- Email: [email protected]
Decomposition and straightening out of the coupled mixed nonlinear Schrödinger flows
Chaos, Solitons and Fractals 20 (2004) 311–321 www.elsevier.com/locate/chaos
Decomposition and straightening out of the coupled €dinger flows mixed nonlinear Schro Xianguo Geng b
a,*
, H.H. Dai
b
a Department of Mathematics, Zhengzhou University, Zhengzhou, Henan 450052, People’s Republic of China Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, People’s Republic of China
Accepted 22 July 2003 Communicated by Prof. M. Wadati
Abstract The nonlinearization approach of Lax pairs is extended to the investigation of a soliton hierarchy proposed by Wadati, Konno and Ichikawa, in which the first nontrivial equation is the coupled mixed nonlinear Schr€ odinger equation. Under a constraint between the potentials and eigenfunctions, solutions of the soliton hierarchy are decomposed into a class of new finite-dimensional Hamiltonian systems. The generating function of the integrals of motion is presented, by which the class of finite-dimensional Hamiltonian systems are further proved to be completely integrable in the Liouville sense. Based on the decomposition and the theory of algebraic curve, the Abel–Jacobi coordinates are introduced to straighten out the corresponding flows. As an application, the compatible solutions of the various flows in Abel–Jacobi coordinates are explicitly obtained. Ó 2003 Elsevier Ltd. All rights reserved.
1. Introduction Soliton equations are nonlinear partial differential equations having specific solutions with nice mathematical and physical properties [1–3]. However it is very difficult to solve them due to their high nonlinearity. Usually one considers the multidimensional problems to be solved in such a way as splitting into several lower-dimensional ones, which are easier to be treated with some available tools. The nonlinearization approach of Lax pairs [4–6] makes it possible to decompose the soliton equations into the compatible Hamiltonian systems of ordinary differential equations. There are at least two important applications of the nonlinearization approach. First, a considerable number of new finitedimensional integrable systems can be generated from the known soliton hierarchies. These finite-dimensional integrable systems inherit many properties of the corresponding soliton equations and enrich the theory of integrable system itself. Second, it provides a way of solving the soliton equation by separation of spatial and temporal variables. Explicit solutions of the soliton equation, which include soliton solution, periodic solution, quasi-periodic solution and other ones [7–14], can be obtained through solving the compatible system of ordinary differential equations. In this paper, our main aim is to study the decomposition of the soliton hierarchy proposed by Wadati, Konno and Ichikawa [3] and straightening out of the corresponding flows based on the nonlinearization approach of Lax pairs [4– 6]. We obtain a class of new finite-dimensional completely integrable systems in the Liouville sense and the compatible solutions of the corresponding flows in Abel–Jacobi coordinates. The present paper is organized as follows. In Section 2, for the sake of convenience we shall reconstruct this hierarchy of soliton equations with a linear term by using the Lenard gradient sequence. In Section 3, the nonlinearization of Lax pairs for the hierarchy of soliton equations leads to
*
Corresponding author. Fax: +852-2788-8561.
0960-0779/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0960-0779(03)00385-0
312
X. Geng, H.H. Dai / Chaos, Solitons and Fractals 20 (2004) 311–321
a class of new finite-dimensional Hamiltonian systems under the Bargmann constraint between the potentials and eigenfunctions. The generating function of the integrals of motion is found, by which the class of finite-dimensional Hamiltonian systems are further proved to be completely integrable in the Liouville sense. In Section 4, we introduce a set of new conserved integrals, from which solutions for the hierarchy of soliton equations are separated to solving the compatible Hamiltonian systems of ordinary differential equations. In Section 5, a hyperelliptic Riemann surface of genus N 1 and Abel–Jacobi coordinates are defined to straighten out the corresponding flows. As an application, the compatible solutions of the corresponding flows in Abel–Jacobi coordinates is explicitly obtained.
