PUY$1GAI ELSEVIER Physica D 94 (1996) i 16-134

Non-perturbative non-integrability of non-homogeneous nonlinear lattices induced by non-resonance hypothesis

K e n U m e n o i Laborato D'for Information Representation. Frontier Research Program. RIKEN. 2-i Hirosawa, Wako, Saitama 35 I-0 I, Japan

Received 19 June 1995: revised 27 November 1995: accepted 3 December 1995 Communicated by H. Flaschka

Abstract

We succeeded in applying Ziglin's test on non-integrability to non-homogeneous nonlinear lattices [Fermi-Pasta-Uiam ~FPU) lattices]. By explicit calculations of the eigenvalues of the monodromy matrices concerning the normal variational equations of Lan~ type and with the use of the non-resonance hypothesis about the eigenvalues, we obtained a theorem proving the non-existence of additional analytic conserved quantities other than the Hamiltonian itself for the FPU lattices in the low energy limit. Furthermore, after introducing a concept of degree of non-integrability, we investigated the classification of non-homogeneous nonlinear lattices using a transformation :It from a non-homogeneous nonliqear lattice to another non- homogeneous nonlinear lauice, which preserves their degree of non-integrability.

Ke3.'words: Non-integrability; Nonlinear lattices: Singularity analysis: Monodromy matrices: Non-resonance condition

1. Introduction

Presently, we know two major sources of detecting non-integrability of the Hamiltonian systems. One is Poincar~'s theorem [8] of the !9th century and the other is Zigl in 's analysis [ 19] relatively recently found in connection with

the classical singularity analysis [9] by Fuchs, Kowalevskaya and Painlev~. What matters here is the following: The

two different methods have significantly different predictive capacities to tell non-integrab~lity. While Poincar~'s

theorem is asserting the non-existence of additional conserved quantities such as

~o({q~,}) +/z,/~l ({q,., p~,}) +/z2~2({qv. Pv}) + - - - , (1.1)

for a Hamiltonian system perturbed from an integrable system,

H = Ho({qv}) + #Hl ( Iqv . Pv}) + #2H2({qv. Pv)) + ' " , ( i .2)

I E-mail: chaosken@giraffe.riken.go.jp.

0167-2789/96/515.0~) © 1996 Elsevier Science B.V. All rights reserved PIi

K. Umeno/Physica D 94 fl996) 116-134 [[7

where/~ is a coupting constant whose explicit value is unknown, Ziglin's type theorem can wove the ram-existence of additional analytic integral for an explicit Hamiltonian. To illustrate the differences between them more con- cretely, we give the following examples. The restricteo three-body problem was proved to be no~inggra~e by Whittaker [ 14] in 1904 based on Poincar6"s theorem, while Ziglin's theorem has never proved its non-imegrab~l~ty because of the problem that a Hamiltonian of the restricted ~ree-bedy problem cannot be defined wigmm the infinitesimal operation (m --* 0), and we do not have any explicit Hamiltonian which is necessary to perfocm sot/d Ziglin's analysis. On the other hand, Poincar~'s theorem assumes the condition on the Hessian of Hamilmc/aa, Le., the Hessian of Hamiltonian should not vanish,

i]2H0 Oqvaq~ • O. ( | 3 )

By this assumption, Poincar6"s theorem cannot tell anything about the non-integrability of nonlinear hgtices consisting of harmonic terms H0 = ½ Y~-7=l q~ plus unharmonic terms such as the nonlinear lattices of Ferm/- Pasta-Ulam type (FPU lattices). Therefore, in the present paper, we examine the possibility of nen-intcgra~IRy proof of the FPU la:tices via Ziglin's analysis. It should also be mentioned that Ziglin's analysis has some tm:haical difficulties, as will be illustrated later, and no non-integrability proof has been so far studied either for realistic Hamiltonian systems such as the FPU lattices with non-homogeneous potential functions, except few rare instances such as the Henon-Heiles systems with two degrees of freedom [5,6,20], or for Hamiltonian systems with many degrees of freedom besides the case of Harniltonian systems with global and symmetric coupling [ 13].

The purpose of the present paper is to show that this Ziglin's analysis can be performed for checking the mm- integrability of nonlinear lattices including the FPU lattices and that these nonlinear lattices have no other analytic conserved quantities.

In Section 2, we show that singularity analysis can be done for a class of the FPU lattices by giving the exp/icit eigenvalnes of the monodromy matrices about a special solution in terms of elliptic functions. In Section 3, we prove a non-integrability theorem for the FPU I~tices based on the singularity analysis of Section 2, which sbows thal each FPU lattice has no additional analytic conserved quantities in the low energy limit. In Section 4, we discuss the classification of non-homogeneous no~linear lattices via the degree of non-integrability. In Section 5, we give a brief summary and open questions about the present analysis.

2. Singularity analysis of nonlinear lattices

Consider the following one-dimensional lattice:

n+l l p2 + y ~ ~,~q~_~ _ q~). ~2.1)

H = ~ i=I i=l

If this Hamiltonian admits the reflective invariance, namely, if it is invariant under the involutive symplectic diffeo- morphism,

JrH = H, (2.2)

where

Jr : C 2n -"* C ?'n, Jr : (ql . . . . . qn, Pl . . . . . Pn) ~ (--ql . . . . . --qn, --Pl . . . . . --Pn), (2.3)

the potential function v ( X ) must satisfy the condition given by

v ( X ) = v ( - X ) . (2.4)

118 K. Umeno/Physica D 94 (1996) 116--134

In fl~e case of harmonic lauices, we have u ( X ) = +/22 X2- Thus more general nonlinear lattices with the reflection symmetry can be defined by the following polynomial function of even degree in X:

_~ , V 4 . 4 /~2m X2m. v t X ) = X- + -~--x + - - - + ~ (2.5)

In this paper, however, we assume that this potential function is a polynomial function of degree four, namely, we

consider the FPU lattices [31

n+l /~1 n+l , -4-g4 ~-~.(qi-Ii=l -qi)4" (2.6) 1 n , la2 ~ - ~ . ( q i - I - q i ) " +

nzeL, = -5 P': + - T "= i=1

Imposing the fixed boundary condition as

qo = qn+l = O. n = odd. (2.7)

the equations of motion are given by

ql = --/z2(2ql -- q2) --/.t4(qP + (ql -- q2)3), q2 = --~2(2q2 -- ql -- q3) --/-t4((q2 -- ql) 3 + (q2 -- q3)3). . (2.8)

q.n = - I t 2 (2qn - q n - ; ) - IX4 ((qn - q n - I )3 + q3).

