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A new hierarchy of (1 + 1)-dimensional soliton equations and its quasi-periodic solutions
Chaos, Solitons and Fractals 35 (2008) 692–704 www.elsevier.com/locate/chaos
A new hierarchy of (1 + 1)-dimensional soliton equations and its quasi-periodic solutions Shan Xue
a,b
, Dianlou Du
a,*
a b
Department of Mathematics, Zhengzhou University, Zhengzhou Henan 450052, PR China Henan Communication Vocational Technology College, Zhengzhou Henan 450000, PR China Accepted 20 June 2007
Abstract A new spectral problem is proposed, from which a hierarchy of (1 + 1)-dimensional soliton equations is derived. With the help of nonlinearization approach, the soliton systems in the hierarchy are decomposed into two new compatible Hamiltonian systems of ordinary differential equations. The generating function flow method is used to prove the involutivity and the functional independence of the conserved integrals. The Abel–Jacobi coordinates are introduced to straighten out the associated flows. Using the Riemann–Jacobi inversion technique, the explicit quasi-periodic solutions for the (1 + 1)-dimensional soliton equations are obtained. Ó 2007 Published by Elsevier Ltd.
1. Introduction The nonlinearization approach [1–3] is a powerful tool for revealing the links between soliton equations and finitedimensional integrable systems. There are at least two important applications of the nonlinearization approach. Firstly, a considerable number of new finite-dimensional integrable systems can be generated from the known soliton hierarchies. Secondly, it provides a way of solving the soliton equations by separating the spatial and temporal variables. Explicit solutions of the soliton equations, which include soliton solutions, periodic solutions, quasi-periodic solutions and other ones, can be obtained through solving the compatible systems of ordinary differential equations. There are several systematic methods to construct quasi-periodic solutions of (1 + 1)-dimensional soliton equations, for instance, the algebro-geometric approach [4,6–13], the alternative elementary algebraic approach [5], the nonlinearization of Lax pairs [1–3] and others. Quasi-periodic solutions for many (1 + 1)-dimensional soliton equations have been derived such as the KdV equation, nonlinear Schro¨dinger equation, mKdV equation, sine-Gordon equation and Toda lattice equation, etc. In this paper, our main aim is to develop further the nonlinearization approach of Lax pairs to construct quasi-periodic solutions of the (1 + 1)-dimensional soliton equations. The present paper is organized as follows. In Section 2, we shall derive a hierarchy of (1 + 1)-dimensional soliton equations associated with the spectral problem (1) by using the Lenard gradient sequence. In Section 3, the spectral problem (1) is nonlinearized into a new finite-dimensional *
Corresponding author. E-mail address: [email protected] (D. Du).
0960-0779/$ - see front matter Ó 2007 Published by Elsevier Ltd. doi:10.1016/j.chaos.2007.06.095
S. Xue, D. Du / Chaos, Solitons and Fractals 35 (2008) 692–704
693
Hamiltonian system under the constraint between the potentials and eigenfunctions. And the integrals of motion for the finite-dimensional Hamiltonian system are obtained and proved to be in involution in pairs. In Section 4, we introduce the elliptic coordinates and quasi-Abel–Jacobi coordinates, by which the functional independence of the integrals of motion is proved. In Section 5, a new set of polynomial integrals is introduced. In Section 6, by the new set of polynomial integrals, we arrive at the decomposition of the soliton hierarchy. In Section 7, a hyperelliptic Riemann surface of genus g = N and the Abel–Jacobi coordinates are defined to straighten out the associated flows. In the last part, Riemann–Jacobi inversion problem is discussed, from which the quasi-periodic solutions of the (1 + 1)-dimensional soliton equations are expressed explicitly in terms of the Riemann theta functions.
2. (1 + 1)-Dimensional soliton hierarchy Firstly, we introduce a new spectral problem k þ u v ux ¼ U u; U ¼ ; vk þ av k u
ð1Þ
where k, a 2 R, k 5 1, and u, v are potentials. Based on the zero-curvature equation, we have the fundamental identity V x ½U ; V ¼ rfðK kJ Þcg; T
ð2Þ 3
where c = (C, B, A) , r is a map: R ! slð2; RÞ defined by 0 1 C A B B C r@ B A ¼ ; C A A and (o = ox) V ¼
A C
B ; A
0 B K¼@
o þ 2u
0
0 v
o 2u vk þ av
2ðvk þ avÞ 2v o
1 C A;
0
2 B J ¼ @0 0
1 0 0 C 2 0 A: 0 0
ð3Þ
The Lenard gradient {gj} is defined recursively by Kgj1 ¼ Jgj ;
g1 ¼ ð0; 0; 1ÞT ;
jP0
ð4Þ
which means that (K kJ)gk = 0 for the generating function 1 X gk ¼ gj1 kj : j¼0
It is easy to see that det r(gk) = 1. The first few numbers of the Lenard gradient {gj} are 1 0 k a1 01 ðo þ 2uÞ vk þ av v þv 2 B C B C 1 g 0 ¼ @ v A; g 1 ¼ @ ð2u oÞv A; 2 1 k a v v þv 0 2 0 k a 1 kþ1 1 k a v þ v xx þ u v þ v x þ u2 þ u2x v 2 þa vk þ av 4 C B kþ1 C; g2 ¼ B A @ 14 ðvxx 2ux v 4uvx þ 4u2 vÞ ðv 2þaÞv vx v k a k a k a ðv þ Þx 4 v þ v þ uv v þ v 1 0 41 k va v þ v xxx þ 34 u vk þ av xx þ 32 u2 þ 34 ux 34 v vk þ av vk þ av x 8 B C B þ 14 uxx þ u3 þ 32 uux 32 uv vk þ av vk þ av C B C B 1 vxxx þ 3 uvxx þ 3 ux vx 3 u2 vx þ 3 vvx vk þ a þ 1 uxx v 3 ux uv C B C 4 v 4 2 g3 ¼ B 8 3 43 2 k4 a 2 C: B þu v 2 uv v þ v C B C B 3 u2 v 3 v2 vk þ a 3 uv þ 1 v vk þ a þ 1 vvk þ a C x @ 2 A 8 v 4 8 xx v 8 v xx 3 þ 4 uv 18 vx vk þ av x
ð5Þ
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Let n X
Gn ¼
gj1 knjþ1 þ gn
j¼0
with gn ¼ ðg1n ; g2n ; g3n þ dn ÞT , where dn ¼
kvk1 va2 2 1 1 g þ g : ðk þ 1Þvk nþ1 ðk þ 1Þvk nþ1
Let V(n) = r(Gn). Then ðnÞ ¼ rðU tn K gn Þ ¼ 0 U tn V ðnÞ x þ ½U ; V
is equivalent to a hierarchy of (1 + 1)-dimensional soliton equations ðu; vÞTtn ¼ X n ; where
0 Xn ¼
ð6Þ h
2 i1 a gnþ1 vkþ2 A: 2 og3nþ1 ðkþ1Þvk
g1 1 o nþ1 vk kþ1 @
þ
k
v
ð7Þ
The relation between Xn and gn+1 is X n ¼ PTgnþ1 ;
ð8Þ T
T
where P is the projective map (c1, c2, c3) ! (c1, c2) , and 1 0 1 o vk o v1k kvk1 va2 0 1 B 2 C: T ¼ @ 0 0 oA vk kþ1 0 0 0 Then corresponding to (5), we have ux X0 ¼ ; vx h 0 1 1 ux o 2k a2 vk1 ðln vÞxx þ 14 ðk 2 þ kÞðln vÞ2x þ avk1 þ 1k kþ1 2 B C C; 2 kþ1 kþ1 X1 ¼ B @ þðk þ 1Þu 2 v A 2 k1 k a kþ1 o v v ðln vÞ þ uv þ au x x ðkþ1Þvk 4 2 0 1 n 2 2 1 1 3 k 3a 3 k ðln vÞ o kðk 1Þv þ ðln vÞ þ ðk þ kÞv ðln vÞ k x xx x C 8 4v 4 B kþ1 v B C B 3a u ðln vÞ þ 1 ðk þ 1Þkðk 1Þvk ðln vÞ3 þ 3 ux ðvk þ a Þ C x xx 2 v 8 4 v x B C
B C uxx 3 3 k a k 3 k 2a þ 4 þ u 2 uvðv þ vÞ ðk þ 1Þv þ 2 uux ð1 kÞv þ v C: X2 ¼ B B C h B 2 C 2 3 a 1 k k kþ1 B ðkþ1Þvk o kþ1 C v v v ðv þ Þ þ kðk þ 1Þv ðln vÞ xx x x 8 8 v x 8 B C i @ A 2 3 k 2a 3 2 kþ1 3 kþ1 þ 4 uvx ðk 1Þv v þ 2 u ðv þ aÞ 8 ðv þ aÞ Specifically, if k = 0, (6) is reduced to the generalized coupled KdV hierarchy [12]. The first two members are ! u ux o u2x þ auv x þ u2 2v 2va ðln vÞxx u : ; ¼ ¼ v t1 vx 2o uðv þ aÞ v4x a2 ðln vÞx v t0 If k = 0 and a = 0, (6) is reduced to the coupled KdV hierarchy [3]. The first two members are 2uux þ u2xx v2x u ux u ; : ¼ ¼ 1 2ðuvÞx 2 vxx v t1 vx v t0 If k = 1, (6) is reduced to the generalized TD hierarchy [13]. The first two members are
ð9Þ
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u ux ; ¼ v t0 vx
u ¼ v t1
1 o auv2x þ 2u2 2 1 o uv2 þ au v
v2 2va2 ðln vÞxx þ v2vxx a2 ðln vÞx
695
! :
And if k = 0 and a = 0, (6) is reduced to the TD hierarchy [18]. The first two members are ! 2 u u ux o u2 v2 þ v4vxx ; ¼ ¼ : v t0 v t1 vx 2uvx þ ux v
3. Nonlinearization of the eigenvalue problem Consider N copies of Eq. (1) kj þ u v uj ; ujx ¼ vk þ av kj u
j ¼ 1; . . . ; N
ð10Þ
with uj = (pj, qj)T and distinct eigenvalues k = k1, . . ., kN. Then we have the following fundamental identity: Proposition 3.1 ðK kJ ÞGk ¼ J g1
N X
! rkj
j¼1
with Gk ¼ g1 k þ g0 þ
PN
rkj j¼1 kkj ;
rkj ¼ ðq2j ; p2j ; pj qj ÞT .