2. The hierarchy In this section, we construct the hierarchy of soliton equations associated with the spectral problem [3] nðkÞ gðkÞu ; vx ¼ U v; U ¼ gðkÞv nðkÞ
ð2:1Þ
where u and v are two potentials, k a constant spectral parameter, nðkÞ ¼ ak2 þ 2bk, gðkÞ ¼ ak þ b. We first introduce recursive equations 2Bmþ1 ¼ Bm;x þ 2uAm ; 2Cmþ1 ¼ Cm;x þ 2vAm ; m P 0; X X X Aj Ak þ a Bj Ck þ b2 Bj Ck ¼ 0; jþk¼m j;k P 0
ð2:2Þ
mP1
jþk¼m j;k P 0
jþk¼mþ1 j;k P 0
with starting points A0 ¼ 1;
B0 ¼ 0;
C0 ¼ 0:
Then it is easy to see that Am , Bm and Cm are uniquely determined by (2.2), and the first few members are 1 A1 ¼ auv; 2 1 1 2 1 1 B2 ¼ ux au v; C2 ¼ vx auv2 ; 2 2 2 2 1 3 2 2 2 1 2 A2 ¼ aðuvx ux vÞ þ a u v b uv; 4 8 2
ð2:3Þ
1 1 1 3 1 B3 ¼ uxx aðu2 vÞx þ auðuvx ux vÞ þ a2 u3 v2 b2 u2 v; 4 4 4 8 2 1 1 1 3 2 2 3 1 2 2 2 C3 ¼ vxx þ aðuv Þx þ avðuvx ux vÞ þ a u v b uv : 4 4 4 8 2
ð2:4Þ
B1 ¼ u;
C1 ¼ v;
Differentiating the third expression of (2.2) with respect to x and using its first and second expressions, we arrive at X X X ðAj Ak;x þ Ak Aj;x Þ þ 2a ðvBj Ak uAj Ck Þ þ 2b2 ðvBj Ak uCk Aj Þ ¼ 0; ð2:5Þ jþk¼m j;k P 0
jþk¼m j;k P 0
jþk¼mþ1 j;k P 0
in view of identities X ðCk Bjþ1 Bj Ckþ1 Þ ¼ 0; jþk¼mþ1 j;k P 0
X
ðCk Bjþ1 Bj Ckþ1 Þ ¼ 0:
ð2:6Þ
jþk¼m j;k P 0
Noticing B0 ¼ C0 ¼ 0, Eq. (2.5) can be written as m X
Amj ½Aj;x þ aðvBjþ1 uCjþ1 Þ þ b2 ðvBj uCj Þ ¼ 0;
ð2:7Þ
j¼0
which implies by induction that Aj;x þ aðvBjþ1 uCjþ1 Þ þ b2 ðvBj uCj Þ ¼ 0;
j P 0:
ð2:8Þ
X. Geng, H.H. Dai / Chaos, Solitons and Fractals 20 (2004) 311–321
313
e j by For the sake of convenience, we introduce A e jþ1 þ b2 A e j; Aj ¼ a A
e 0 ¼ 0; A
ð2:9Þ
j P 0;
which gives rise to e j;x ¼ uCj vBj ; A
ð2:10Þ
j P 0:
From (2.10), (2.3) and (2.4), we have e 2 ¼ 1 uv b2 a2 ; A 2 1 3 e A 3 ¼ ðuvx ux vÞ þ au2 v2 þ b4 a3 : 4 8
e 1 ¼ a1 ; A
ð2:11Þ
Substituting (2.9) into the first and second expressions of (2.2) yields e j ¼ 2Bjþ1 2au A e jþ1 ; Bj;x þ 2b2 u A 2 e e jþ1 : Cj;x 2b v A j ¼ 2Cjþ1 þ 2av A
ð2:12Þ
Eqs. (2.10) and (2.12) can be written as the Lenard recursive form Kgj1 ¼ Jgj ;
ð2:13Þ
jP0
e j ÞT and two operators where gj1 ¼ ðCj ; Bj ; A 0 1 0 0 0 o 2b2 u K ¼ @ o 0 2b2 v A; J ¼ @ 2 u u v o
K and J are defined by (o ¼ ox ) 1 2 2au 0 2av A: v o
ð2:14Þ
The first several Lenard gradients are 0
g1
0 1 0 ¼ @ 0 A; 0
0
1 v g0 ¼ @ u A; a1
1 1 1 2 v auv B 2 x 2 C B C B 1 C 1 2 B g1 ¼ B ux au v C C; 2 2 B C @ 1 A uv b2 a2 2
0
1 1 1 1 3 2 2 3 1 2 2 2 v aðuv avðuv a b þ Þ þ u vÞ þ u v uv xx x x x B4 C 4 4 8 2 B C B1 C 1 1 3 1 2 2 C 2 2 3 2 B g2 ¼ B uxx aðu vÞx þ auðuvx ux vÞ þ a u v b u v C: 4 4 4 8 2 B C @ A 1 3 ðuvx ux vÞ þ au2 v2 þ b4 a3 4 8 Assume that the time dependence of v for the spectral problem (2.1) obeys the equation ! ðmÞ ðmÞ V11 V12 ðmÞ ðmÞ ¼ vtm ¼ V v; V ðmÞ ðmÞ V21 V11
ð2:15Þ
with ðmÞ
V11 ¼
m X
e j gðkÞ2 nðkÞmj ; A
j¼0
ðmÞ
V12 ¼
m X j¼0
Bj gðkÞnðkÞmj ;
ðmÞ
V21 ¼
m X
Cj gðkÞnðkÞmj :
ð2:16Þ
j¼0