Define M' = {(q,p) E C 2n I qt =/: 0 o r p z ~ 0}, ,VI = M ' / J and the canonical projection Jr : M' ---> M.

An additional complex analytic integral in C 2n. if it exists, also induces the corresponding analytic integral on M. Therefore it is sufficient to check the non-integrability of the reduced equations of the FPU lattices on M in order to confirm the non-integrability of the original equations of the FPU lattices.

Consider the following one-parameter family of the solution for (2.8):

F ( E . t ) : q j = / ( ( - - l ) J -- l ) ( - l ) ( J + t ) / 2 C ( b ( t ) (2.9)

with the initial condition

~ 0 ) = I, ~(0) = 0. HFPU(0) = ,~. (2.10)

This is a complex solution in the complex time pl&-.e as

ql = Cop( t ) . q2 = O. q3 = - C c ~ ( t ) . . . . . q n - i = O. qn = ( - I ) m - I ) / 2 C d p ( t ) , (2.11)

which is depicted in Fig. !. Thus, the underlying equation of ¢ (t) is reduced to the following Hamiltonian system wifla one degree of freedom:

+ 2tt2~ + 2p4C2~ 3 = 0. (2.12)

whose Hamiltonian is given by

H(O. ~P) = ½(~)2 + p21~2 -k ½/.t4C2~ 4 = const. (2.13)

Then the original tohal energy 6 is written as follows:

,F = H = H(c / : , .d )½(n + I)C 2 : +(n + I)C2(bt2 + ½//4C 2) (2.14)

K. Umeno/Physica D 94 (1996) 116-134 ~ E9

I un~,na~

~ Rea l

i=O i=1 i=2 i=3 i--4 i=5 i=6 i=-7 i--n+1=8

Fig. I. The initial lattice displacements in the complex plane are depicted for the nonlinear lattices with the refl~:tion symmmtty (n ----- 7) for/~2 > O./*4 > O. ~ < 0 in Eq. (2.26). In this case. the initial lattice displacements are pure imaginary.

by the initial condition (2.10). Eq. (2.14) enables us to choose C as the following complex quantity:

l~/ . ~ + ( ~ / ( . + t ) ) . ~ - . 2 C = (2.15)

Therefore. H(¢ , ~) becomes

H(~,/~) = ~ p + t12e} 2 + ~ /z 2 + ~--~/a4 - / 2 2 ~ 4 (2.16)

where ~ = ~,/3 = #L By combining (2.13) with (2.14), we obtain the differential equation of ¢( t ) as

/(~)2 = F2(l --~b 2) + 1}'4(I _ ¢ 4 )

with the system parameters

Y2 =- ]g2, ]/4 ~ ll4 C2.

Thus, the phase curve F(t , E) (2.9) is given by the elliptic integral

¢,(t)

f d¢~ t, ,(0)=l ~ /2 ;y2(1 - ~2) + ½Y4(i - ~4)

and the solution of the differential equation (2.17) with the condition

r2,,=4 # 0

is given explicitly by the formula

$( t ) = cn(k; a t ) ,

where

ot = x/2y2 + 2y4, k = ¥ J ' 2 Y4 y2"~ 2y4 ~

(2.t7)

~2.tS)

(2A9)

(2.20)

(2.2l)

(2.22)

120 K. Umeno/Physica D 94 (1996) 116-134

cn(k: at) is the Jacobi cn elliptic function, and k is the modulus of the elliptic integral, because of the equalities

(-dd~z)2={--asn(k:czt)dn(k:ct,,}2

= ot2{(-2k 2 + !)(1 - cn2(k; at)) + k2( 1 - cn4(k; at))}

= 2 ~ ( i - ¢~2) + F4(I - ~4) (2.23)

in Eq. (2A7). We remark here that the relation

/ , ~_ + y4 = #2 + C2#4 = ; + ~ - - - ~ # 4 > 0, (2.24)

holds for #4 > 0, #2 > 0, e > 0. This means that the modulus of the elliptic function k satisfies the relation

/( / ! 4E#4 ! O < k = I - I l + i n + l ) . u ] < ~ (2.25)

and C is a real quantity. On the contrary, if the relations

#2 > O. #4 > O, ~ < 0 (2.26)

are satisfied, the modulus of the elliptic function k becomes pure imaginary, i.e., k 2 < 0. C becomes also pure

imaginary as in Fig. i. We note here that even in this unphysical case, there is no barrier to perform the following

singularity analysis. However, in the following we consider the physic~ case with the condition as #2 > O,

#4 > O, ~ > O. According to the reduction xr : M' --* /0 = M ' / J by the reflective symmetry J , the solution (2.11) of the induced FPU lattices on ~ / f o r #4 > O, #2 > 0 is a single-valued, meromorphic and doubly periodic fanction in t, having two fundamental periods in the complex time plane as follows (see Fig. 2):

2K(k) 2K(k) + 2iK' (k) Tl (E, # ) = ~ ~ ( ~ , # ) = (2.27)

ot ct

Here K(k) and K'(k) are the complete elliptic integrals of the first kind:

] ] de o K'(k) = dv t2.28) K(k) = v/( i - v2)(I - k2v 2) v/(! - v2)(i - - (I - k 2 ) v 2 ) "

0 0

Note here that these solutions have a single pole at t = r, where

r = + i ..... (rood T~, T2)

in the parallelogram of each period ceil: the phase curves r ( ~ , t) are punctured tori. If we consider the solution (2.11)

not on the reduced phase space I(4 = M ' / J but on the original phase space M' , the solution has the fundamental periods as

