ot ( [vl< 1) and q: R ~ R N which satisfies q(s) ~ g (g') as s ~ oo (_oo) and dq(s)/ds -~ 0 as s ~ -+ oo. Here the ground states g and g' (which may coincide) are the points in R N where Vtakes its minimal value V(g) = V(g'). In the following, we set the kink velocity o equal to zero, which can be always satisfied by a suita- ble Lorentz transformation.

Formally, the stability or the instability o f the so- lution ~OK(X , t) can be determined if we solve eq. (2) in the form

~ x , t) = ~OK(X, t) + 6 ~ x , t ) , (3)

with the initial conditions ~ x , 0) = ~OK(X, 0) + 6Q(x)

and ~(x, 0) = 6P(x) where 6Q(x) and 6P(x) are arbi- trary functions with compact supports, and investigate

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whether the solution deviates from ~0K(X , t) as t in- creases. If we restrict ourselves to the linear regime, where t ( > 0 ) , la~01, and la~l are sufficiently small, we might expand the hamiltonian as

a 2~(stat f + dy

+ 0 ((6~0)3) . (4)

and solve the linearized equation

6 2~(stat ~ ~(~V, t) . (5) 6(o'(x, t) = - fdy 6¢(x) 6 ~o(y ) ~OK Then we find that a perturbation mode 6Q(x) corre- sponding to the negative (positive) eigenvalue of the operator 6 2~stat/6~02 grows exponentially (does not grow) as t increases, irrespective o f the values of 6P(x). Thus if some (none) of the eigenvalues are negative, then the corresponding (none of the) initial perturba- tions are the unstable perturbations, and the kink so- lution ~0K(X, t) is linearly unstable (stable) [1 ] .

It should be noted that a direct calculation of the above eigenvalue problem is generally quite difficult, and can hardly be executed unless the field variable is o f one-component (i.e., N = 1). In the following, we are going to propose a new treatment of the problem which makes use of a dynamical system description o f the system and the Morse's index theorem.

Let us write down the equation for q(s) by substi- tuting the relation ~0K(X , t) = q(x) to eq. (2).

d2q(s)[ds 2 = d V(q )/dq[q=q(s) . (6)

This equation can be regarded as a Newton's equation for a point-particle whose position at "t ime" s is q(s) in FiN. And the trajectory q(s); _ o o < s < ~ connects two unstable fixed points g and g'. (Note that the sign of potential has been changed.) The crucial point in this interpretation is that the action o f this dynamical system

$o

S(se;q) = f ds [{(dq/ds) 2 + V(q)] , (7) -oo

exactly coincides with the static part of the hamilto- nian ~fstat o f the original system if we let s e ~ ~ .

Now we will describe Morse's results on the varia- tional analysis. For a given sequence _oo < sl < s2 <

... < s k < Se, and arbitrary 8qi @ Fi N (i = 1 ..... k), we construct a piecewise linear perturbed path q'(s) so that it satisfies q'(si) = q(si) + 8qi , and coincides with q(s) at s = s e and s = - oo. Then one can expand the action (7) for q ' as

t 1 S(se;q ) = S(se;q ) +~.. ~E#6qiSq j + O((6q)3) , (8)

t,]

where (E#) is a k X k matrix usually called the hessian o f the trajectory q(s); - oo < s ~< s e. Then Morse's in- dex theorem [2,3] asserts that (for a sufficiently large k), the number of negative eigenvalues of (E#) coin- cides with the Morse index ind q(Se) , which is always afinite quantity, and defined as a number of conjugate points (counted with their multiplicities) along the trajectory q(s); - ~ < s < s e. A point s' E (_0% Se ) is said to be a conjugate point with multiplicity m, if the hessian of the trajectory q(s); - oo < s < s' has m zero-eigenvalues, or in other words, if there exist m independent trajectories infinitesimally close to q(s) and coincide with q(s) at s = s' and s = - oo.