It is obvious ðK kJ ÞGk ¼ 0 () g1 þ cg1 ¼
N X
rkj ;
c 2 R:
ð11Þ
j¼1
For the simplicity, we choose c = 0. Then (11) gives the constraints between the potentials and eigenfunctions: 1
v ¼ ð2hp; qi aÞkþ1 ; kþ2 k 1 1 a k hq; qið2hp; qi aÞkþ1 þ hp; pið2hp; qi aÞkþ1 hp; pið2hp; qi aÞkþ1 ; u¼ kþ1 kþ1 kþ1
ð12Þ
where hÆ,Æi denotes the canonical inner product in RN ; p ¼ ðp1 ; . . . ; pN ÞT ; q ¼ ðq1 ; . . . ; qN ÞT . Substituting (12) into (10), we arrive at a finite-dimensional Hamiltonian system: p oH 0 =oq ; ¼ q x oH 0 =op
ð13Þ
where 1 H 0 ¼ hAp; qi vhq; qi þ hp; pihp; qiv1 ; 2
A ¼ diagðk1 ; . . . ; kN Þ:
From (2) and (11), we know that the Lax matrix X N N X k v j j þ ¼: l þ ; Vk ¼ k k k k kj vk þ av k j j¼1 j¼1 where j ¼ rðrkj Þ ¼
pj qj
p2j
q2j
pj qj
ð14Þ
! ;
satisfies Vx [U, V] = 0. Therefore, Fk = det Vk is invariant along the x flow and yields the integrals {Fl} as follows: 1
1
F k ¼ Qk ðq; qÞQk ðp; pÞ Q2k ðp; qÞ þ 2hp; qiQk ðp; pÞð2hp; qi aÞkþ1 þ 2kQk ðp; qÞ Qk ðq; qÞð2hp; qi aÞkþ1 2hp; qi k2 1 X ¼ k2 þ kl1 F l ; ð15Þ l¼0
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where Qk(n, g) = h(kE A)1n, gi, and F 0 ¼ 2H 0 ; X 1 Fl ¼ ½hAi q; qihAj p; pi hAi p; qihAj p; qi þ 2hAlþ1 p; qi þ 2hp; qihAl p; pið2hp; qi aÞkþ1 iþj¼l1
ð16Þ
1 kþ1
hAl q; qið2hp; qi aÞ : In order to prove the involutivity of {Fl}, we introduce the generating function method. Denoting the variable of Fk flow by tk, we get d u ¼ W ðk; kj Þuj ; dtk j
ð17Þ
where W ðk; kj Þ ¼
2 V k þ Dk ; k kj
Dk ¼ ðDijk Þ22 ;
21 D12 k ¼ Dk ¼ 0; 22 1 D11 k ¼ Dk ¼ 2Qk ðp; pÞv
4 2 Qk ðp; pÞhp; qivk2 Qk ðq; qÞvk 2 : kþ1 kþ1
Proposition 3.2 d V l ¼ ½W ðk; lÞ; V l 8k; l 2 C; dtk fF k ; F l g ¼ 0 8k; l 2 C; fF j ; F l g ¼ 0 8j; l ¼ 0; 1; 2; . . . ; :
ð18Þ ð19Þ ð20Þ
Proof. Noticing (17) and the identity d dll 2 Vl ¼ ½Dk ; ll þ ½V k ; lk ll þ ½W ðk; lÞ; V l ; dtk dtk kl a direct calculation gives (18), which implies the invariance of Fk along the tk flow: 0 = dFk/dtk = {Fl, Fk}. Substituting the expansion (15) into (19) gives (20) by comparing the same powers of k, l. h
4. Elliptic coordinates and functional independence 0 21 It is easy to see that each one of F k ; V 12 k and V k , as a rational function of k, has simple poles at kj s (1 6 j 6 N), since 2 the coefficients of (k kj) is zero in Fk. Hence we obtain that bðkÞ ; F k ¼ Qk ðq; qÞQk ðp; pÞ Q2k ðp; qÞ þ 2kQk ðp; qÞ þ 2hp; qiQk ðp; pÞv1 Qk ðq; qÞv1 2hp; qi k2 ¼ aðkÞ nðkÞ ; V 12 ð21Þ k ¼ v Qk ðp; pÞ ¼ v aðkÞ a a mðkÞ k k V 21 ; k ¼ v þ þ Qk ðq; qÞ ¼ v þ v v aðkÞ
where aðkÞ ¼
N Y ðk kj Þ;
bðkÞ ¼
j¼1
mðkÞ ¼
N Y j¼1
N þ2 Y
ðk bj Þ;
j¼1
ðk lj Þ;
nðkÞ ¼
N Y ðk mj Þ: j¼1
The roots {lj} and {mj} are defined as elliptic variables.
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697
Proposition 4.1. The elliptic coordinates satisfy the evolution equations along the tk flow: dl 1 mðkÞ qffiffiffiffiffiffiffiffiffiffiffi j ¼ ; 0 dt ðk l j ÞaðkÞm ðlj Þ 4 Rðlj Þ k
ð22Þ
1 dm nðkÞ pffiffiffiffiffiffiffiffiffiffiffi j ¼ ; 4 Rðmj Þ dtk ðk mj ÞaðkÞn0 ðlj Þ Q þ2 where RðkÞ ¼ aðkÞbðkÞ ¼ 2N j¼1 ðk aj Þ, with aj = kj (j = 1, . . . , N), aN+j = bj (j = 1, . . . , N + 2). Proof. Substituting k = lj, mj, respectively, in (21) , we have qffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi Rðlj Þ Rðmj Þ 11 11 ; V mj ¼ : V lj ¼ aðlj Þ aðmj Þ The second and third components of (18) can be written as d 12 4 11 12 11 ðV 11 V 12 V 12 V ¼ k V l Þ þ 2Dk V l ; dtk l kl k l d 21 4 11 21 21 ðV 21 V 11 V 11 V ¼ k V l Þ 2Dk V l : dtk l kl k l Let l = mj and l = lj, respectively. After some calculations we obtain (22). By means of the interpolation formula, we have (s = 1, . . . , g) g X j¼1
g X
lgs dlj kgs j qffiffiffiffiffiffiffiffiffiffiffi ; ¼ 4 Rðlj Þ dtk aðkÞ
j¼1
h
mgs dmj kgs pjffiffiffiffiffiffiffiffiffiffiffi : ¼ aðkÞ 4 Rðmj Þ dtk
For fixed k0, introduce the quasi-Abel–Jacobi coordinates: /~s ¼
g Z X j¼1
lj
k0
lgs pffiffiffiffiffiffiffiffiffiffi dl; 4 RðlÞ
w~s ¼
g Z X j¼1
mj k0
mgs pffiffiffiffiffiffiffiffiffi dm ðs ¼ 1; . . . ; gÞ: 4 RðmÞ
ð23Þ
Proposition 4.2. (straightening of the Fk flow). d/~s kgs ; ¼ dtk aðkÞ
dw~s kgs : ¼ dtk aðkÞ
ð24Þ
Corollary 4.3 (straightening of the Fl flow). Let tl be the variable of the Fl flow. Then 0 1 1 A1 AN 1 ! B 1 AN 2 C ~ d/ ~ ~ B C d/ d/ C; ; ;...; ¼B . . B .. C .. dt0 dt1 dtN 1 @ A
ð25Þ
1 ~ l ¼ dw=dt ~ l , where Aj’s are the coefficient in the expansion and d/=dt 1 X kN 1 ¼ ¼ 1 þ Aj kj ; aðkÞ ð1 k1 k1 Þ ð1 kN k1 Þ j¼1
ð26Þ
which could be represented through the power sums of kj, sm ¼ km1 þ þ kmN : 0 1 A1 ¼ s1 ;
Am ¼
X C 1B Bsm þ si Aj C A: m@ iþj¼m i;jP1
ð27Þ
698
S. Xue, D. Du / Chaos, Solitons and Fractals 35 (2008) 692–704
Proof. According to the definition of the Poisson bracket, we have 1 1 X X d/~s 1 ~ 1 d/~s ¼ f/~s ; F k g ¼ f/s ; F l g ¼ : lþ1 lþ1 dtk dtl l¼0 k l¼0 k
With the supplementary definition A0 = 1, Aj = 0 (j = 1, 2, 3, . . .), the comparison of the coefficients of kl1 in (25) ~ l ¼ ðAl ; Al1 ; . . . ; Algþ1 ÞT . h yields d/~s =dtl ¼ Alsþ1 , and d/=dt Proposition 4.4. F0, F1, . . . , FN1 given in (16) are functionally independent. Proof. We need only prove the linear independence of the differentials dF0, dF1, . . . , dFN1. Suppose Let H ¼ /~s in the formula [14]
PN 1 l¼0
cl dF l ¼ 0.
fH; F g ¼ x2 ðI dF ; I dH Þ: Then we have 0¼
N 1 X
cl x2 ðI dF l ; I d/~s Þ ¼
N 1 X
l¼0
cl /~s ; F l ¼
l¼0
N 1 X
cl
l¼0
d/~s : dtl
Hence c0 = c1 = cN1 = 0 since the coefficient determinant is equal to 1 by (25).
h
Theorem 4.5. The finite-dimensional Hamiltonian system (13) is completely integrable in the Liouville sense.