T~{E, # ) = 4K(k) T~(~, g ) = 2K(k) + 2iK'(k) (2.29)

which ar~ different from (2.27) and this solution on M' has two poles at t = r ' , where

, 2K(k) .K'(k) 4K(k) + i K ' ( k ) (mod TI', T ~ ) "r ~- +1

K. Umeno/Physica D 94 ¢ 19961 116-134 [2~

. , , ,< . "Tl(,,s,2,~,4)

Real t

T2(~ - , o, ~2, ~,4) = ; ~ m " + i~, /

/ /

g2 - I

-- Base i ~ in l

._c

g' gt _E

Real t T I ( ~ --. 0 , P 2 , # 4 ) - - ~ p 2 1 r

Fig. 2. Tim two fundm, . . . . . I periods TI. T2 arc dcpict¢~ in the complex time plane for tim FPU lanices. The arrow means the traa~,forma~s~o of the period cells into the period cells in the low energy limit.

in the parallelogram of each periodic ceil.

Let us consider the variational equations along the phase curve F(~. t) in (2.9) of the general norJinear la t ices

with the reflection symmetry. The linearized variational equations are written as follo~vs:

k=! ~)qkaqj r ~k

= --(Y2 + 3y4cn2(k; crt))(2~j - ~j_ I - / [ j + l ) for ! _< j < n, (2.30)

where ~o = ~n+t = 7o = T/n~-I = 0 and ~j = ~qj , llj = 8p) (1 < j < n). If we rewrite these linear variationa~ equations (2.30) in the form of the vector variational equation as

" ~ , ( 2 . 3 | ~ - ~ = --(}'2, + 3y4cn2(k: a t ) ) 0 - i - I •

: " i "

0 . - . 0 - I

122 K. Umeno/Physica D 94 (1996) 116-134

and use ~ fact that the eigenvalues of the n x n symmetric matrix

2 - I 0 - - - 0 - l 2 - i - - - 0

G = 0 - l 2 - 1 - - - (2.32)

! i : : :

0 --- 0 - ! 2

are ob~ned as {4sin2(j:r/2(n + i ) ) [ I < j < n} by a normal orthogonal transformation G ~ O G O - I , the variational equations (2.30) can he transformed into the decoupled form:

~ ( t ) = - -4s in 2 ~ (}2_ + 3y4cn2(k:at))~j(t) (! <_ j < n), (2.33)

where ~ ' = 0 ~ . Clearly, these equations are regarded as the vector form of Hill's equations [7]

d2~' + A(t)~' = O, A( t + T) = A( t ) , (2.34) dt 2

where T = TI, T2. For j = ½ (n + 1 ), the relation

~,~+t~/2 = (~l - ~3 +~5 + " " + ( - I )~" - l ) /2~n) (2.35)

holds. The corresponding variational equation

~"m+1)/2 -- -2(72_ + 3?'4 cn2(k: ctt))~n+l)/2(t) (2.36)

has a t ime-dependent integral I (~ , ~: t) = i (~ , ~: t) derived from the variational operation D applied to the Hamiltonian (2.6):

l ( ~ . 7 1 : t ) = D n - - ~ 7 " ~ p + ~ - H = r l ' P + { " Vq

---- C~(r/ i - 173 -4- t/5 + - - - + ( - i)(n-I)/2r/n)

+ 2(CF2 cn(k: at) + Cg4cn3(k: at))(~j - ~3 + ~5 + " " + (-I)~n-I)/2~n) (2.37)

because

I d._l/ = ~(~i - ~3 + " " + ( - l )~n- I ) /2~n) + 2~(F2 + 3y4 cn2(k: a t ) ) (~ l - ~3 + " " + ( - I ) (n - I ) /2~n) ---- O. C dt

(2.38)

Eq. (2.36) is called the tangential variational equation. On the other hand, a (2n - 2) -dimensional normal variational equation (NVE) is defined by Eq. (2.33) with the tangential variational equation (2.36) removed as follows:

~', = - 4 s i n 2 ~ ( ~ + 3y4cn'(k:at))~j . g ' j = r/j (2.39)

for 1 < j ( -# (n + 1)/2) < n.

K. UmenolPhysica D 94 (1996) 116-134 ~23

Let us consider the monodromy matrices g defined by the analytic continuation of the solution ( i f ) = (~ ( tL ~i'l (t) . . . . . ~;n+l)/2(t), ~'(n+l)/2(t) . . . . . ~n(t), r/~(t)) of the NVE (2.39) along the periodic orbits in the phase era-yes f (¢, t) as follows:

¢ ' ( r , ) = 8 ,C ' (0 ) , ¢ ' (T2) = g2~'(0) . (2.40:~

The periods in Eqs. (2.40) are Ti, T2 in F~s. (2.27), respectively. These two fundamental periods Te and T2 na~za]~ form the parallelogram, whose associate monodromy matrices are given by gi g2g l ! g2 t (= g,) . These matrices are naturally endowed with the symplectic structure and the pairing properties of the eigenvaIues, namely [o.i, o. / I, 0.2, 0.21 . . . . . On, 0.,~- I }. In general, the explicit calculation of the eigenvalues of the monodmmy matrices is an unsuccessful business except the rare cases such as Hamiltonian systems with homogeneous p o i y ~ functions [ ! 7], Riemann's equation, and the Jordan-Pochhammer equations [2,4,10]. This is one of the tmavoida[~ difficulties in performing Ziglin's analysis of general dynamical systems. In our case. however, we compute explic~iy the eigenvalues of the commutator g. = g t g 2 g l l g ~ I by using the fact that the normal variational equation (2.39) happens to be in a class of Lam6 equations [ 15]

d2y (El sn2(k: a t ) + E2)y 0. (2.4|) dt 2

where Ei and E2 are constants, and the eigenvalues 0. of the commutator g. = g lg2g l l g~! are known [ 15| to be determined by the indicial equation