Noticing gðkÞ2 ¼ anðkÞ þ b2 , the compatibility condition of (2.1) and (2.16) yields the zero-curvature equation, Utm VxðmÞ þ ½U ; V ðmÞ ¼ 0, which is equivalent to the hierarchy of nonlinear evolution equations ðutm ; vtm ÞT ¼ Xm ;
mP0
ð2:17Þ
with the vector field Xj ¼ PKgj1 ¼ PJgj ;
ð2:18Þ
314
X. Geng, H.H. Dai / Chaos, Solitons and Fractals 20 (2004) 311–321
where P is the projective map c ¼ ðcð1Þ ; cð2Þ ; cð3Þ ÞT ! ðcð1Þ ; cð2Þ ÞT . The first three equations in the hierarchy (2.17) are as follows (X0 ¼ 0) ut1 ¼ ux ; vt1 ¼ vx ; 1 ut2 ¼ uxx aðu2 vÞx 2bu2 v 4b4 a2 u ; 2 1 vt2 ¼ vxx þ aðuv2 Þx 2buv2 4b4 a2 v ; 2 1 3 3 8 ut3 ¼ uxxx aðu2 vÞxx þ aðu2 vx uvux Þx þ a2 ðu3 v2 Þx b2 u 2vux au2 v2 b4 a3 ; 4 2 4 3 1 3 3 8 2 4 3 2 2 2 2 3 2 2 vt3 ¼ vxxx þ aðuv Þxx þ aðuvvx v ux Þx þ a ðu v Þx b v 2uvx þ au v þ b a : 4 2 4 3
ð2:19Þ ð2:20Þ
ð2:21Þ
Eqs. (2.20) and (2.21) are the coupled mixed nonlinear Schr€ odinger equation and its higher-order equation, respectively. The coupled mixed nonlinear Schr€ odinger equation has possesses infinitely many symmetries, which constitute some infinite-dimensional Lie algebra [15]. We now define the generating function of fgk g: gk ¼
1 X
gk nðkÞk1 ;
ð2:22Þ
k¼0
which satisfies ðK nðkÞJ Þgk ¼ 0;
ð1Þ
ð2Þ
ð1Þ
ugk þ vgk þ ogk ¼ 0
with the aid of (2.13) and Jg0 ¼ 0. Assume that ðanðkÞ þ b2 Þcð3Þ gðkÞcð2Þ V ðu; v; kÞ ¼ r½c ¼ : gðkÞcð1Þ ðanðkÞ þ b2 Þcð3Þ
ð2:23Þ
ð2:24Þ
Then we have the fundamental identity Vx ½U ; V ¼ U ½P fðK nðkÞJ Þcg;
ð2:25Þ
where U is the differential of the map ðu; vÞT ! U ðu; vÞ. Therefore, ðK nðkÞJ Þc ¼ 0 implies o detr½c ¼ 0, from which we deduce det r½gk ¼ 1
ð2:26Þ
in view of ð3Þ
ðanðkÞ þ b2 Þgk ¼ 1 þ
1 X
Ak nðkÞk1 :
ð2:27Þ
k¼0
3. The Bargmann constraint and finite-dimensional integrable systems In this section, we shall discuss the nonlinearization of the spectral problem (2.1). Let us consider N copies of the spectral problem (2.1)
pj;x qj;x
¼
nðkj Þ gðkj Þu gðkj Þv nðkj Þ
pj ; qj
16j6N
with distinct eigenvalues k ¼ k1 ; . . . ; kN . A simple calculation gives gðkj Þq2j dkj =du ¼ rkj ¼ 2 dkj =dv gðkj Þpj
ð3:1Þ
X. Geng, H.H. Dai / Chaos, Solitons and Fractals 20 (2004) 311–321
315
up to a constant factor. We introduce the Bargmann constraint [4] N X Pg0 ¼ rkj ; j¼1
that is u ¼ hgðKÞp; pi;
v ¼ hgðKÞq; qi;
ð3:2Þ
where h; i stands for the canonical inner product in RN , K ¼ diagðk1 ; . . . ; kN Þ, p ¼ ðp1 ; . . . ; pN ÞT , q ¼ ðq1 ; . . . ; qN ÞT . Substituting (3.2) into (3.1), we arrive at a finite-dimensional Hamiltonian system oH ; px ¼ nðKÞp hgðKÞp; pigðKÞq ¼ oq ð3:3Þ oH qx ¼ hgðKÞq; qigðKÞp nðKÞq ¼ op with the Hamiltonian 1 H ¼ hnðKÞp; qi þ hgðKÞp; pihgðKÞq; qi: 2
ð3:4Þ
Let Gk ¼
N X j¼1
0 1 0 1 gðkÞ2 Qk ðgðKÞq; qÞ 0 e k a 1 B C r j 2 @0A¼ ; 2 @ gðkÞ Qk ðgðKÞp; pÞ A nðkÞ nðkj Þ gðkÞ2 gðkÞ 2 c0 1 þ Q ðgðKÞ p; qÞ
ð3:5Þ
k
where rekj ¼ ðrkj ; pj qj Þ , c0 ¼ hp; qi a is defined as T
1
Qk ð~ p; q~Þ ¼ hðnðkÞI nðKÞÞ1 p~; q~i ¼