5. The polynomial integrals {Hl} We now introduce a new set of integrals {Hl} recursively by 1 H 0 ¼ F 0; 2
1 H 1 ¼ F 1; . . . ; 2
1 1 X Hl ¼ F l þ H iH j; 2 2 iþj¼l3
ð28Þ
which is put in the equivalent form k2 F k ¼ ðH k k2 Þ2
ð29Þ
with the help of the generating function Hk ¼
1 X
H l kl1 :
l¼0
Proposition 5.1. {Hl} satisfies the Liouville conditions of completely integrability: fH k ; H l g ¼ 0 8k; l 2 C; fH j ; H l g ¼ 0 8j; l ¼ 0; 1; 2 . . . ; (ii) H0, H1, . . . , HN1 are functionally independent. (i)
Proof. The involutivity of {Hl} is based on the equality 1 fH l ; H k g ¼ pffiffiffiffiffiffiffiffiffiffiffi fF l ; F k g ¼ 0: 4 F kF l From (29), we have k2 dF k ¼ 2ðH k k2 Þ dH k ;
ð30Þ
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0
1
dF 0
0
10
1
C B 0 C B C B C B 0 C B C¼B C B H 0 C B C B .. C B . A @
B dF B 1 B B dF 2 1B B 2B B dF 3 B . B . @ .
1
H N 4
dF N 1
0
1
0 .. .
0 .. .
1 .. .
..
H N 3
H N 2
H N 1
.
dH 0
1
CB dH CB 1 CB CB dH 2 CB CB CB dH 3 CB CB . CB . A@ .
C C C C C C: C C C C A
1
699
dH N 1
Thus the linear independence of dH0, . . . , dHN1 is equivalent to that of dF0, . . . , dFN1, which completes the proof of the functional independence of H0, . . . , HN1. h
6. Relation of Hl and Xl Acting with J1K upon g1 ¼ N X
PN
j¼1 rkj l
times and noting ker J ¼ fcg1 j8c 2 Cg, we arrive at
klj rkj ¼ glþ1 þ c1 gl2 þ þ cl g1
ð31Þ
j¼1
since each time there is an extra term cig1. Hence we have 0¼
N X
aðkj Þrkj ¼ gN þ1 þ cN 0 gN þ þ cNN g0 þ cN ;N þ1 g1 ;
ð32Þ
j¼1
where aðkÞ ¼
N Y
ðk kj Þ ¼
N X
j¼1
cN 0 ¼ a1 ;
aN l kl
ða0 ¼ 1Þ;
l¼0
cN 1 ¼ a2 ;
cNN ¼
N 1 X
aN l1 cl ;
cN ;N þ1 ¼
l¼1
cN ;N i ¼ aN iþ1 þ
N 1 X
aN l1 cli
N 1 X
aN l1 clþ1 ;
ð33Þ
l¼0
ði ¼ 1; . . . ; N 2Þ:
l¼iþ1
Proposition 6.1. Let (p(x), q(x))T be a solution of the finite-dimensional Hamiltonian system (13). Then 0 kþ2 1 k 1 a hq; qið2hp; qi aÞkþ1 þ kþ1 hp; pið2hp; qi aÞkþ1 kþ1 B C u 1 C k ¼ f ðp; qÞ ¼ B kþ1 hp; pið2hp; qi aÞkþ1 @ A v 1 kþ1 ð2hp; qi aÞ
ð34Þ
solves the stationary soliton equation X N þ cN 0 X N 1 þ þ cNN X 1 ¼ 0: Proof. Operating PT upon (32) yields (35).
ð35Þ h
Proposition 6.2. gk ¼
k Gk : k Hk 2
Proof. Define the generating function of the coefficients {cj} in (31) by ck ¼ k þ
1 X j¼1
cj kj1 :
ð36Þ
700
S. Xue, D. Du / Chaos, Solitons and Fractals 35 (2008) 692–704
Multiplied by kj1 and summed with respect to j from 0 to 1, (31) becomes G k ¼ ck g k : 2
Thus r(Gk) = ckr(gk) and F k ¼ c2k due to det r(gk) = 1, which yields ck ¼ k
H k k
and (36) by comparing with (29).
h
Proposition 6.3 f ðIrH k Þ ¼ kPTgk jM ; k 2 C; f ðIrH l Þ ¼ X l jM ; l ¼ 0; 1; 2 . . . :
ð37Þ
Proof. From (12) and (17), we get dv 4 2 hp; qiQk ðp; pÞvk1 þ Qk ðq; qÞvkþ1 ; ¼2 dtk kþ1 kþ1 du 1 f4a2 hp; piQk ðp; pÞv2k4 4ahq; qiQk ðp; pÞv2k2 þ 4Qk ðp; qÞ 4ahp; piQk ðq; qÞv2k2 ¼ dtk k þ 1 4hq; qiQk ðq; qÞv2k 8ahp; piQk ðp; pÞvk3 4a½kQk ðp; pÞ hp; pivk2 þ 8aQk ðp; qÞvk1 4½hq; qi kQk ðq; qÞvk þ 4khp; piQk ðp; pÞv2 2k½2hp; pi 2kQk ðp; pÞv1 4kQk ðp; qÞg: And a direct calculation gives
! o v1k 2hp; qiv1 þ Qk ðq; qÞ þ kvk1 va2 ½v Qk ðp; pÞ 1 1 d u PTGk ¼ ¼ : k þ 1 v1k o½2k þ 2Qk ðp; qÞ 2 dtk v Thus f ðIrF k Þ ¼ f
d dtk
p d d u ¼ ¼ 2PTGk : ðf ðp; qÞÞ ¼ dtk dtk v q
By (29) and (36), we have f ðIrH k Þ ¼
k2 f ðIrF k Þ ¼ kPTgk : 2ðk H k Þ 2
Hance we have the first part of (37). The second part is obtained by comparing the coefficients of the same power kl1. h Corollary 6.4. Let (p(x, tm), q(x,tm))T be a compatible solution of the H0 and Hm flow. Then (u, v)T = f(p, q) solves the (1 + 1)-dimensional soliton equation ðu; vÞTtm ¼ X m :
ð38Þ
7. The Abel–Jacobi coordinates The shape of (23) suggests the consideration of the holomorphic differential kgs dk ~ s ¼ pffiffiffiffiffiffiffiffiffiffi ; x 4 RðkÞ
s ¼ 1; 2; . . . ; g;
ð39Þ
on the hyperelliptic curve C: n2 16RðkÞ ¼ 0;
ð40Þ
with genus g = N since deg R = 2N + 2. For the same k, there are two points on different sheets of the Riemann surface C: pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi qþ ¼ k; 4 RðkÞ ; q ¼ k; 4 RðkÞ :
S. Xue, D. Du / Chaos, Solitons and Fractals 35 (2008) 692–704
701
At infinity, the affine equation (40) is transformed into ðz ¼ k1 ; ^ n ¼ zN þ1 nÞ: ^n2 16R ðzÞ ¼ 0 2N þ2
1
ð41Þ Q2N þ2
with R ðzÞ ¼ z Rðz Þ ¼ j¼1 ð1 aj zÞ. The two infinities are represented as 1l ¼ ðz ¼ 0; ^n ¼ ð1Þm 4Þ;
m ¼ 1; 2:
ð42Þ
Take the canonical basis of cycles on C: a1, . . . , ag; b1, . . . , bg. Let C = (Cjs) be the inverse of the periodic matrix (Asl): Z ~ s: C ¼ ðAsl Þ1 ; A ¼ ð43Þ x sl gg al
Then for the normalized holomorphic differential xj ¼
g X
~ x ¼ ðx1 ; . . . ; xg ÞT ¼ C x;
~ s; C js x
s¼1
we have Z
xj ¼ djl ;
al
Z
xj ¼ Bjl :
ð44Þ
bl
According to the Riemannian bilinear relation, the matrix B = (Bjl) is symmetric and has positive definite imaginary part, and is used to construct the Riemannian theta function of C: X pffiffiffiffiffiffiffi hðfÞ ¼ exp p 1ðhBz; zi þ 2hf; ziÞ; f 2 Cg : z2Z g
For fixed k0, the Abel–Jacobi coordinates are defined as g Z qðll Þ g Z qðml Þ X X x; w ¼ x: /¼ l¼1
qðk0 Þ
l¼1
P2N þ2 l kj . Then the coefficients of Lemma 7.1. Let sl ¼ j¼1 1 X 1 pffiffiffiffiffiffiffiffiffiffiffi ¼ K l zl R ðzÞ l¼0 satisfy the recursive formula [6]: K0 ¼ 1;
1 K1 ¼ s1 ; . . . ; 2
ð45Þ
qðk0 Þ
0 Kl ¼
ð46Þ 1
X C 1B B sl þ si Kj C @ A: 2l iþj¼l
ð47Þ
i;jP1
Let C1, . . . , Cg be the column vectors of C. Then by direct calculations, the coefficients in 1 X 1 pffiffiffiffiffiffiffiffiffiffiffi ðC 1 z þ C 2 z2 þ þ C g zg Þ ¼ Xl zl 4 R ðzÞ l¼1
ð48Þ
are Xl ¼ 14ðKl1 C 1 þ þ Klg C g Þ; l P 1, with additional defined Ks = 0 (s = 1, 2. . .). Proposition 7.2. The sl flow is straightened by the Abel–Jacobi coordinates: d/ ¼ 2Xlþ1 ; dsl
dw ¼ 2Xlþ1 : dsl
ð49Þ
Proof. By (21), and (29), we have pffiffiffiffiffiffiffiffiffiffi k RðkÞ ¼ aðkÞðk2 H k Þ:
ð50Þ
Since the derivative of a function along a Hamiltonian flow is equal to its Poisson bracket with the Hamiltonian, (29) implies
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d k2 d ¼ : 2 dsk 2ðk H k Þ dtk
ð51Þ
Therefore, from (24), (50) and (48), we get 1 ~ X d/ d/ kg ¼C ¼ pffiffiffiffiffiffiffiffiffiffi ðC 1 k0 þ C 2 k1 þ þ C g kgþ1 Þ ¼ 2 X l zl : dsk dsk 2 RðkÞ l¼1