A 2 -- A - (~2k2)E| = 0, 0. = exp(2~iA) (2.42)

with the singular point (pole) r located at the center of L.t~. paraiielogram:

T~ + T2 2K(k) + iK'(k) T (2.43)

2

If we apply the indicial equation (2.42) to the normal variational equations (2.39) of the FPU lattices, the exptments of the eigenvalues of g, are given by

z~ 2 - z~ - i z - ~ s m ~ ~ = O. (2 .44)

Noting

y4/et2k 2 = 1 (2.45)

from (2.22), we finally obtain the eigenvalues of the commutator as

for 1 < j ( # ½(n + 1)) < n, because there is only one pole singularity inside the parallelogram formed by

counterclockwise closed loop of the monodromy group glg2g-(Ig~ I. Even for more general nonlinear lauices given by potential functions (2.5) of degree 2m we can also compute the phase factors. Let us consider singuiax solutions ~(t) and ~j(t) whose singularities are located at t = r in the complex time plane as follows:

¢~(t) = C'(t - r)/~, p < 0, ~j(t) = C"(t - r ) v~. (2.47~

124 K. Umeno/Physica D 94 (1996t 116-134

where C' is a particular constant. C" is an arbitrary constam. Thus, we can easily obtain the |brmulas for the relations

--I f l(f l-- !) , _ m ( 2 m - l)sin2 ( ja" ) f l = m - !" (C')2m-2-- 2~m v 7 - t ~ / - 2 ( m - !) 2 ~ = 0 (2.48)

by the equations of motion. With the use of the fact that the phase factor is obtained by means of the analytic continuation along the closed loop around the singular point tvJ. we obtain the phase factors as

exp(2:ri~/) = exp {2~i~ ( I 4-~/I + '-m(2m - 1) sin2 ( ~ ) ) J ( m - ! )2

{ ~ _ m ( 2 m - I ) s i n 2 ( jzr ) 1 = - e x p -t-~i I + ~ ( m - I ) 2 ~ , (2.49)

where 2m is the degree of the potential polynomial of v (X) ;,n (2.5). it is easy to confirm that in the case that 2m = 4, formula (2.49) recovers the eigenvalues (2.46) of the monodromy matrices obtained from the Jacobi elliptic function. However, in the other case that 2m > 4, we cannot compute the eigenvalues of monodromy matrices only from the phase factors in general, because the normal variational equations do not belong to a class of Lamd equations. To summarize, we can say that it is essential to perform the singularity analysis towards non-homogeneous nonlinear lattices using the explicit eigenvalues of the monodromy matrices that the special solutions can he written in terms of e|lipfic functions, and the normal variational equations are in a type of Lamd equations.

3. Non-integrabiliO, theorem

3. I. Non-integrabili~." proof in the integrable limit

The main purpose of this section is to give a theorem stating the non-integrability by using the monodromy matrices o b ~ n e d in Section 2. We consider monodromy matrices g. If the eigenvalues {oj, o 71 . . . . . an, o n I } of monodromy matrices g do not .satisfy the relation

II I', In cr l a 2 - - - - a A = 1 (3.1)

for any set of integers {lj . . . . . I,,} except the trivial case !1 = 12 . . . . . in = O. we c~.ll the monodromy mauices g non-resonant. It is already known that the existence of a non-resonant monodromy matrix is a basic assumption in order to perform Ziglin's analysis [191. Moreover. if there are straight-line sohaions such as (2. ! !) whose monodromy matrices are non-resonant and if the variational equations can be diagonalized into a decoupled form like (2.33). Ziglin's theorem can he generalized to Yoshida's theorem for Hamiltonian systems composed of kinetic energy terms and potential energy terms [16].

Yoshida's theorem asserts the following in terms of two different monodromy matrices {g,,. gb}: suppose that ~here exists an additional complex analytic integral, which is holomorphic along the solution (2.9). and that one of fi~e mon,xlromy mau'/ces ga is non-resonant. Then it is necessary that one of the two cases, namely, (I) gb(;v) must preserve the eigendirection of go O.j). i.e.. go (~-.i) must commute with gb(~.j).

d | ) gb(~.j)must permute the eigendirection of g,, O.j). i.e.. gb(Xj)is written by [ 0/~ ° ~ ] i n the base of go having

the eigenvalues i and - i

K. Umeno/Physica D 94 (1996) 116-134 J25

for some suffix j , at least occurs for any other monodromy matrix gb represented in the basis of go- As for ~ e non-resonance condition of the monodromy matrix g , = gug2g~tg2 !. the following lemma is a'~ready known:

Lemma I [I 1,12]. The n quantities 1~/25 - 2 4 c o s ( j l r / ( n + i)) [J = I . . . . . n} are rationally independent.

We remark that the rational independency of the set {,~'~5 - 24 cos( j Tr / ( n + ! )) I J = ! . . . . . n} guaxantees that the commutator g , ----- gl g2g; ' lg21 whose eigenvalues are given in Eq. (2.46) is non-resonant.

For connection with the algebraic number theory on the cyclotomicfield Q(exp (~i / (n + !))) over O~ ~ [ | 2~ Thus, we can regard that the non-resonance hypothesis ho!ds for ~ c monodromy matrix g, . Now. using Yoshida's theorem and hh¢ variational analysis in Section 2, we obtain a theorem which proves the non-integrability of~m FPU

lattice foran arbitrary set o f ~ e system parameters [n, / t2. ~t4 I n (> 3) odd. tt2 > O, ~t4 > O} as follows:

Theorem I. An FPU lattice (2.6) which is characterized by an arbitrary set of the system parameters {n./z2./~:~ ~ n > 3, n odd, g2 > O, g4 > 0} has no analytic first integrals besides the Hamiltonian itself for fully small energy ~ ( ~ O).