is a conserved integral of the x-flow and the bilinear function Qk ð~ p; q~Þ on RN
N X j¼1
p~j q~j : nðkÞ nðkj Þ
Then a direct verification shows that ðK nðkj ÞJ Þ rekj ¼ 0, which implies ðK nðkÞJ ÞGk ¼ 0:
ð3:6Þ
Utilizing (2.24), we introduce 1 þ Qk ðgðKÞ2 p; qÞ gðkÞQk ðgðKÞp; pÞ Vk ¼ r½Gk ¼ ; gðkÞQk ðgðKÞq; qÞ 1 Qk ðgðKÞ2 p; qÞ
ð3:7Þ
whose the determinant, det Vk , is invariant under the action of the x-flow. Resorting to the identities gðkÞ2 ¼ anðkÞ þ b2 ;
Qk ðgðKÞ3 p~; q~Þ ¼ gðkÞ2 Qk ðgðKÞ~ p; q~Þ ahgðKÞ~ p; q~i;
the generating function of integrals for the finite-dimensional Hamiltonian system (3.3) can be written as 1 þ Qk ðgðKÞ2 p; qÞ gðkÞQk ðgðKÞp; pÞ ; Fk ¼ det Vk ¼ gðkÞQk ðgðKÞq; qÞ 1 Qk ðgðKÞ2 p; qÞ
ð3:8Þ
ð3:9Þ
that is
Q ðgðKÞ3 p; pÞ Qk ðgðKÞ2 p; qÞ Fk ¼ 1 2Qk ðgðKÞ2 p; qÞ þ ahgðKÞp; piQk ðgðKÞq; qÞ þ k Qk ðgðKÞ2 p; qÞ Qk ðgðKÞq; qÞ 1 X ¼ 1 þ nðkÞm1 Fm ;
ð3:10Þ
m¼0
where F0 ¼ 2hgðKÞ2 p; qi þ ahgðKÞp; pihgðKÞq; qi ¼ 2aH 2b2 hp; qi; 3 j 2 mj1 m1 X p; qi hgðKÞ nðKÞ p; pi hgðKÞ nðKÞ 2 m m Fm ¼ 2hgðKÞ nðKÞ p; qi þ ahgðKÞp; pihgðKÞnðKÞ q; qi þ : 2 j hgðKÞnðKÞmj1 q; qi j¼0 hgðKÞ nðKÞ p; qi
ð3:11Þ
316
X. Geng, H.H. Dai / Chaos, Solitons and Fractals 20 (2004) 311–321
In what follows, we prove that the involutivity of fFm g, that is fFm ; Fl g ¼ 0 for m; l P 0. Here the Poisson bracket of smooth functions in the symplectic space ðR2N ; dp ^ dqÞ is defined as N X of og of og of og of og ¼ ff ; gg ¼ ; ; : oqj opj opj oqj oq op op oq j¼1 By using (3.9), a direct calculation shows that fF1 ; Fk g ¼ 0;
81; k 2 C
ð3:12Þ
with the help of equality E D ðnð1ÞI nðKÞÞ1 ðnðkÞI nðKÞÞ1 gðKÞl p~; q~ ¼
h i 1 Qk ðgðKÞl p~; q~Þ Q1 ðgðKÞl p~; q~Þ : nð1Þ nðkÞ
ð3:13Þ
Therefore, we have the following fact. Proposition 3.1. The functions Fm defined by (3.11) are in involution in pairs, fFm ; Fl g ¼ 0;
8m; l P 0:
ð3:14Þ
Theorem 3.1. The finite-dimensional Hamiltonian systems p sm ¼
oFm ; oq
q sm ¼
oFm ; op
mP0
ð3:15Þ
are completely integrable in the Liouville sense. Proof. By Proposition 3.1, we need only prove that the conserved integrals fFmþl g; 0 6 l 6 N 1, are functionally independent. In fact, from (3.11) we have oFmþl ¼ 2gðKÞ2 nðKÞmþl q: op p¼0 Therefore, we arrive oFm oFmþ1 op1 op1 oF m oFmþ1 op2 op2 oF m oFmþ1 opN opN
at
oFmþN 1 op1 N oFmþN 1 Y Y N gðkj Þ2 nðkj Þm qj ðnðkj Þ nðki ÞÞ 6 0: op2 ¼ ð2Þ 1 6 i
ð3:16Þ
Then the N 1-forms dFmþl , 0 6 l 6 N 1, are linearly independent over some region of R2N . Therefore, Fm ; Fmþ1 ; . . . ; FmþN 1 are functionally independent. The proof is completed. h View the generating function Fk as a Hamiltonian in the symplectic space (R2N ; dp ^ dq), whose canonical equation is pj;sk ¼ qj;sk
oFk 2gðkÞ2 gðkj Þqj Qk ðgðKÞp; pÞ þ 2gðkj Þ2 pj þ 2gðkj Þ2 pj Qk ðgðKÞ2 p; qÞ ; ¼ oqj nðkÞ nðkj Þ
oFk 2gðkÞ2 gðkj Þpj Qk ðgðKÞq; qÞ 2gðkj Þ2 qj 2gðkj Þ2 qj Qk ðgðKÞ2 p; qÞ : ¼ ¼ opj nðkÞ nðkj Þ
ð3:17Þ
Through tedious calculations, we arrive at ðusk ; vsk ÞT ¼ 2gðkÞ2 PJGk ; in view of (3.17) and (3.2).