Hence we obtain the first part of (49) after comparing the coefficients of zl, while the second part is obtained similarly. h P P The straightened (49) are easily integrated by quadratures: / ¼ /0 þ 2 Xlþ1 sl ; w ¼ w0 2 Xlþ1 sl . Under the Abel–Jacobi coordinates, the evolution picture of the corresponding flows becomes very simple: flow H l : / ¼ /0 þ 2Xlþ1 sl ; w ¼ w0 2Xlþ1 sl ; flow X l : / ¼ /0 þ 2X1 x þ 2Xlþ1 sl ; w ¼ w0 2X1 x 2Xlþ1 sl :
ð52Þ
8. Quasi-periodic solutions The Abel map A : DivðCÞ ! JðCÞ ¼ Cg =T is defined by Z P X X nl AðP l Þ; AðP Þ ¼ x; A nl P l ¼
ð53Þ
P0
where P0 = q(k0) is fixed, Div(C) is the divisor group, and the Lattice T is spanned by the periodic vectors {dj; Bj}, which are the column vectors of E and (Bjl) defined by (44). The definition of Abel–Jacobi coordinates are rewritten as ( ) ( ) g g X X qðlj Þ ; w ¼ A qðmj Þ : ð54Þ /¼A j¼1
j¼1
In the neighborhood of 1m (m = 1, 2), since the two-valued function kN 1 have (z = k1) pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi kN 1 RðkÞ ¼ ð1Þm R ðzÞ:
pffiffiffiffiffiffiffiffiffiffi RðkÞ tends to (1)m due to (42), we
By (39), we get ~ ¼ ð1Þm1 x ¼ Cx
dz pffiffiffiffiffiffiffiffiffiffiffi ðC 1 z þ þ C g zg Þ: 4z R ðzÞ
With the help of (48), we obtain 1 X Xl zl1 dz; x ¼ ð1Þml
ð55Þ
l¼1
Aðqðz1 ÞÞ ¼ gm þ ð1Þm1 where gm ¼
Z
1 X 1 X l zl ; l l¼1
ð56Þ
P0
x: 1m
Comparing the coefficient of the Laurent expansions on both sides of (21), we obtain N X v ðmj kj Þ ¼ hp; pi; j¼1
N a X v þ ðkj lj Þ ¼ hq; qi: v j¼1 k
ð57Þ
S. Xue, D. Du / Chaos, Solitons and Fractals 35 (2008) 692–704
703
By using (11) and (57), we have N X 1 ox lnðvkþ1 þ aÞ ¼ ðmj lj Þ; 2 j¼1
ð58Þ N N X X 1 1 k1 k1 u¼ kj þ kÞ mj þ 1Þ lj : ðav ðav kþ1 kþ1 j¼1 j¼1 j¼1 P P In order to calculate lj , mj , we use the Riemann theorem [15,16], which asserts that there exists a constant vector M such that N X
(i) hðAðqðkÞÞ / MÞ has exactly g zeros at k = l1, . . . , lg; (ii) hðAðqðkÞÞ w MÞ has exactly g zeros at k = m1, . . . , mg . Thus by a standard treatment [6,15,17], we have g X
lj ¼ I 1 ðCÞ
2 X
j¼1
m¼1
g X
2 X
1 h1 Resk¼1m kd ln hðAðqðkÞÞ / MÞ ¼ I 1 ðCÞ þ ox ln ; h2 2
1 h mj ¼ I 1 ðCÞ Resk¼1m kd ln hðAðqðkÞÞ w MÞ ¼ I 1 ðCÞ ox ln 1 ; 2 h2 j¼1 m¼1 Pg R where I 1 ðCÞ ¼ j¼1 aj kxj and (m = 1, 2) X hm ¼ hð/ þ M þ gm Þ ¼ h 2 Xl sl þ /0 þ M þ gm ; X Xl sl w0 M gm : hm ¼ hðw M gm Þ ¼ h 2
ð59Þ
By using (58) and (59), we have ox lnðvkþ1 þ aÞ ¼ ox ln u¼
N X j¼1
kj þ
h1 h1 ; h2 h2
1 1 h 1 1 h1 ðavk1 kÞ I 1 ðCÞ ox ln 1 ðavk1 þ 1Þ I 1 ðCÞ þ ox ln : kþ1 2 kþ1 2 h2 h2
ð60Þ
Therefore, noticing that the theta function is an even function we obtain by (60) and the evolution of H0 flow that quasiperiodic solutions of the stationary (35) 1 kþ1 hð2X1 x þ a2 Þhð2X1 x þ b2 Þhða1 Þhðb1 Þ vðxÞ ¼ ðvkþ1 ð0Þ þ aÞ a ; hð2X1 x þ a1 Þhð2X1 x þ b1 Þhða2 Þhðb2 Þ ð61Þ 1 hð2X1 x þ b1 Þ 1 hð2X1 x þ a1 Þ ðavk1 ðxÞ kÞox ln ðavk1 ðxÞ þ 1Þox ln uðxÞ ¼ M 0 2ðk þ 1Þ hð2X1 x þ b2 Þ 2ðk þ 1Þ hð2X1 x þ a2 Þ with constants am ¼ /0 þ M þ gm ;
bm ¼ w0 M gm ;
M0 ¼
N X
kj I 1 ðCÞ;
j¼1
which are called the finite-band potentials of the spectral problem (1). Consider the H0, Hm flows and the evolution of Xm flow. Then (1 + 1)-dimensional equations (6) have solutions hð2X1 x þ 2Xmþ1 tm þ a2 Þhð2X1 x þ 2Xmþ1 tm þ b2 Þ vðx; tm Þ ¼ ½vkþ1 ð0; tm Þ þ a hð2X1 x þ 2Xmþ1 tm þ a1 Þhð2X1 x þ 2Xmþ1 tm þ b1 Þ 1 kþ1 hð2Xmþ1 tm þ a1 Þhð2Xmþ1 tm þ b1 Þ a ; hð2Xmþ1 tm þ a2 Þhð2Xmþ1 tm þ b2 Þ ð62Þ 1 hð2X1 x þ 2Xmþ1 tm þ b1 Þ 1 ðavk1 ðx; tm Þ kÞox ln ðavk1 ðx; tm Þ þ 1Þ uðx; tm Þ ¼ M 0 2ðk þ 1Þ hð2X1 x þ 2Xmþ1 tm þ b2 Þ 2ðk þ 1Þ hð2X1 x þ 2Xmþ1 tm þ a1 Þ : ox ln hð2X1 x þ 2Xmþ1 tm þ a2 Þ
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S. Xue, D. Du / Chaos, Solitons and Fractals 35 (2008) 692–704
Acknowledgments This work was supported by National Natural Science Foundation of China (project No. 10471132) and basic research project (072300410080) of Henan. The authors also thank the Youth Teacher Foundation and Natural Science Foundation (2004110006) of Henan Education Department for financial support.
References [1] Cao CW, Geng XG. Classical integrable systems generated through nonlinearization of eigenvalue problems. In: Proceedings of the conference on nonlinear physics (Shanghai 1989), Research Reports in Physics. Berlin: Springer; 1990. p. 66–78. [2] Cao CW. Nonlinearization of the Lax system for AKNS hierarchy. Sci China A 1990;33:528–36. [3] Cao W, Geng XG. C. Neumann and Bargmann systems associated with the coupled KdV soliton hierarchy. J Phys A 1990;23:4117–25. [4] Belokolos ED, Bobenko AI, Enolskii VZ, Its AR, Matveev VB. Algebro-geometric approach to nonlinear integrable equations. Berlin: Springer; 1994. [5] Gesztesy F, Ratnaseelan R. An alternative approach to algebro-geometric solutions of the AKNS hierarchy. Rev Math Phys 1998;10:345–91. [6] Cao CW, Wu YT, Geng XG. Relation between the Kadometsev–Petviashvili equation and the confocal involutive system. J Math Phys 1999;40:3948–70. [7] Geng XG, Wu YT, Cao CW. Quasi-periodic solutions of the modified Kadomtsev–Petviashvili equation. J Phys A 1999;32:3733–42. [8] Cao CW, Geng XG, Wu YT. From the special 2 + 1 Toda lattice to the Kadomtsev–Petviashvili equation. J Phys A 1999;32:8059–78. [9] Geng XG, Cao CW. Quasi-periodic solutions of the (2 + 1)-dimensional modified Korteweg-de Vries equation. Phys Lett A 1999;261:289–96. [10] Geng XG, Cao CW. Decomposition of the (2 + 1)-dimensional Gardner equation and its quasi-periodic solutions. Nonlinearity 2001;14:1433. [11] Hao YH, Du DL. Two (2 + 1)-dimensional soliton equations and their quasi-periodic solutions. Chaos, Solitons & Fractals 2005;26:979–96. [12] Li XM, Geng XG. Lax matrix and a generalized coupled KdV hierarchy. Phys A 2003;327:357–70. [13] Li XM, Zhang JS. A generalized TD hierarchy through the Lax matrix and their explicit solutions. Phys Lett A 2004;321:14–23. [14] Arnold VI. Mathematical methods of classical mechanics. Berlin: Springer; 1978. [15] Griffiths P, Harris J. Principles of algebraic geometry. New York: Wiley; 1999. [16] Mumford D. Tata lectures on theta. Boston: Birkha¨user; 1984. [17] Zhou RG. The finite-band solution of the Jaulent–Miodek equation. J Math Phys 1997;38:2535–46. [18] Tu GZ, Meng DZ. The trace identity, a powerful tool for constructing the hamiltonian. Structure of integrable system. Acta Math Appl Sin 1989;5:89–96.