Proof. From the non-resonance hypothesis on g , , we can apply the Yoshida theorem [16] to the FPU kaiticeso According to the above argument, if we prove that at least one monodromy matrix g~ ¢ {gt, g2} does not have [[~

following properties: Ca) gs (~.j) preserves the eigendirection of g , (~.j) arid (b) gs (£j) permutes the eigeadirecti¢r: of g , (~.j) (the eigenvalues of gs (£j) are i, - i ) , at once for any suffix j (I < j ( ~ ~(n + I)) < n) of gsO.j), then the FPU lattices [31 are concluded go have no other analytic conserved quantities besides the Hamiitonian itself, i.e., the as~rt ion of the present theorem holds. In the following, we will show that gl (~-)) for any j ( ~ ½ in + I )) has neither property (a) nor property {b).

When we take the limit E --* O, the relations

Y4 = ~4 C2 . . . . . Y2m = g2mC 2m-2 ~ O, ot --* VVfl*2, k ~ 0 (3.2)

hold and we can compute the periods of the monodromy matrices g b g2 in the limit as

I I r~ - , ~ , ~ , ~ - , ~ + io~ (3.3)

by formula (2.27). To compute the monodromy matrix g iOq) in this limit, we rewrite the variational equations

(2.39) in terms of the modulus of the elliptic integral k as

2k 2 ~ t t ) ) ~ = O . j = 1 . . . . n (3.4) ~',j + 4 , 2 sin2 ( ~ ) ( i + . ~ _ . ~ c n 2,~.11¢, .

by virtue of Eqs. (2.15) and (2.22). For arbitrary k ~ [0, I[, we have a fundamental system of s o l u ~ as

{¢~(k; t), ~jb(k: t)} of (3.4) satisfying

d a I tT(,;o) = l, ~ ; (*:'l,=o, =~7 (*:°) =o. (3.5)

¢~(,:o)=o. ~ ( , : , ) , _ o = ~ ( k : o ) = ! .

Because cn2(k: a t ) in Eqs. (3.4) has the following Taylor expansions [ I [ at t¢ _= k 2 = O:

c¢~ I • 2 [ o t ~ \ 2/Ot~r \ cn2 (k :a t ) = c o s 2 x ,(~-~'t'] - ~tcsm t~-Kt)COS t ~ t ) + . . . (3.6)

126 K. Ur~nolPhysica D 94 ( 19961116-134

and analytic in Jc at ic = O, the fundamental system of solutions {~a ~b} are also analytic in Ic at x = 0 [6] as

~71 ' : , ) = ~:o( , ) + ~,(,)~ + ~7.,(,) ,~" + - - - . 13.7~ = + + ' " .

where the unperturbed parts {*~.o, ~0} are obtained by

j~r

~ o ( t ) = I sin (2sin (jTr/2(n + I)) x/-~2t) • 2,¢~_sin( jTr/2(n + I))

from the initial conditions (3.5). The monodromy matrix gl O.j) is given by

[ ~,a(k: T.) ~ ( k : Ti , ]

g,(Zj) = L~7 (k: Ti) ~b(k: T i ) J " (3.9)

and with the use of the fact that the fundamental system of solutions (3.7) are analytic in r at r = 0, we show that gt (Z j) for j = i . . . . . n have also the Taylor expansions as

gl (Z)) = gl.0~,Xj) + gl.I (Zi)K + gl.2(~.j)K 2 + - - - . (3.10)

Thus now, when we consider the low energy limit (6 --~ 0, r ~ 0). we obtain the formula

ja" sin (x/'2n sin ( j~ "~1 cos (V/'2n " sin ( ~ ) ) I 2sin(j~t/2(n + l ) ) v f ~ \ 2 (n + l ) ] ] J

• " j Y r

g l ( , ) -2 sin ( ~ ) ~ / - V - 2 sin (,/~a" sin ( ~ ) ) cos(~/2a • sin ( ~ - ~ n - ~ ) a " ) /

(3.11)

This means that the eigenvalues of gt (Zj) tend to

j : r ( - i , ' r - f2 sin j~r

and gi (Zj) for any j (q: ~ (n + I )) does not have property (b). Nov,. assume that g2(Zj) for some j (#- / (n + l)) has property (a). Then if gl(kj) as well as g2(~.j) has also

property (aL we have

g.O,j ) = gl (kj )g2(Aj ) g l i i~.j )g21(3,j ) = id, 13.13)

where i d denotes the 2 x 2 identity matrix. Hem relation (3.13) means that g t (~.j) and g2(kj) commute each other and clearly contradicts the non-resonance hypothesis of g.. Consider the other case where gl (~.j) has property (a) and g2(~q ) does not have property (a). However, in the representation of g tO.j) and g2(Zj) in the basis of g.(Zj) as

g l ( ~ j l = I / l Z • g 2 l ~ . j ) = C

the relation

[ ad - iL2bc ab(pL 2 -- I) q g.(3.j) = gl(Zj)g2(Zj}gl l (3 . j )g21(Xj) = L c d ( i / e 2 _ !) ad - bc /~ 2 J (3.15)

K. Umeno/Physica D ~4 (1996) 116-134 [27

must be satisfied. Since g,(Xj) is assumed to have a diagonal representation as g,(Xj) = diog|aj, a/-~| and from relation (3.15). we obtain

a=O, d = O , b c = - I , (3.16)

when g,(Xj) # id; g20-j) must have property (b). Therefore, in the basis of g,O.j), we have

Relations (3.17) result in

[ar(~ ~rJ -1o ] =g.(.~.j) -.~ gl(3,j)g2(~,j)gll(3.j)g21(~.j)=g2(3.j), (3,|8)

where aj = -- exp{Jri~/25 -- 24 cos(j~/(n + I))}. However, relation (3.18) causes again a conwadiction with the fact that the eigenvalues of g~O.j) approach to

lexp (i~2~t2 sin ( ~ ) ) . exp (-iTr2~Cr2 sin ( ~ ) 1 ] (3.19)

. . q ~ .

in the limit ¢ --, 0 and there is a difference between the elgenvalues of g, (~.)) and the elgenvalues of g[ (~)) as win be shown in Appendix A.

We have seen that gt()~j) for any j (-~ ~(n + i)) has neither property Ca) nor property (b). Now the theorem holds.