ð3:18Þ
X. Geng, H.H. Dai / Chaos, Solitons and Fractals 20 (2004) 311–321
317
4. Decomposition of soliton equations In order to decompose the soliton equations (2.17) into the Hamiltonian systems of ordinary differential equations, we define a new set of integrals fHk g recursively by F1 ¼ 2aH2 þ 2b2 H1 a2 H12 ;
F0 ¼ 2aH1 ;
F2 ¼ 2aH3 þ 2b2 H2 2a2 H1 H2 2ab2 H12 ; Fm ¼ 2aHmþ1 þ 2b2 Hm a2
m X
Hj Hmj 2ab2
j¼0
m1 X
Hj Hm1j b4
j¼0
m2 X
ð4:1Þ Hj Hm2j ;
m P 3;
j¼0
which is put in the equivalent form Fk ¼ ð1 gðkÞ2 Hk Þ2
ð4:2Þ
with the aid of the generating function Hk ¼
1 X
Hk nðkÞk1 :
ð4:3Þ
k¼1
The involutivity of fHk g is based on the equality fH1 ; Hk g ¼
1 gðkÞ2 gð1Þ2
pffiffiffiffiffiffiffiffiffi fF1 ; Fk g ¼ 0: Fk F1
The constraint (3.2) can be written as N X
rekj ¼ g0 þ d0
ð4:4Þ
j¼1
with the constant vector d0 ¼ ð0; 0; c0 ÞT . Operating with J 1 K upon (4.4) k times and noting ker J ¼ f.g0 j 8.ðconstantÞg give N X
nðkj Þk rekj ¼ gk þ c1 gk1 þ c2 gk2 þ þ ck g0 þ ð1Þk ak b2k d0 ;
kP1
ð4:5Þ
j¼1
in view of ðK nðkj ÞJ Þ rekj ¼ 0 and (2.13), where cj ’s are constants of integration. Resorting to (3.5) and (4.5), we arrive at ( ) N 1 N X X X ad0 rekj k1 k e kþ1 k 2k ¼ nðkÞ nðkj Þ r kj þ ð 1Þ a b d0 Gk ¼ nðkÞ nðkj Þ anðkÞ þ b2 j¼1 k¼0 j¼1 ¼
1 X
nðkÞk1 fgk þ c1 gk1 þ c2 gk2 þ þ ck g0 g ¼ ck gk
ð4:6Þ
k¼0
with ck ¼ 1 þ
1 X
ck nðkÞk :
ð4:7Þ
k¼1
From (2.26) and (3.7), we have Vk ¼ rðkÞ½ck gk ¼ ck rðkÞ½gk ;
ð4:8Þ
Fk ¼ c2k ;
ð4:9Þ
which implies by comparing (4.2) and (4.9) that ck ¼ 1 gðkÞ2 Hk ; c1 ¼ aH1 ;
ck ¼ aHk b2 Hk1 ;
ð4:10Þ k P 2:
ð4:11Þ
318
X. Geng, H.H. Dai / Chaos, Solitons and Fractals 20 (2004) 311–321
Denote the variables of the Hk -flow and Hk -flow by tk and tk , respectively. From (4.2), we have d 1 d ¼ : dtk 2gðkÞ2 ck dsk
ð4:12Þ
Using (3.18), (4.6), (2.23) and (2.18), we obtain (X0 ¼ 0) 1 X 1 1 utk usk ¼ ¼ PJGk ¼ PJgk ¼ Xk nðkÞk1 : 2 vtk v ck sk 2gðkÞ ck k¼1
ð4:13Þ
Therefore, we obtain the assertion. Theorem 4.1. Let ðpðx; tk Þ; qðx; tk ÞÞT be a compatible solution of the H -flow and Hk -flow. Then uðx; tk Þ ¼ hgðKÞpðx; tk Þ; pðx; tk Þi;
vðx; tk Þ ¼ hgðKÞqðx; tk Þ; qðx; tk Þi
ð4:14Þ
solves the kth soliton equation ðutk ; vtk ÞT ¼ Xk ðu; vÞ:
ð4:15Þ
Proof. We need only verify that fH ; Hk g ¼ 0. A direct calculation gives fhp; qi; Fk g ¼ 0, which leads to fhp; qi; Fj g ¼ 0 and fhp; qi; Hk g ¼ 0. This implies fH ; Hk g ¼ 0 because of (3.14) and F0 ¼ 2aH 2b2 hp; qi. Therefore, there exists the compatible solution of the H -flow and Hk -flow [16]. h
5. Straightening out of the flows Noting (3.8) and (3.9), we prove easily that Fk ¼ Vk12 Vk21 þ ðVk11 Þ2
h i RðnÞ ¼ 1 þ 2Qk ðgðKÞ2 p; qÞ Qk ðgðKÞp; pÞ ahgðKÞq; qi þ Qk ðgðKÞ3 q; qÞ þ Qk ðgðKÞ2 p; qÞ2 ¼ ; aðnÞ2
ð5:1Þ
mðnÞ ; aðnÞ nðnÞ ; ¼ gðkÞQk ðgðKÞq; qÞ ¼ gðkÞhgðKÞq; qi aðnÞ
ð5:2Þ
Vk12 ¼ gðkÞQk ðgðKÞp; pÞ ¼ gðkÞhgðKÞp; pi Vk21
where n ¼ nðkÞ, and N Y ½n nðkk Þ; aðnÞ ¼
bðnÞ ¼
k¼1
N Y
½n nðkkþN Þ;
k¼1
RðnÞ ¼ aðnÞbðnÞ ¼
2N Y
½n nðkj Þ;
ð5:3Þ
j¼1
mðnÞ ¼
N 1 Y
½n nðlj Þ;
nðnÞ ¼
j¼1
N 1 Y
½n nðmj Þ:
j¼1
From (5.3), we obtain N 1 X hgðKÞnðKÞp; pi ¼ c1 nðlj Þ; hgðKÞp; pi j¼1
N 1 X hgðKÞnðKÞq; qi ¼ c1 nðmj Þ; hgðKÞq; qi j¼1
1 hgðKÞ2 p; qi ¼ auv þ c2 ; 2 where c1 ¼
N X j¼1
nðkj Þ;
N 1 1X c2 ¼ c1 nðkjþN Þ: 2 2 j¼1
ð5:4Þ
ð5:5Þ
X. Geng, H.H. Dai / Chaos, Solitons and Fractals 20 (2004) 311–321
319
Resorting to (3.2) and (3.3), we have ux ¼ 2hgðKÞnðKÞp; pi 2uhgðKÞ2 p; qi;
ð5:6Þ
vx ¼ hgðKÞnðKÞq; qi þ 2vhgðKÞ2 p; qi; which, together with (5.4) and (5.5), implies o lnu ¼ 2 o lnv ¼ 2
N 1 X
nðlj Þ þ auv þ 2ðc1 c2 Þ;
j¼1 N 1 X