A new hierarchy of (1 + 1)-dimensional soliton equations and its quasi-periodic solutions Shan Xue
a,b
, Dianlou Du
a,*
a b
Department of Mathematics, Zhengzhou University, Zhengzhou Henan 450052, PR China Henan Communication Vocational Technology College, Zhengzhou Henan 450000, PR China Accepted 20 June 2007
Abstract A new spectral problem is proposed, from which a hierarchy of (1 + 1)-dimensional soliton equations is derived. With the help of nonlinearization approach, the soliton systems in the hierarchy are decomposed into two new compatible Hamiltonian systems of ordinary differential equations. The generating function flow method is used to prove the involutivity and the functional independence of the conserved integrals. The Abel–Jacobi coordinates are introduced to straighten out the associated flows. Using the Riemann–Jacobi inversion technique, the explicit quasi-periodic solutions for the (1 + 1)-dimensional soliton equations are obtained. Ó 2007 Published by Elsevier Ltd.
1. Introduction The nonlinearization approach [1–3] is a powerful tool for revealing the links between soliton equations and finitedimensional integrable systems. There are at least two important applications of the nonlinearization approach. Firstly, a considerable number of new finite-dimensional integrable systems can be generated from the known soliton hierarchies. Secondly, it provides a way of solving the soliton equations by separating the spatial and temporal variables. Explicit solutions of the soliton equations, which include soliton solutions, periodic solutions, quasi-periodic solutions and other ones, can be obtained through solving the compatible systems of ordinary differential equations. There are several systematic methods to construct quasi-periodic solutions of (1 + 1)-dimensional soliton equations, for instance, the algebro-geometric approach [4,6–13], the alternative elementary algebraic approach [5], the nonlinearization of Lax pairs [1–3] and others. Quasi-periodic solutions for many (1 + 1)-dimensional soliton equations have been derived such as the KdV equation, nonlinear Schro¨dinger equation, mKdV equation, sine-Gordon equation and Toda lattice equation, etc. In this paper, our main aim is to develop further the nonlinearization approach of Lax pairs to construct quasi-periodic solutions of the (1 + 1)-dimensional soliton equations. The present paper is organized as follows. In Section 2, we shall derive a hierarchy of (1 + 1)-dimensional soliton equations associated with the spectral problem (1) by using the Lenard gradient sequence. In Section 3, the spectral problem (1) is nonlinearized into a new finite-dimensional *
Corresponding author. E-mail address: [email protected] (D. Du).
0960-0779/$ - see front matter Ó 2007 Published by Elsevier Ltd. doi:10.1016/j.chaos.2007.06.095
S. Xue, D. Du / Chaos, Solitons and Fractals 35 (2008) 692–704
693
Hamiltonian system under the constraint between the potentials and eigenfunctions. And the integrals of motion for the finite-dimensional Hamiltonian system are obtained and proved to be in involution in pairs. In Section 4, we introduce the elliptic coordinates and quasi-Abel–Jacobi coordinates, by which the functional independence of the integrals of motion is proved. In Section 5, a new set of polynomial integrals is introduced. In Section 6, by the new set of polynomial integrals, we arrive at the decomposition of the soliton hierarchy. In Section 7, a hyperelliptic Riemann surface of genus g = N and the Abel–Jacobi coordinates are defined to straighten out the associated flows. In the last part, Riemann–Jacobi inversion problem is discussed, from which the quasi-periodic solutions of the (1 + 1)-dimensional soliton equations are expressed explicitly in terms of the Riemann theta functions.
2. (1 + 1)-Dimensional soliton hierarchy Firstly, we introduce a new spectral problem k þ u v ux ¼ U u; U ¼ ; vk þ av k u
ð1Þ
where k, a 2 R, k 5 1, and u, v are potentials. Based on the zero-curvature equation, we have the fundamental identity V x ½U ; V ¼ rfðK kJ Þcg; T
ð2Þ 3
where c = (C, B, A) , r is a map: R ! slð2; RÞ defined by 0 1 C A B B C r@ B A ¼ ; C A A and (o = ox) V ¼
A C
B ; A
0 B K¼@
o þ 2u
0
0 v
o 2u vk þ av
2ðvk þ avÞ 2v o
1 C A;
0
2 B J ¼ @0 0
1 0 0 C 2 0 A: 0 0
ð3Þ
The Lenard gradient {gj} is defined recursively by Kgj1 ¼ Jgj ;
g1 ¼ ð0; 0; 1ÞT ;
jP0
ð4Þ
which means that (K kJ)gk = 0 for the generating function 1 X gk ¼ gj1 kj : j¼0
It is easy to see that det r(gk) = 1. The first few numbers of the Lenard gradient {gj} are 1 0 k a1 01 ðo þ 2uÞ vk þ av v þv 2 B C B C 1 g 0 ¼ @ v A; g 1 ¼ @ ð2u oÞv A; 2 1 k a v v þv 0 2 0 k a 1 kþ1 1 k a v þ v xx þ u v þ v x þ u2 þ u2x v 2 þa vk þ av 4 C B kþ1 C; g2 ¼ B A @ 14 ðvxx 2ux v 4uvx þ 4u2 vÞ ðv 2þaÞv vx v k a k a k a ðv þ Þx 4 v þ v þ uv v þ v 1 0 41 k va v þ v xxx þ 34 u vk þ av xx þ 32 u2 þ 34 ux 34 v vk þ av vk þ av x 8 B C B þ 14 uxx þ u3 þ 32 uux 32 uv vk þ av vk þ av C B C B 1 vxxx þ 3 uvxx þ 3 ux vx 3 u2 vx þ 3 vvx vk þ a þ 1 uxx v 3 ux uv C B C 4 v 4 2 g3 ¼ B 8 3 43 2 k4 a 2 C: B þu v 2 uv v þ v C B C B 3 u2 v 3 v2 vk þ a 3 uv þ 1 v vk þ a þ 1 vvk þ a C x @ 2 A 8 v 4 8 xx v 8 v xx 3 þ 4 uv 18 vx vk þ av x
ð5Þ
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S. Xue, D. Du / Chaos, Solitons and Fractals 35 (2008) 692–704
Let n X
Gn ¼
gj1 knjþ1 þ gn
j¼0
with gn ¼ ðg1n ; g2n ; g3n þ dn ÞT , where dn ¼
kvk1 va2 2 1 1 g þ g : ðk þ 1Þvk nþ1 ðk þ 1Þvk nþ1
Let V(n) = r(Gn). Then ðnÞ ¼ rðU tn K gn Þ ¼ 0 U tn V ðnÞ x þ ½U ; V
is equivalent to a hierarchy of (1 + 1)-dimensional soliton equations ðu; vÞTtn ¼ X n ; where
0 Xn ¼
ð6Þ h
2 i1 a gnþ1 vkþ2 A: 2 og3nþ1 ðkþ1Þvk
g1 1 o nþ1 vk kþ1 @
þ
k
v
ð7Þ
The relation between Xn and gn+1 is X n ¼ PTgnþ1 ;
ð8Þ T
T
where P is the projective map (c1, c2, c3) ! (c1, c2) , and 1 0 1 o vk o v1k kvk1 va2 0 1 B 2 C: T ¼ @ 0 0 oA vk kþ1 0 0 0 Then corresponding to (5), we have ux X0 ¼ ; vx h 0 1 1 ux o 2k a2 vk1 ðln vÞxx þ 14 ðk 2 þ kÞðln vÞ2x þ avk1 þ 1k kþ1 2 B C C; 2 kþ1 kþ1 X1 ¼ B @ þðk þ 1Þu 2 v A 2 k1 k a kþ1 o v v ðln vÞ þ uv þ au x x ðkþ1Þvk 4 2 0 1 n 2 2 1 1 3 k 3a 3 k ðln vÞ o kðk 1Þv þ ðln vÞ þ ðk þ kÞv ðln vÞ k x xx x C 8 4v 4 B kþ1 v B C B 3a u ðln vÞ þ 1 ðk þ 1Þkðk 1Þvk ðln vÞ3 þ 3 ux ðvk þ a Þ C x xx 2 v 8 4 v x B C
B C uxx 3 3 k a k 3 k 2a þ 4 þ u 2 uvðv þ vÞ ðk þ 1Þv þ 2 uux ð1 kÞv þ v C: X2 ¼ B B C h B 2 C 2 3 a 1 k k kþ1 B ðkþ1Þvk o kþ1 C v v v ðv þ Þ þ kðk þ 1Þv ðln vÞ xx x x 8 8 v x 8 B C i @ A 2 3 k 2a 3 2 kþ1 3 kþ1 þ 4 uvx ðk 1Þv v þ 2 u ðv þ aÞ 8 ðv þ aÞ Specifically, if k = 0, (6) is reduced to the generalized coupled KdV hierarchy [12]. The first two members are ! u ux o u2x þ auv x þ u2 2v 2va ðln vÞxx u : ; ¼ ¼ v t1 vx 2o uðv þ aÞ v4x a2 ðln vÞx v t0 If k = 0 and a = 0, (6) is reduced to the coupled KdV hierarchy [3]. The first two members are 2uux þ u2xx v2x u ux u ; : ¼ ¼ 1 2ðuvÞx 2 vxx v t1 vx v t0 If k = 1, (6) is reduced to the generalized TD hierarchy [13]. The first two members are
ð9Þ
S. Xue, D. Du / Chaos, Solitons and Fractals 35 (2008) 692–704
u ux ; ¼ v t0 vx
u ¼ v t1
1 o auv2x þ 2u2 2 1 o uv2 þ au v
v2 2va2 ðln vÞxx þ v2vxx a2 ðln vÞx
695
! :
And if k = 0 and a = 0, (6) is reduced to the TD hierarchy [18]. The first two members are ! 2 u u ux o u2 v2 þ v4vxx ; ¼ ¼ : v t0 v t1 vx 2uvx þ ux v
3. Nonlinearization of the eigenvalue problem Consider N copies of Eq. (1) kj þ u v uj ; ujx ¼ vk þ av kj u
j ¼ 1; . . . ; N
ð10Þ
with uj = (pj, qj)T and distinct eigenvalues k = k1, . . ., kN. Then we have the following fundamental identity: Proposition 3.1 ðK kJ ÞGk ¼ J g1
N X
! rkj
j¼1
with Gk ¼ g1 k þ g0 þ
PN
rkj j¼1 kkj ;
rkj ¼ ðq2j ; p2j ; pj qj ÞT .