4. Classification of non-honmgeneous nonlinear lattices

Here we consider the classification of these non-integrable nonlinear lattices via the degree of n o n - i ~ l i t y , We define the degree of non-integrability as follows.

Definition !. We write a relation

{H,~} ~ {H',~.'! (4.|)

if and only if the Hamiltonian systems with n degrees of freedom H and H' have the same number of additional analytic integrals which are functionally independent together with the Hamiitoc.ians H and H', respectively. I-left ¢ (6') denotes the total energy of the Hamiltonian systems H (H').

If we associate the positive integer

~? = 2r • 3 n-r - ! (4.2)

with a Hamiitonian system H with n degrees of freedom which has r additi~hal analytic integrals which ~e functionally independent together with the Hamiitonian H, we can classify the Hamiltonian systems via the of non-integrability as follows:

{H,e} ~ {H',6'} ¢==~ 0({H,¢}) = o({H',e'}). (4.3)

1 2 8 K. Umeno/Phys ica D 94 (1996) 116-134

For a non-homogeneous nonlinear lattice

i t l -~ m n

= t=! 9-~ (qi - I - qi) z~ (4.4)

of degree 2m. consider tim following transformation :~1 from a non-homogeneous nonlinear lattice to another non- homogeneous nonlinear lattice.

Defln#ion 2. We call a transformation :~1 from HI,, ~4 zz~ (q, P; t) to Hi,, ;"4 ~' (q ' ' p ' : t ' ) a homogenizer if . . . . . . . 2 . . . . . . 2 m

the relations

! 1 t '= att. q'= ~ q - n'= ~-;;-~r-~P.

! ! ! . . . . ~2,n = ~2m.

(4.5)

are ~atisfied, where ct is a real quantity larger than unity (I < at < oc).

R e m a r k !. T h e transformation :~ in F..qs. (4.5) can he regarded as a generalization of the scaling transformation for homogeneous nonlinear lattices Hu2=o.....t,~_z=o.uz, . . Moreover, the equations of motion itself remain invariant under the transformation :11, while a Hamiitonian is changed according to the variations of coupling constants

{t~zr t 1 < }, < m}. Thus, we can say that :~I preserves the degree of non-integrability as

{:~H. : ~ } ~ {H. E}. (4.6)

Remark 2. The operation :~ corresponds to a kind of renorrnalization group along the time axis from the viewpoint of the coarse graining in statistical physics and it is easy to see that the fixed point of:~l is the homogeneous nonlinear lattice H*. i.e..

H* = :~H*. (4.7)

where

H* = lim :}~tH = I /22m ~'-~, p~ + - - i~=l~qi-I -- qi) 2m (4.8) 5 l--* ~c - i=! 2 m

and 9t t denotes i iterations of the operation :~l. This is the reason why the :~l is called a homogenizer here.

Remark 3. Under the transformation :R, the total energy ~ is changed into ~' by the formula

1 ~' = :ll~ -- at2m/, ._l ,~. (4.9)

By Remarks i - 3 we can conclude that

{ H,, I/~,~ ,.._.l,~. ( i /a t4 l ),~ } ~ { H~.. .~.p ~: } (4 .10)

foran arbitrary FPU lattice {Hu,_4, ~. ~}. If we take the limit ot / --* oc preserving the relation ¢/at41 ~ O(1), LHS of

(4.1 O) approaches a homogeneous nonlinear lattice of degree four, which means that the degree of non-integrability of a homogeneous nonlinear lattice is the same as the degree of non-integrability of an FPU lattice in the high energy

K. Umeno / Physic a D 94 ( 1996 ) I 16-134 |29

limit ¢ --, ~ . In relation to the non-imegrability of the homogeneous nonlinear lattice, there is a known resuR by Yoshida [ 17] stating as follows:

Consider an n number of distinct exponents

8m(2m-- I ) s in2 ( j~r ) Aj - - i + ] - ~ _ - - ] ~ . ~ ( f o r j = 1 . . . . . n) , ( 4 . | [ ~

or the n representative of the Kowalevski exponents [ 18] of the homogeneous nonlinear lattice of degree 2,'n. If the n quantities {Aj } in (4. ! !) are rationally independent, then the homogeneous nonlinear lattices have no analytic lust integrals except the Hamiltonian itself for 2m (_> 4) and for odd n (> 3).

Note that the n quantities (4. I 1) equal the phase factors (2.49) of the eigenvalues of the c o m m ~ g , Thus, the non-resonauce hypothesis proving the non-integrability in the low energy limit • ---, 0 induces also the n t ~ integrability of nonlinear lattices with a non-homogeneons potential function in the high energy limit e ---, o¢. The reason for this uniqueness of the non-resonance hypothesis lies in the fact that after a conformal mapping z =

- I - I O4(t), the closed loop of the commutator glg2gi g2 becomes a homotopy equivalent loop which correspomis to Yoshida's construction of the monodromy matrix G = (GoGI)16 giving the same non-resonauce hypothesis in

terms of Kowalevski exponents. Here, Go corresponds to a counterclockwise closed loop around the singulatRy z = 0 and G I corresponds to another counterclockwise closed loop around the singularity z = I in the complex z plane, where z satisfies the following Gauss hypergeometric equation:

d2~; [ _ 3 5 ] d ~ ' + 3 y 4 s i n 2 ( ~ ~ ; = 0 (4. |2) Z ( I - z ) - ~ Z 2 + L ¢ - 4 z j dz ~" \ ~ , n - e t ~ /

into which the conformal mapping z = 4~ 4 transforms the variational equation (2.33) with the condition y2 = Oaml Go, G I are the two fundamental monodromy groups of the Gauss hypergeometric equation. See also Fig. 3.