nðmj Þ auv 2ðc1 c2 Þ;
j¼1 N 1 X
o lnuv ¼ 2
nðmj Þ
j¼1
N 1 X
ð5:7Þ
! nðlj Þ :
j¼1
From (3.7), (3.17) and (4.12), we obtain i dVl12 4gðlÞ h gðkÞVl11 Vk12 gðlÞVk11 Vl12 ; ¼ nðlÞ nðkÞ dsk 21 i dVl 4gðlÞ h gðlÞVk11 Vl21 gðkÞVl11 Vk21 : ¼ nðlÞ nðkÞ dsk Using (5.1), we have pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rðnðlk ÞÞ Vl11 ; ¼ k aðnðlk ÞÞ
Vm11 ¼ k
ð5:8Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rðnðmk ÞÞ : aðnðmk ÞÞ
ð5:9Þ
Substituting l ¼ lk ; mk into the first expression and the second expression of (5.8), respectively, yields 1 dnðlk Þ 4gðkÞ2 mðnðkÞÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ¼ aðnðkÞÞ½nðkÞ nðlk Þm0 ðnðlk ÞÞ Rðnðlk ÞÞ dsk 1 dnðmk Þ 4gðkÞ2 nðnðkÞÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ¼ aðnðkÞÞ½nðkÞ nðlk Þn0 ðnðlk ÞÞ Rðnðmk ÞÞ dsk
ð5:10Þ
After some calculations we obtain by the interpolation formula for polynomials that N 1 X nðlk ÞN 1j dnðlk Þ 4gðkÞ2 nðkÞN 1j pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ¼ ; aðnðkÞÞ Rðnðlk ÞÞ dsk k¼1 N 1 X nðmk ÞN 1j dnðmk Þ 4gðkÞ2 nðkÞN 1j pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ¼ aðnðkÞÞ Rðnðmk ÞÞ dsk k¼1
1 6 j 6 N 1:
ð5:11Þ
Here nðlk Þ, nðmk Þ are called the elliptic coordinate of the finite-dimensional Hamiltonian system (3.3). Let us consider the hyperelliptic curve C defined by the affine equation, f2 ¼ RðnÞ, (n ¼ nðkÞ), with genus N 1 and the usual holomorphic differentials nN 1j dn ~ j ¼ pffiffiffiffiffiffiffiffiffiffi ; 1 6 j 6 N 1: x ð5:12Þ RðnÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Denote qðnðlk ÞÞ ¼ n ¼ nðlk Þ; f ¼ Rðnðlk ÞÞ , qðnðmk ÞÞ ¼ n ¼ nðmk Þ; f ¼ Rðnðmk ÞÞ 2 C. Let P0 2 C be fixed. We introduce the quasi-Abel–Jacobi coordinates by N 1 Z qðnðlk ÞÞ N 1 Z qðnðmk ÞÞ X X ~ j ; w~j ¼ ~ j ; 1 6 j 6 N 1: ð5:13Þ /~j ¼ x x k¼1
P0
k¼1
P0
Then (5.11) can be put in the form d/~j 4gðkÞ2 nðkÞN 1j ¼ ; dsk aðnðkÞÞ
dw~j 4gðkÞ2 nðkÞN 1j ¼ ; dsk aðnðkÞÞ
1 6 j 6 N 1:
ð5:14Þ
320
X. Geng, H.H. Dai / Chaos, Solitons and Fractals 20 (2004) 311–321
Let a1 ; b1 ; . . . ; aN 1 ; bN 1 be the canonical basis of cycles on the hyperelliptic curve C, and C ¼ ðAjk Þ1 ðN 1ÞðN 1Þ ;
Ajk ¼
Z
ð5:15Þ
~j: x ak
Define the normalized holomorphic differential by xs ¼
N 1 X
~j; Csj x
~: x ¼ ðx1 ; . . . ; xN 1 ÞT ¼ C x
ð5:16Þ
xs ¼ Bsk :
ð5:17Þ
j¼1
Then we have [17,18] Z
xs ¼ dsk ; ak
Z bk
The Abel map AðP Þ and the Abel–Jacobi coordinates are defined as Z P
X X nk AðPk Þ: x; A nk Pk ¼ AðP Þ ¼
ð5:18Þ
P0 N 1 X
/¼A
! qðnðlk ÞÞ
k¼1 N 1 X
w¼A
¼ ¼
k¼1
N 1 Z X k¼1
qðnðlk ÞÞ
qðnðmk ÞÞ
k
ð5:19Þ xj ¼ C w~:
P0
b ðz1 Þ ¼ Let Sk ¼ nðk1 Þ þ þ nðk2N Þ and R k
xj ¼ C /~;
P0
k¼1
! qðnðmk ÞÞ
N 1 Z X
Q2N
j¼1 ð1
nðkj Þz1 Þ. Then the coefficients in (see [7])
1 X 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Xk zk 1 b R ðz Þ k¼0
ð5:20Þ
are determined recursively by 1 K1 ¼ S1 ; 2 0 1 X 1 BSk þ Si Xj C Xk ¼ @ A: iþj¼k 2k
X0 ¼ 1;
ð5:21Þ
i;j P 1
From (4.10), (4.2) and (5.1) we get pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aðnðkÞÞck ¼ RðnðkÞÞ:
ð5:22Þ
Let Ck be the kth column vector of the matrix C. By (5.19), (5.14), (4.12) and (5.22) we have d/ d/~ 4gðkÞ2 nðkÞN 1 ðC1 nðkÞ1 þ þ CN 1 nðkÞN þ1 Þ; ¼C ¼ aðnðkÞÞ dsk dsk d/ 2nðkÞN 1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðC1 nðkÞ1 þ þ CN 1 nðkÞN þ1 Þ ¼ dtk RðnðkÞÞ ¼2
1 X
2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðC1 nðkÞ1 þ þ CN 1 nðkÞN þ1 Þ b ðnðkÞ1 Þ nðkÞ R
Xk nðkÞk1 ðC1 nðkÞ1 þ þ CN 1 nðkÞN þ1 Þ ¼
k¼0
1 X
e k nðkÞk1 ; X
ð5:23Þ
k¼1
with the constants (Ck ¼ 0, k P N ) e k ¼ 2ðXk1 C1 þ Xk2 C2 þ þ X0 Ck Þ; X