It is obvious ðK kJ ÞGk ¼ 0 () g1 þ cg1 ¼
N X
rkj ;
c 2 R:
ð11Þ
j¼1
For the simplicity, we choose c = 0. Then (11) gives the constraints between the potentials and eigenfunctions: 1
v ¼ ð2hp; qi aÞkþ1 ; kþ2 k 1 1 a k hq; qið2hp; qi aÞkþ1 þ hp; pið2hp; qi aÞkþ1 hp; pið2hp; qi aÞkþ1 ; u¼ kþ1 kþ1 kþ1
ð12Þ
where hÆ,Æi denotes the canonical inner product in RN ; p ¼ ðp1 ; . . . ; pN ÞT ; q ¼ ðq1 ; . . . ; qN ÞT . Substituting (12) into (10), we arrive at a finite-dimensional Hamiltonian system: p oH 0 =oq ; ¼ q x oH 0 =op
ð13Þ
where 1 H 0 ¼ hAp; qi vhq; qi þ hp; pihp; qiv1 ; 2
A ¼ diagðk1 ; . . . ; kN Þ:
From (2) and (11), we know that the Lax matrix X N N X k v j j þ ¼: l þ ; Vk ¼ k k k k kj vk þ av k j j¼1 j¼1 where j ¼ rðrkj Þ ¼
pj qj
p2j
q2j
pj qj
ð14Þ
! ;
satisfies Vx [U, V] = 0. Therefore, Fk = det Vk is invariant along the x flow and yields the integrals {Fl} as follows: 1
1
F k ¼ Qk ðq; qÞQk ðp; pÞ Q2k ðp; qÞ þ 2hp; qiQk ðp; pÞð2hp; qi aÞkþ1 þ 2kQk ðp; qÞ Qk ðq; qÞð2hp; qi aÞkþ1 2hp; qi k2 1 X ¼ k2 þ kl1 F l ; ð15Þ l¼0
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S. Xue, D. Du / Chaos, Solitons and Fractals 35 (2008) 692–704
where Qk(n, g) = h(kE A)1n, gi, and F 0 ¼ 2H 0 ; X 1 Fl ¼ ½hAi q; qihAj p; pi hAi p; qihAj p; qi þ 2hAlþ1 p; qi þ 2hp; qihAl p; pið2hp; qi aÞkþ1 iþj¼l1
ð16Þ
1 kþ1
hAl q; qið2hp; qi aÞ : In order to prove the involutivity of {Fl}, we introduce the generating function method. Denoting the variable of Fk flow by tk, we get d u ¼ W ðk; kj Þuj ; dtk j
ð17Þ
where W ðk; kj Þ ¼
2 V k þ Dk ; k kj
Dk ¼ ðDijk Þ22 ;
21 D12 k ¼ Dk ¼ 0; 22 1 D11 k ¼ Dk ¼ 2Qk ðp; pÞv
4 2 Qk ðp; pÞhp; qivk2 Qk ðq; qÞvk 2 : kþ1 kþ1
Proposition 3.2 d V l ¼ ½W ðk; lÞ; V l 8k; l 2 C; dtk fF k ; F l g ¼ 0 8k; l 2 C; fF j ; F l g ¼ 0 8j; l ¼ 0; 1; 2; . . . ; :
ð18Þ ð19Þ ð20Þ
Proof. Noticing (17) and the identity d dll 2 Vl ¼ ½Dk ; ll þ ½V k ; lk ll þ ½W ðk; lÞ; V l ; dtk dtk kl a direct calculation gives (18), which implies the invariance of Fk along the tk flow: 0 = dFk/dtk = {Fl, Fk}. Substituting the expansion (15) into (19) gives (20) by comparing the same powers of k, l. h
4. Elliptic coordinates and functional independence 0 21 It is easy to see that each one of F k ; V 12 k and V k , as a rational function of k, has simple poles at kj s (1 6 j 6 N), since 2 the coefficients of (k kj) is zero in Fk. Hence we obtain that bðkÞ ; F k ¼ Qk ðq; qÞQk ðp; pÞ Q2k ðp; qÞ þ 2kQk ðp; qÞ þ 2hp; qiQk ðp; pÞv1 Qk ðq; qÞv1 2hp; qi k2 ¼ aðkÞ nðkÞ ; V 12 ð21Þ k ¼ v Qk ðp; pÞ ¼ v aðkÞ a a mðkÞ k k V 21 ; k ¼ v þ þ Qk ðq; qÞ ¼ v þ v v aðkÞ
where aðkÞ ¼
N Y ðk kj Þ;
bðkÞ ¼
j¼1
mðkÞ ¼
N Y j¼1
N þ2 Y
ðk bj Þ;
j¼1
ðk lj Þ;
nðkÞ ¼
N Y ðk mj Þ: j¼1
The roots {lj} and {mj} are defined as elliptic variables.
S. Xue, D. Du / Chaos, Solitons and Fractals 35 (2008) 692–704
697
Proposition 4.1. The elliptic coordinates satisfy the evolution equations along the tk flow: dl 1 mðkÞ qffiffiffiffiffiffiffiffiffiffiffi j ¼ ; 0 dt ðk l j ÞaðkÞm ðlj Þ 4 Rðlj Þ k
ð22Þ
1 dm nðkÞ pffiffiffiffiffiffiffiffiffiffiffi j ¼ ; 4 Rðmj Þ dtk ðk mj ÞaðkÞn0 ðlj Þ Q þ2 where RðkÞ ¼ aðkÞbðkÞ ¼ 2N j¼1 ðk aj Þ, with aj = kj (j = 1, . . . , N), aN+j = bj (j = 1, . . . , N + 2). Proof. Substituting k = lj, mj, respectively, in (21) , we have qffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi Rðlj Þ Rðmj Þ 11 11 ; V mj ¼ : V lj ¼ aðlj Þ aðmj Þ The second and third components of (18) can be written as d 12 4 11 12 11 ðV 11 V 12 V 12 V ¼ k V l Þ þ 2Dk V l ; dtk l kl k l d 21 4 11 21 21 ðV 21 V 11 V 11 V ¼ k V l Þ 2Dk V l : dtk l kl k l Let l = mj and l = lj, respectively. After some calculations we obtain (22). By means of the interpolation formula, we have (s = 1, . . . , g) g X j¼1
g X
lgs dlj kgs j qffiffiffiffiffiffiffiffiffiffiffi ; ¼ 4 Rðlj Þ dtk aðkÞ
j¼1
h
mgs dmj kgs pjffiffiffiffiffiffiffiffiffiffiffi : ¼ aðkÞ 4 Rðmj Þ dtk
For fixed k0, introduce the quasi-Abel–Jacobi coordinates: /~s ¼
g Z X j¼1
lj
k0
lgs pffiffiffiffiffiffiffiffiffiffi dl; 4 RðlÞ
w~s ¼
g Z X j¼1
mj k0
mgs pffiffiffiffiffiffiffiffiffi dm ðs ¼ 1; . . . ; gÞ: 4 RðmÞ
ð23Þ
Proposition 4.2. (straightening of the Fk flow). d/~s kgs ; ¼ dtk aðkÞ
dw~s kgs : ¼ dtk aðkÞ
ð24Þ
Corollary 4.3 (straightening of the Fl flow). Let tl be the variable of the Fl flow. Then 0 1 1 A1 AN 1 ! B 1 AN 2 C ~ d/ ~ ~ B C d/ d/ C; ; ;...; ¼B . . B .. C .. dt0 dt1 dtN 1 @ A
ð25Þ
1 ~ l ¼ dw=dt ~ l , where Aj’s are the coefficient in the expansion and d/=dt 1 X kN 1 ¼ ¼ 1 þ Aj kj ; aðkÞ ð1 k1 k1 Þ ð1 kN k1 Þ j¼1
ð26Þ
which could be represented through the power sums of kj, sm ¼ km1 þ þ kmN : 0 1 A1 ¼ s1 ;
Am ¼
X C 1B Bsm þ si Aj C A: m@ iþj¼m i;jP1
ð27Þ
698
S. Xue, D. Du / Chaos, Solitons and Fractals 35 (2008) 692–704
Proof. According to the definition of the Poisson bracket, we have 1 1 X X d/~s 1 ~ 1 d/~s ¼ f/~s ; F k g ¼ f/s ; F l g ¼ : lþ1 lþ1 dtk dtl l¼0 k l¼0 k
With the supplementary definition A0 = 1, Aj = 0 (j = 1, 2, 3, . . .), the comparison of the coefficients of kl1 in (25) ~ l ¼ ðAl ; Al1 ; . . . ; Algþ1 ÞT . h yields d/~s =dtl ¼ Alsþ1 , and d/=dt Proposition 4.4. F0, F1, . . . , FN1 given in (16) are functionally independent. Proof. We need only prove the linear independence of the differentials dF0, dF1, . . . , dFN1. Suppose Let H ¼ /~s in the formula [14]
PN 1 l¼0
cl dF l ¼ 0.
fH; F g ¼ x2 ðI dF ; I dH Þ: Then we have 0¼
N 1 X
cl x2 ðI dF l ; I d/~s Þ ¼
N 1 X
l¼0
cl /~s ; F l ¼
l¼0
N 1 X
cl
l¼0
d/~s : dtl
Hence c0 = c1 = cN1 = 0 since the coefficient determinant is equal to 1 by (25).
h
Theorem 4.5. The finite-dimensional Hamiltonian system (13) is completely integrable in the Liouville sense.