Taking into account the correspondences ~0K(X , t) = q(x) and limse~o . S(s e; q) = ~fstat(~0), we find that the hessian (E~) in (8) is nothing but a finite dimen- sional counterpart of the functional derivative 5 2~stat] 6~02 in eq. (4). Therefore we see that though the func- tional derivative 6 2~Cstat/6~02 itself is a formal k -+ oo (and then) s e ~ oo limit of (Ev), the number of its negative eigenvalues is a finite quantity which charac- terizes the trajectory q(s). This fact enables us to justify the linear stability analysis rigorously, and interpret the number of negative eigenvalues as that of indepen- dent unstable perturbations. It is also important to note that though we have restricted the basic solutions to functions o f s = x - ot, we have considered all possi- ble field configurations 6 ~ x , t) as perturbations to the solutions.

Consequently; we arrive at the following rigorous stability theorem in an arbitrary nonlinear Kle in- Gordon type equation ,2. The numberofindependent unstable perturbation modes to a kink solution is al- ways finite, and coincides with the Morse index ind ~o K = lims_.~ind q(s).

, 2 It is easily found that the end point s = ~ is always a con- jugate point, since any kink solution has a zero-eigenvalued mode, usually called the Goldstone mode (or translational mode).

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Examples. We briefly discuss two elementary ap- plications of the above general theory. The details and the analysis of other examples will be reported elsewhere [41.

First, consider a one-component system (i.e.,N = 1) with an arbitrary potential V(¢). (The familiar exam- ples are the ~04 and sine-Gordon systems.) In this case, where the corresponding dynamical system is that of one dimension, it can be easily shown that the con]u-

q

g

a - V

g g

_ _ _ S

b l -V

gate points are nothing but the turning points where the velocity dq(s)/ds vanishes. Therefore, we imme- diately conclude the generally accepted fact that the kink solutions with q(_oo) =# q(oo) (topological kinks) are always stable, and those with q(_oo) = q(oo) (non- topological kinks) are unstable to a single perturbation mode (fig. 1).

The second example, the anisotropic I~0t 4.system, is described by the two-component field variable ~0 = (~o 1 , ~02), ~oi(x, t) E R, and the potential

1 1 , V(~0) = - ~ h012 +¼ I~0t 4 +~ o2(~02) 2

(0 < o < 1), (9)

with two ground states gl = ( - 1 , 0 ) and g2 = (1,0). In this system, the kink solutions with q(_oo) = gl are completely determined in refs. [5-7] and classi- fied as (TK1) a topological kink (i.e., q(~) = g2) corre- sponding to a segment connecting gl and g2, (TK2) two topological kinks forming an ellipse including gl and g2, and (NTK) a set of nontopological kinks (i.e., q(oo) = gl) parametrized by a single parameter. More- over, the TK1 solution and all the solutions in NTK pass through one of the foci of the ellipse formed by TK2 solutions (fig. 2). This implies that the above fo- cus is a conjugate point of the corresponding dynami- cal system, and thus the kink solutions in TK1 and NTK are unstable to (at least) one perturbation mode. By a detailed analysis using the parametrization of ref. [7], we can also prove [4] that the kink solutions in TK2 are stable. These facts have been conjectured numerically, but are proved here for the first time.

~'2

gl ~ g 2 ~ ~°1

Fig. 1. Trajectories o f a topological kink solution (a) and a nontopological kink solut ion (b) in one-component systems.

Fig. 2. Trajectories o f topological kink solut ions and some nontopological kink solutions in the anisotropic 1~ol 4 system.

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We wish to thank Professor Y. Wada for useful discussions and a careful reading of the manuscript , Professor M. Suzuki, Dr. Y. Ono, Dr. S. Takesue, and Y. Ohfuti for stimulating discussions.

References

[1] R. Jackiw, Rev. Mod. Phys. 49 (1977) 681. [2] M. Morse, The calculus of variations in the large (Am.

Math. Soc., Providence, 1934).

[3] J. Milnor, Morse theory (Princeton Univ. Press, Princeton, 1962).

[4] H. Ito and H. Tasaki, in preparation. [5] K.R. Subbaswamy and S.E. Trullinger, Physica 2D

(1981) 379. [6] E. Magyari and H. Thomas, Phys. Lett. 100A (1984) 11. [7] H. Ito,Phys. Lett. l12A (1985) 119. [8] C.H. Taubes, Commun. Math. Phys. 86 (1982) 257; 86

(1982) 299. [9] N.S. Manton, Phys. Rev. D28 (1983) 2019;

P. Forg~tes and Z. Horv~th, Phys. Lett. 138B (1984) 397.

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