So far, we have succeeded in checking the non-integrability of the FPU lattices both in the low energy limit and

in the high energy limit from a single non-resonance hypothesis. However, there still remain some problems about

the integrability of the FPU lattices for a finite energy 0 < ~ < ~x~. To consider the problem, we introduce a real parameter

Z -- - - > 0. (4 . |3) (n + l )p. 2

where Z is found to he a dimensionless parameter, because

dimension of Z = [L2/MT2] [ i /MT2L 2] = [ I I (4.14) |ll [I/M2T 4]

by the standard unit system as [M] = [kg], [TI = [see], [L] = [ml. We remark here that Z is invariant under the transformation :R preserving the degree of non-integrability. This means that it is sufficient for us to classify the degree of non-integrability of the FPU lattices by using this dimensionless parameter Z- Furthemm~, with the use of the parameter Z, we can replace the formula of the modulus of the elliptic function k in F,q. (2.25) by

k=~ ,n+4x (4.15)

We note here that k ~ 0 if and only if Z --" 0 and this limit can be realized for each set of {~,/~2,/.L410 < e, /~2,/~4 < o~} when we take n (odd) a fully great number. If we recall that in Theorem 1 it is essential for proving

130 K. Umeno/Physica D 94 (1996) ! 16-134

' / /: _/_:, e- C~: ;r) g = ~ o I ¢ / - ? , / , - I ~ -,~'=o ~'=

e l / , /" ,

L =o K ;_ 0(-i . . i t

( f = -f' ) Z = ~4

F/g. 3. The arrow means the transformation from the closed loop defining the commutator glg2gflg21 in the complex time plane into the closed loop (GoGI)i6 in the complex z plane by the mapping z = ¢)4.

the non-integrability of the FPU lattices to take lhe limit k ---* 0 (X -* 0) so that the eigenvalues o f g l (kj) approach

quantities in Eq. (3.12), we have the following theorem.

Theorem 2. An FPU lauice (2.6) characterized by an arbitrary set of the system parameters {6, g2, P4 I 0 < ~ <

~¢, 0 < P2 < or , 0 < /~4 < e¢} has no additional analytic integrals of motion besides the Hamiltonian itself for a

fully great number o f degrees of freedom n (n odd).

This result shows that we need a fully great number n of degrees of freedom to guarantee the strongest non-

integrability o ( { H ~ u . ~ }) = 3 n - ! of each FPU lattice HFPU with the systems parameters {/.t2, ~4 I O < ~2 < oo,

0 < P4 < 0¢} for 0 < e < 0¢ and a problem remains still open as to whether we can prove the non-integrability of

each FPU lattices with the system parameters {t~2. P4, n I 0 < p2 < oo, 0 < p4 < oo, n odd, n >_ 3} for a given

finite energy ~ (0 < ¢ < c¢). Clearly, Theorems 1 and 2 have the reciprocal conditions in terms of the system

parameters {¢. p2./~4, n} and we can unify them into a single theorem as follows.

K. Umeno/Physica D 94 (1996) 116--134 I3|

Theorem 3. An FPU lattice (2.6) characterized by the system parameters {~, #2, #4, n J 0 < ~ < ~ . 0 < #2 < ~ , 0 < #4 < c~. n > 3, n odd} has no additional analytic integrals of motion besides the Hamiltonian itself when the condition

f#4 X = ~ ~ 0 (4.16~

(n + I)#22

is satisfied.

$. Summary and discussion

We got the non-integrability proof of the FPU lattices by tracing the following steps. In Section 2, we have shown that the singularity analysis can be performed towards the non-homogeneous nonlinear lattices by finding the special straight-line solutions in terms of elliptic functions. Especially, it is shown there tha,. the normal variational equations can be decoupled into separated equations and each of them happens to be a ~ equation and cousequemly we can compute the eigenvalnes of the monodromy matrices associated with the counterclockwise loop a~ng the period cell of the special solutions. In Section 3, with the use of Yoshida's theorem whose validity is guaranteed when the normal variational equations can be decoupled, we can prove the non-existence of the additional analytic conserved quantities besides the Hamiltonian itself for the FPU lattices in the low energy limit. In Section 4, we have considered the classification of the FPU lattices via the degree of non-integrability. There, by ~ n g a transformation .~l from a non-homogeneous nonlinear lattice into another non-homogeneous nonlinear lattice, which preserves the equations of motion, we have shown that the degree of non-integrability of each FPU lattice in the high energy limit is the same as that of the homogeneous nonlinear lattice whose non-integrability is known to be proven by the analysis using Kowalevski exponents. In case of the FPU lattices with arbitrary energy in 0 < ~ < ~ , we have a theorem showing the non-integrability for a fully great number of degrees of freedom by httreducmg a dimensionless parameter which characterizes the normal variational equations. This theorem (Theorem 2) and Theorem ! have the reciprocal sufficient conditions for the non-integrability of the ~ lattices and finally dk-'y a~e unified into a single theorem (Theorem 3) by the dimensionless parameter Z.

There remains an open problem to be solved: Can we have a theo~m about the non-integrability of tno~ general nonlinear lattices of degree 2m > 4? More specifically speaking, can we have a theorem asserting the non- integrability of the non-homogeneous nonlinear lattices of degree 2m > 4 such as Theorem~ ! and 2? The present neck is in a fact that the normal variational equations along the special solutions of hyper-ellipticfunctions do _.riot

belong to a class of Lam~ equations, though they can always be decoupled into n - ! separated variational equations, and we cannot compute the eigenvalues of the monodromy matrices associated with these variational equations in general and therefore we cannot perform the Ziglin analysis completely. However, the problem on the degree of non-integrability for more general non-homogeneous nonlinear lattices remains an interesting open question to be solved, because the present analysis on the non-integrability of the FPU lattices, toget_l~er with. Yosb,.'~,~'s ~ , , - - g ~ about the homogeneous nonlinear lattice in terms of Kowalevski exponents, strongly suggests universality about the non-integrability of more general non-homogenoous nonlinear lattices in Eq. (4.4).

Acknowledgements

The majority of this work was carried out to complete the doctoral dissertation at Department of Physics, University of Tokyo. 1 would like to thank Prof. Masuo Suzuki, Prof. Miki Wadati and Prof. Hanto Yoshida for var.uable discussions. This work was supported by the Special Re,archer 's Program towards Basic Science at the RIKEN and the Program of the Complex Systems I1 at the Internationai institute for Advanced Study (llAS-Kyogo). ! would like to thank Prof. Shun-ichi Amari for his continual encouragement.