5. The polynomial integrals {Hl} We now introduce a new set of integrals {Hl} recursively by 1 H 0 ¼ F 0; 2
1 H 1 ¼ F 1; . . . ; 2
1 1 X Hl ¼ F l þ H iH j; 2 2 iþj¼l3
ð28Þ
which is put in the equivalent form k2 F k ¼ ðH k k2 Þ2
ð29Þ
with the help of the generating function Hk ¼
1 X
H l kl1 :
l¼0
Proposition 5.1. {Hl} satisfies the Liouville conditions of completely integrability: fH k ; H l g ¼ 0 8k; l 2 C; fH j ; H l g ¼ 0 8j; l ¼ 0; 1; 2 . . . ; (ii) H0, H1, . . . , HN1 are functionally independent. (i)
Proof. The involutivity of {Hl} is based on the equality 1 fH l ; H k g ¼ pffiffiffiffiffiffiffiffiffiffiffi fF l ; F k g ¼ 0: 4 F kF l From (29), we have k2 dF k ¼ 2ðH k k2 Þ dH k ;
ð30Þ
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0
1
dF 0
0
10
1
C B 0 C B C B C B 0 C B C¼B C B H 0 C B C B .. C B . A @
B dF B 1 B B dF 2 1B B 2B B dF 3 B . B . @ .
1
H N 4
dF N 1
0
1
0 .. .
0 .. .
1 .. .
..
H N 3
H N 2
H N 1
.
dH 0
1
CB dH CB 1 CB CB dH 2 CB CB CB dH 3 CB CB . CB . A@ .
C C C C C C: C C C C A
1
699
dH N 1
Thus the linear independence of dH0, . . . , dHN1 is equivalent to that of dF0, . . . , dFN1, which completes the proof of the functional independence of H0, . . . , HN1. h
6. Relation of Hl and Xl Acting with J1K upon g1 ¼ N X
PN
j¼1 rkj l
times and noting ker J ¼ fcg1 j8c 2 Cg, we arrive at
klj rkj ¼ glþ1 þ c1 gl2 þ þ cl g1
ð31Þ
j¼1
since each time there is an extra term cig1. Hence we have 0¼
N X
aðkj Þrkj ¼ gN þ1 þ cN 0 gN þ þ cNN g0 þ cN ;N þ1 g1 ;
ð32Þ
j¼1
where aðkÞ ¼
N Y
ðk kj Þ ¼
N X
j¼1
cN 0 ¼ a1 ;
aN l kl
ða0 ¼ 1Þ;
l¼0
cN 1 ¼ a2 ;
cNN ¼
N 1 X
aN l1 cl ;
cN ;N þ1 ¼
l¼1
cN ;N i ¼ aN iþ1 þ
N 1 X
aN l1 cli
N 1 X
aN l1 clþ1 ;
ð33Þ
l¼0
ði ¼ 1; . . . ; N 2Þ:
l¼iþ1
Proposition 6.1. Let (p(x), q(x))T be a solution of the finite-dimensional Hamiltonian system (13). Then 0 kþ2 1 k 1 a hq; qið2hp; qi aÞkþ1 þ kþ1 hp; pið2hp; qi aÞkþ1 kþ1 B C u 1 C k ¼ f ðp; qÞ ¼ B kþ1 hp; pið2hp; qi aÞkþ1 @ A v 1 kþ1 ð2hp; qi aÞ
ð34Þ
solves the stationary soliton equation X N þ cN 0 X N 1 þ þ cNN X 1 ¼ 0: Proof. Operating PT upon (32) yields (35).
ð35Þ h
Proposition 6.2. gk ¼
k Gk : k Hk 2
Proof. Define the generating function of the coefficients {cj} in (31) by ck ¼ k þ
1 X j¼1
cj kj1 :
ð36Þ
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Multiplied by kj1 and summed with respect to j from 0 to 1, (31) becomes G k ¼ ck g k : 2
Thus r(Gk) = ckr(gk) and F k ¼ c2k due to det r(gk) = 1, which yields ck ¼ k
H k k
and (36) by comparing with (29).
h
Proposition 6.3 f ðIrH k Þ ¼ kPTgk jM ; k 2 C; f ðIrH l Þ ¼ X l jM ; l ¼ 0; 1; 2 . . . :
ð37Þ
Proof. From (12) and (17), we get dv 4 2 hp; qiQk ðp; pÞvk1 þ Qk ðq; qÞvkþ1 ; ¼2 dtk kþ1 kþ1 du 1 f4a2 hp; piQk ðp; pÞv2k4 4ahq; qiQk ðp; pÞv2k2 þ 4Qk ðp; qÞ 4ahp; piQk ðq; qÞv2k2 ¼ dtk k þ 1 4hq; qiQk ðq; qÞv2k 8ahp; piQk ðp; pÞvk3 4a½kQk ðp; pÞ hp; pivk2 þ 8aQk ðp; qÞvk1 4½hq; qi kQk ðq; qÞvk þ 4khp; piQk ðp; pÞv2 2k½2hp; pi 2kQk ðp; pÞv1 4kQk ðp; qÞg: And a direct calculation gives
! o v1k 2hp; qiv1 þ Qk ðq; qÞ þ kvk1 va2 ½v Qk ðp; pÞ 1 1 d u PTGk ¼ ¼ : k þ 1 v1k o½2k þ 2Qk ðp; qÞ 2 dtk v Thus f ðIrF k Þ ¼ f
d dtk
p d d u ¼ ¼ 2PTGk : ðf ðp; qÞÞ ¼ dtk dtk v q
By (29) and (36), we have f ðIrH k Þ ¼
k2 f ðIrF k Þ ¼ kPTgk : 2ðk H k Þ 2
Hance we have the first part of (37). The second part is obtained by comparing the coefficients of the same power kl1. h Corollary 6.4. Let (p(x, tm), q(x,tm))T be a compatible solution of the H0 and Hm flow. Then (u, v)T = f(p, q) solves the (1 + 1)-dimensional soliton equation ðu; vÞTtm ¼ X m :
ð38Þ
7. The Abel–Jacobi coordinates The shape of (23) suggests the consideration of the holomorphic differential kgs dk ~ s ¼ pffiffiffiffiffiffiffiffiffiffi ; x 4 RðkÞ
s ¼ 1; 2; . . . ; g;
ð39Þ
on the hyperelliptic curve C: n2 16RðkÞ ¼ 0;
ð40Þ
with genus g = N since deg R = 2N + 2. For the same k, there are two points on different sheets of the Riemann surface C: pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi qþ ¼ k; 4 RðkÞ ; q ¼ k; 4 RðkÞ :
S. Xue, D. Du / Chaos, Solitons and Fractals 35 (2008) 692–704
701
At infinity, the affine equation (40) is transformed into ðz ¼ k1 ; ^ n ¼ zN þ1 nÞ: ^n2 16R ðzÞ ¼ 0 2N þ2
1
ð41Þ Q2N þ2
with R ðzÞ ¼ z Rðz Þ ¼ j¼1 ð1 aj zÞ. The two infinities are represented as 1l ¼ ðz ¼ 0; ^n ¼ ð1Þm 4Þ;
m ¼ 1; 2:
ð42Þ
Take the canonical basis of cycles on C: a1, . . . , ag; b1, . . . , bg. Let C = (Cjs) be the inverse of the periodic matrix (Asl): Z ~ s: C ¼ ðAsl Þ1 ; A ¼ ð43Þ x sl gg al
Then for the normalized holomorphic differential xj ¼
g X
~ x ¼ ðx1 ; . . . ; xg ÞT ¼ C x;
~ s; C js x
s¼1
we have Z
xj ¼ djl ;
al
Z
xj ¼ Bjl :
ð44Þ
bl
According to the Riemannian bilinear relation, the matrix B = (Bjl) is symmetric and has positive definite imaginary part, and is used to construct the Riemannian theta function of C: X pffiffiffiffiffiffiffi hðfÞ ¼ exp p 1ðhBz; zi þ 2hf; ziÞ; f 2 Cg : z2Z g
For fixed k0, the Abel–Jacobi coordinates are defined as g Z qðll Þ g Z qðml Þ X X x; w ¼ x: /¼ l¼1
qðk0 Þ
l¼1
P2N þ2 l kj . Then the coefficients of Lemma 7.1. Let sl ¼ j¼1 1 X 1 pffiffiffiffiffiffiffiffiffiffiffi ¼ K l zl R ðzÞ l¼0 satisfy the recursive formula [6]: K0 ¼ 1;
1 K1 ¼ s1 ; . . . ; 2
ð45Þ
qðk0 Þ
0 Kl ¼
ð46Þ 1
X C 1B B sl þ si Kj C @ A: 2l iþj¼l
ð47Þ
i;jP1
Let C1, . . . , Cg be the column vectors of C. Then by direct calculations, the coefficients in 1 X 1 pffiffiffiffiffiffiffiffiffiffiffi ðC 1 z þ C 2 z2 þ þ C g zg Þ ¼ Xl zl 4 R ðzÞ l¼1
ð48Þ
are Xl ¼ 14ðKl1 C 1 þ þ Klg C g Þ; l P 1, with additional defined Ks = 0 (s = 1, 2. . .). Proposition 7.2. The sl flow is straightened by the Abel–Jacobi coordinates: d/ ¼ 2Xlþ1 ; dsl
dw ¼ 2Xlþ1 : dsl
ð49Þ
Proof. By (21), and (29), we have pffiffiffiffiffiffiffiffiffiffi k RðkÞ ¼ aðkÞðk2 H k Þ:
ð50Þ
Since the derivative of a function along a Hamiltonian flow is equal to its Poisson bracket with the Hamiltonian, (29) implies
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S. Xue, D. Du / Chaos, Solitons and Fractals 35 (2008) 692–704
d k2 d ¼ : 2 dsk 2ðk H k Þ dtk
ð51Þ
Therefore, from (24), (50) and (48), we get 1 ~ X d/ d/ kg ¼C ¼ pffiffiffiffiffiffiffiffiffiffi ðC 1 k0 þ C 2 k1 þ þ C g kgþ1 Þ ¼ 2 X l zl : dsk dsk 2 RðkÞ l¼1