k P 1:
ð5:24Þ
X. Geng, H.H. Dai / Chaos, Solitons and Fractals 20 (2004) 311–321
321
Theorem 5.1 (Straightening out of the flow) 1 d/ X e k nðkÞk1 ; X ¼ dtk k¼1
d/ e ¼ Xk ; dtk
dw e k; ¼ X dtk
1 X dw e k nðkÞk1 ; X ¼ dtk k¼1
ð5:25Þ ð5:26Þ
k P 1:
Proof. In a way similar to the calculation of (5.24), we can derive the second expression of (5.25). Comparing the coefficients of kk1 in both sides of (5.25) yields (5.26). As an application of Theorem 5.1, we obtain the compatible solutions of flows Xk in Abel–Jacobi coordinates: e 1x þ X e k tk ; / ¼ /0 þ X
e 1x X e k tk ; w ¼ w0 X
kP2
ð5:27Þ
which can be inverted into explicit solutions in the original coordinates with the help of the Jacobi inversion technique [7,10,17]. h
Acknowledgements Project 10071075 was supported by National Natural Science Foundation of China. This work was supported by the Special Funds for Chinese Major State Basic Research Project ‘‘Nonlinear Science’’ and a grant from the Research Grants Council of the HKSAR, China (Project no. 9040644).
References [1] Ablowitz MJ, Segur H. Solitons and the inverse scattering transform. Philadelphia: SIAM; 1981. [2] Novikov SP, Manakov SV, Pitaevskii LP, Zakharov VE. Theory of solitons, the inverse scattering methods. New York: Consultants Bureau; 1984. [3] Wadati M, Konno K, Ichikawa YH. J Phys Soc Japan 1979;46:1965. [4] Cao CW, Geng XG. In: Nonlinear physics, research reports in physics. Berlin: Springer; 1990. p. 68–78. [5] Cao CW. Sci China A 1990;33:528. [6] Cao CW, Geng XG. J Phys A: Math Gen 1990;23:4117. [7] Cao CW, Wu YT, Geng XG. J Math Phys 1999;40:3948. [8] Dai HH, Geng XG. J Phys Soc Japan 1999;68:2878. [9] Cao CW, Geng XG, Wu YT. J Phys A: Math Gen 1999;32:8059. [10] Geng XG, Cao CW, Dai HH. J Phys A: Math Gen 2001;34:989. [11] Zeng YB. J Phys A: Math Gen 2000;33:L115. [12] Zhou RG. J Math Phys 1997;38:2535. [13] Geng XG. Physica A 2003;317:270. [14] Qiao ZJ. Rev Math Phys 2001;13:545. [15] Geng XG. Ann Diff Eqs 1989;5:397. [16] Arnold VI. Mathematical methods of classical mechanics. Berlin: Springer; 1978. [17] Griffiths P, Harris J. Principles of algebraic geometry. New York: Wiley; 1994. [18] Mumford D. Tata lectures on theta II. Boston: Birkh€auser; 1984.
Decomposition and straightening out of the coupled €dinger flows mixed nonlinear Schro Xianguo Geng b
a,*
, H.H. Dai
b
a Department of Mathematics, Zhengzhou University, Zhengzhou, Henan 450052, People’s Republic of China Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, People’s Republic of China
Accepted 22 July 2003 Communicated by Prof. M. Wadati
Abstract The nonlinearization approach of Lax pairs is extended to the investigation of a soliton hierarchy proposed by Wadati, Konno and Ichikawa, in which the first nontrivial equation is the coupled mixed nonlinear Schr€ odinger equation. Under a constraint between the potentials and eigenfunctions, solutions of the soliton hierarchy are decomposed into a class of new finite-dimensional Hamiltonian systems. The generating function of the integrals of motion is presented, by which the class of finite-dimensional Hamiltonian systems are further proved to be completely integrable in the Liouville sense. Based on the decomposition and the theory of algebraic curve, the Abel–Jacobi coordinates are introduced to straighten out the corresponding flows. As an application, the compatible solutions of the various flows in Abel–Jacobi coordinates are explicitly obtained. Ó 2003 Elsevier Ltd. All rights reserved.