132 K. Umeno/Physica D 94 (1996J 116--134

Apl~mdix A

In this appendix, we show the difference between the eigenvalues of g. (~.j) and the eigenvalues of g2 O-j) for any j (# !,(n + ! )). We prove this by reductio ad absurdum. First, we assume that the equality

j:r exp (-iJr2,d~ sin ( ~ ) ) Spec(g~O.j )) -- {exp ( in '2 ,~ sin ( ~ ) ) , }

= { - exp (~ri~/25 - 24c°s n ~ l ) , - exp (-~ri~/25 - 24cos n ~ l ) }

= Spec(g.Oq)) (A.I)

would hold for some j (# ½ (n + I)). Our purpose here is to show that relation (A. I) is false. By (A. 1), one of the following equalities

1 j~r ~ 2 5 - 2 4 c o s ( n ~ l ) + 1 + 2m + = 2v~2 sin ( ~ ) (A.2)

~/ j~" jTr 2 5 - 24cos ( ~ - ~ ) + I +2m_ = - 2 v r 2 sin ( ~ ) (A.3)

holds for m +. m_ ~ Z and j :# ½ (n + I ) Because of the inequalities

1 < 25 -24cos ~ <7. 0 < 2 , f 2 s i n ( j:r ~ <2,f2 , (A.4) \ 2 ( n + 1)]

it is sufficient to consider equalities (A.2) and (A.3) for m+ ~ {-3. -2 , - !, 0} and m_ ~ {-5, -4, -3 , -2}. If we re,a-ire sin(j:r/2(n + 1 )) as X, Eqs. (A.2) and (A.3) are changed into the equations

40X~ + (i + 2m+)4v'2~X+ + I - ( ! +2m+) 2 = 0. (A.5)

and

40X 2 - - (! + 2m_)4q'2X_ + i -- (1 +2m_) z =0 . (A.6)

respectively. By considering X (= X+. X_ > 0). we can easily obtain a series of the solutions as follows:

X+(m+ = O) = O.

x + ( ~ + = - l , = ~ov2..

X+(m+ = - 2 ) = l/v'2,.

X+(m+ -3) = ~ ( 5 J , ~ + ~ ) > ! • = ~ ( 7 + 2 , / ~ ) = ~ > i . i X_{m_ = - 2 ) = 3,/2,

x _ ( , , , _ = - 3 ) = ; c ¢ - 5 - , / 2 + 2,/N-6).

X__(m_ = - 4 ) = !/4~2.

X_(m_ = -5) = ~(-9-f22 + v'~-2).

(A.7)

K. Umeno/Physica D 94 (1996) i ¢6-134 [33

Furthermore, because of the following relations 0 < X+ < I, X± :fi i/,¢t2 for j :fi ½(n + !), we can ~ a r d the .solutions X+(m+ = 0), X+(m+ = - 2 ) , X+(m+ = - 3 ) , and X _ ( m _ = -4 ) . For the solutions X+(m+ =

- I ) , X _ ( m _ = - 2 ) , we have the equalities as

cos = I - 2 ( X ÷ ( m + = - l ) ) 2 = ~ and cos = l - 2 ( X _ ( m _ = - 2 ) ) 2 = 2-'5"

~A.8)

respectively. However, these relations (A.8) contradict with a fact [ ! ! ] that

(1- ()- 0 = o r 2 o r " ( A . 9 ) c o s ~ ~ Q ~ c o s ~ 2

Thus, we can also discard the solutions X+ (m + = - ! ) and X _ (m_ = - 2). Let us consider the solutions X_ (m_ =

- 3 ) and X _ ( m _ = -5 ) . If we rewrite e x p ( j z r / n + i) as ( , the relation

25( 4 + 70( 3 - 46( 2 + 70( + 25 = 0 (A.|O)

is satisfied for the solution X_ (m_ = - 3 ) , while the relation

25( 4 + 462( 3 + 626( 2 + 462( + 25 = 0 (A.| l )

is satisfied for the solution X_ (m- = - 5 ) . Since ¢ is one of 2(n + ! )th root of unity and 2(n + l)(mod 4) = 0, the

minimal polynomial P ( ( ) with coefficients of rational integers which has a solution ( for P ( ( ) = 0 must be otto of the cyclotomic polynomials ~ l ( Y ) E Z,[Y]:

~(I) q~l(Y) = I - [ ( Y - ( i ) , (A.12)

i=l

where 2(n + l)(mod l) = 0, {(i } is a set of all primitive lth roots of unity and ~0(I) is the number of positive integers which are less than or equal to I and relatively prime to I and this function 9 is called Euler's 9-fumiott . A~

cyclotomic polynomials q~t (Y) are known to be irreducible over Q and we can check that cyclotomic polynomials

OI(Y) whose degrees are less than or equal to four are restricted to the case [ 12] that

R = {¢~t(Y)12 < Deg ~t(Y) < 4}

= {q~3(Y) = y2 + y + I, q54(g) = y2 + 1, @5(Y) = y4 + y3 + y2 + y + 1, ~6(Y) = y2 _ y + l,

~s(Y) = y4 + I, ~I2(Y) = y4 _ y2 + !}. (A.13)

Using this fact, we can say that a cyclotomic polynomial q~t(Y) E ~ must divide either the polynomial

PI(Y) -- 25Y 4 + 70y3 - 46Y 2 + 70Y + 25 (A.!4)

in Eq. (A.10) or the polynomial

P2(Y) ----- 25Y 4 + 462Y 3 + 626Y 2 + 462Y + 25 (A.|5)

in Eq. (A. I 1). It is easy to check that this is impossible. Thus now, we can conclude that assumption (A. 1) is false; i.e., we have shown that for any j(=/: ~(n + 1)),

Specfg~0.j)) :/: Specfg,(;,.j)). (A. |6)

134

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