Hence we obtain the first part of (49) after comparing the coefficients of zl, while the second part is obtained similarly. h P P The straightened (49) are easily integrated by quadratures: / ¼ /0 þ 2 Xlþ1 sl ; w ¼ w0 2 Xlþ1 sl . Under the Abel–Jacobi coordinates, the evolution picture of the corresponding flows becomes very simple: flow H l : / ¼ /0 þ 2Xlþ1 sl ; w ¼ w0 2Xlþ1 sl ; flow X l : / ¼ /0 þ 2X1 x þ 2Xlþ1 sl ; w ¼ w0 2X1 x 2Xlþ1 sl :
ð52Þ
8. Quasi-periodic solutions The Abel map A : DivðCÞ ! JðCÞ ¼ Cg =T is defined by Z P X X nl AðP l Þ; AðP Þ ¼ x; A nl P l ¼
ð53Þ
P0
where P0 = q(k0) is fixed, Div(C) is the divisor group, and the Lattice T is spanned by the periodic vectors {dj; Bj}, which are the column vectors of E and (Bjl) defined by (44). The definition of Abel–Jacobi coordinates are rewritten as ( ) ( ) g g X X qðlj Þ ; w ¼ A qðmj Þ : ð54Þ /¼A j¼1
j¼1
In the neighborhood of 1m (m = 1, 2), since the two-valued function kN 1 have (z = k1) pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi kN 1 RðkÞ ¼ ð1Þm R ðzÞ:
pffiffiffiffiffiffiffiffiffiffi RðkÞ tends to (1)m due to (42), we
By (39), we get ~ ¼ ð1Þm1 x ¼ Cx
dz pffiffiffiffiffiffiffiffiffiffiffi ðC 1 z þ þ C g zg Þ: 4z R ðzÞ
With the help of (48), we obtain 1 X Xl zl1 dz; x ¼ ð1Þml
ð55Þ
l¼1
Aðqðz1 ÞÞ ¼ gm þ ð1Þm1 where gm ¼
Z
1 X 1 X l zl ; l l¼1
ð56Þ
P0
x: 1m
Comparing the coefficient of the Laurent expansions on both sides of (21), we obtain N X v ðmj kj Þ ¼ hp; pi; j¼1
N a X v þ ðkj lj Þ ¼ hq; qi: v j¼1 k
ð57Þ
S. Xue, D. Du / Chaos, Solitons and Fractals 35 (2008) 692–704
703
By using (11) and (57), we have N X 1 ox lnðvkþ1 þ aÞ ¼ ðmj lj Þ; 2 j¼1
ð58Þ N N X X 1 1 k1 k1 u¼ kj þ kÞ mj þ 1Þ lj : ðav ðav kþ1 kþ1 j¼1 j¼1 j¼1 P P In order to calculate lj , mj , we use the Riemann theorem [15,16], which asserts that there exists a constant vector M such that N X
(i) hðAðqðkÞÞ / MÞ has exactly g zeros at k = l1, . . . , lg; (ii) hðAðqðkÞÞ w MÞ has exactly g zeros at k = m1, . . . , mg . Thus by a standard treatment [6,15,17], we have g X
lj ¼ I 1 ðCÞ
2 X
j¼1
m¼1
g X
2 X
1 h1 Resk¼1m kd ln hðAðqðkÞÞ / MÞ ¼ I 1 ðCÞ þ ox ln ; h2 2
1 h mj ¼ I 1 ðCÞ Resk¼1m kd ln hðAðqðkÞÞ w MÞ ¼ I 1 ðCÞ ox ln 1 ; 2 h2 j¼1 m¼1 Pg R where I 1 ðCÞ ¼ j¼1 aj kxj and (m = 1, 2) X hm ¼ hð/ þ M þ gm Þ ¼ h 2 Xl sl þ /0 þ M þ gm ; X Xl sl w0 M gm : hm ¼ hðw M gm Þ ¼ h 2
ð59Þ
By using (58) and (59), we have ox lnðvkþ1 þ aÞ ¼ ox ln u¼
N X j¼1
kj þ
h1 h1 ; h2 h2
1 1 h 1 1 h1 ðavk1 kÞ I 1 ðCÞ ox ln 1 ðavk1 þ 1Þ I 1 ðCÞ þ ox ln : kþ1 2 kþ1 2 h2 h2
ð60Þ
Therefore, noticing that the theta function is an even function we obtain by (60) and the evolution of H0 flow that quasiperiodic solutions of the stationary (35) 1 kþ1 hð2X1 x þ a2 Þhð2X1 x þ b2 Þhða1 Þhðb1 Þ vðxÞ ¼ ðvkþ1 ð0Þ þ aÞ a ; hð2X1 x þ a1 Þhð2X1 x þ b1 Þhða2 Þhðb2 Þ ð61Þ 1 hð2X1 x þ b1 Þ 1 hð2X1 x þ a1 Þ ðavk1 ðxÞ kÞox ln ðavk1 ðxÞ þ 1Þox ln uðxÞ ¼ M 0 2ðk þ 1Þ hð2X1 x þ b2 Þ 2ðk þ 1Þ hð2X1 x þ a2 Þ with constants am ¼ /0 þ M þ gm ;
bm ¼ w0 M gm ;
M0 ¼
N X
kj I 1 ðCÞ;
j¼1
which are called the finite-band potentials of the spectral problem (1). Consider the H0, Hm flows and the evolution of Xm flow. Then (1 + 1)-dimensional equations (6) have solutions hð2X1 x þ 2Xmþ1 tm þ a2 Þhð2X1 x þ 2Xmþ1 tm þ b2 Þ vðx; tm Þ ¼ ½vkþ1 ð0; tm Þ þ a hð2X1 x þ 2Xmþ1 tm þ a1 Þhð2X1 x þ 2Xmþ1 tm þ b1 Þ 1 kþ1 hð2Xmþ1 tm þ a1 Þhð2Xmþ1 tm þ b1 Þ a ; hð2Xmþ1 tm þ a2 Þhð2Xmþ1 tm þ b2 Þ ð62Þ 1 hð2X1 x þ 2Xmþ1 tm þ b1 Þ 1 ðavk1 ðx; tm Þ kÞox ln ðavk1 ðx; tm Þ þ 1Þ uðx; tm Þ ¼ M 0 2ðk þ 1Þ hð2X1 x þ 2Xmþ1 tm þ b2 Þ 2ðk þ 1Þ hð2X1 x þ 2Xmþ1 tm þ a1 Þ : ox ln hð2X1 x þ 2Xmþ1 tm þ a2 Þ
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S. Xue, D. Du / Chaos, Solitons and Fractals 35 (2008) 692–704
Acknowledgments This work was supported by National Natural Science Foundation of China (project No. 10471132) and basic research project (072300410080) of Henan. The authors also thank the Youth Teacher Foundation and Natural Science Foundation (2004110006) of Henan Education Department for financial support.
References [1] Cao CW, Geng XG. Classical integrable systems generated through nonlinearization of eigenvalue problems. In: Proceedings of the conference on nonlinear physics (Shanghai 1989), Research Reports in Physics. Berlin: Springer; 1990. p. 66–78. [2] Cao CW. Nonlinearization of the Lax system for AKNS hierarchy. Sci China A 1990;33:528–36. [3] Cao W, Geng XG. C. Neumann and Bargmann systems associated with the coupled KdV soliton hierarchy. J Phys A 1990;23:4117–25. [4] Belokolos ED, Bobenko AI, Enolskii VZ, Its AR, Matveev VB. Algebro-geometric approach to nonlinear integrable equations. Berlin: Springer; 1994. [5] Gesztesy F, Ratnaseelan R. An alternative approach to algebro-geometric solutions of the AKNS hierarchy. Rev Math Phys 1998;10:345–91. [6] Cao CW, Wu YT, Geng XG. Relation between the Kadometsev–Petviashvili equation and the confocal involutive system. J Math Phys 1999;40:3948–70. [7] Geng XG, Wu YT, Cao CW. Quasi-periodic solutions of the modified Kadomtsev–Petviashvili equation. J Phys A 1999;32:3733–42. [8] Cao CW, Geng XG, Wu YT. From the special 2 + 1 Toda lattice to the Kadomtsev–Petviashvili equation. J Phys A 1999;32:8059–78. [9] Geng XG, Cao CW. Quasi-periodic solutions of the (2 + 1)-dimensional modified Korteweg-de Vries equation. Phys Lett A 1999;261:289–96. [10] Geng XG, Cao CW. Decomposition of the (2 + 1)-dimensional Gardner equation and its quasi-periodic solutions. Nonlinearity 2001;14:1433. [11] Hao YH, Du DL. Two (2 + 1)-dimensional soliton equations and their quasi-periodic solutions. Chaos, Solitons & Fractals 2005;26:979–96. [12] Li XM, Geng XG. Lax matrix and a generalized coupled KdV hierarchy. Phys A 2003;327:357–70. [13] Li XM, Zhang JS. A generalized TD hierarchy through the Lax matrix and their explicit solutions. Phys Lett A 2004;321:14–23. [14] Arnold VI. Mathematical methods of classical mechanics. Berlin: Springer; 1978. [15] Griffiths P, Harris J. Principles of algebraic geometry. New York: Wiley; 1999. [16] Mumford D. Tata lectures on theta. Boston: Birkha¨user; 1984. [17] Zhou RG. The finite-band solution of the Jaulent–Miodek equation. J Math Phys 1997;38:2535–46. [18] Tu GZ, Meng DZ. The trace identity, a powerful tool for constructing the hamiltonian. Structure of integrable system. Acta Math Appl Sin 1989;5:89–96.