1. Introduction Soliton equations are nonlinear partial differential equations having specific solutions with nice mathematical and physical properties [1–3]. However it is very difficult to solve them due to their high nonlinearity. Usually one considers the multidimensional problems to be solved in such a way as splitting into several lower-dimensional ones, which are easier to be treated with some available tools. The nonlinearization approach of Lax pairs [4–6] makes it possible to decompose the soliton equations into the compatible Hamiltonian systems of ordinary differential equations. There are at least two important applications of the nonlinearization approach. First, a considerable number of new finitedimensional integrable systems can be generated from the known soliton hierarchies. These finite-dimensional integrable systems inherit many properties of the corresponding soliton equations and enrich the theory of integrable system itself. Second, it provides a way of solving the soliton equation by separation of spatial and temporal variables. Explicit solutions of the soliton equation, which include soliton solution, periodic solution, quasi-periodic solution and other ones [7–14], can be obtained through solving the compatible system of ordinary differential equations. In this paper, our main aim is to study the decomposition of the soliton hierarchy proposed by Wadati, Konno and Ichikawa [3] and straightening out of the corresponding flows based on the nonlinearization approach of Lax pairs [4– 6]. We obtain a class of new finite-dimensional completely integrable systems in the Liouville sense and the compatible solutions of the corresponding flows in Abel–Jacobi coordinates. The present paper is organized as follows. In Section 2, for the sake of convenience we shall reconstruct this hierarchy of soliton equations with a linear term by using the Lenard gradient sequence. In Section 3, the nonlinearization of Lax pairs for the hierarchy of soliton equations leads to
*
Corresponding author. Fax: +852-2788-8561.
0960-0779/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0960-0779(03)00385-0
312
X. Geng, H.H. Dai / Chaos, Solitons and Fractals 20 (2004) 311–321
a class of new finite-dimensional Hamiltonian systems under the Bargmann constraint between the potentials and eigenfunctions. The generating function of the integrals of motion is found, by which the class of finite-dimensional Hamiltonian systems are further proved to be completely integrable in the Liouville sense. In Section 4, we introduce a set of new conserved integrals, from which solutions for the hierarchy of soliton equations are separated to solving the compatible Hamiltonian systems of ordinary differential equations. In Section 5, a hyperelliptic Riemann surface of genus N 1 and Abel–Jacobi coordinates are defined to straighten out the corresponding flows. As an application, the compatible solutions of the corresponding flows in Abel–Jacobi coordinates is explicitly obtained.
2. The hierarchy In this section, we construct the hierarchy of soliton equations associated with the spectral problem [3] nðkÞ gðkÞu ; vx ¼ U v; U ¼ gðkÞv nðkÞ
ð2:1Þ
where u and v are two potentials, k a constant spectral parameter, nðkÞ ¼ ak2 þ 2bk, gðkÞ ¼ ak þ b. We first introduce recursive equations 2Bmþ1 ¼ Bm;x þ 2uAm ; 2Cmþ1 ¼ Cm;x þ 2vAm ; m P 0; X X X Aj Ak þ a Bj Ck þ b2 Bj Ck ¼ 0; jþk¼m j;k P 0
ð2:2Þ
mP1
jþk¼m j;k P 0
jþk¼mþ1 j;k P 0
with starting points A0 ¼ 1;
B0 ¼ 0;
C0 ¼ 0:
Then it is easy to see that Am , Bm and Cm are uniquely determined by (2.2), and the first few members are 1 A1 ¼ auv; 2 1 1 2 1 1 B2 ¼ ux au v; C2 ¼ vx auv2 ; 2 2 2 2 1 3 2 2 2 1 2 A2 ¼ aðuvx ux vÞ þ a u v b uv; 4 8 2
ð2:3Þ
1 1 1 3 1 B3 ¼ uxx aðu2 vÞx þ auðuvx ux vÞ þ a2 u3 v2 b2 u2 v; 4 4 4 8 2 1 1 1 3 2 2 3 1 2 2 2 C3 ¼ vxx þ aðuv Þx þ avðuvx ux vÞ þ a u v b uv : 4 4 4 8 2
ð2:4Þ
B1 ¼ u;
C1 ¼ v;
Differentiating the third expression of (2.2) with respect to x and using its first and second expressions, we arrive at X X X ðAj Ak;x þ Ak Aj;x Þ þ 2a ðvBj Ak uAj Ck Þ þ 2b2 ðvBj Ak uCk Aj Þ ¼ 0; ð2:5Þ jþk¼m j;k P 0
jþk¼m j;k P 0
jþk¼mþ1 j;k P 0
in view of identities X ðCk Bjþ1 Bj Ckþ1 Þ ¼ 0; jþk¼mþ1 j;k P 0
X
ðCk Bjþ1 Bj Ckþ1 Þ ¼ 0:
ð2:6Þ
jþk¼m j;k P 0
Noticing B0 ¼ C0 ¼ 0, Eq. (2.5) can be written as m X
Amj ½Aj;x þ aðvBjþ1 uCjþ1 Þ þ b2 ðvBj uCj Þ ¼ 0;
ð2:7Þ
j¼0
which implies by induction that Aj;x þ aðvBjþ1 uCjþ1 Þ þ b2 ðvBj uCj Þ ¼ 0;
j P 0:
ð2:8Þ
X. Geng, H.H. Dai / Chaos, Solitons and Fractals 20 (2004) 311–321
313
e j by For the sake of convenience, we introduce A e jþ1 þ b2 A e j; Aj ¼ a A
e 0 ¼ 0; A
ð2:9Þ
j P 0;
which gives rise to e j;x ¼ uCj vBj ; A
ð2:10Þ
j P 0:
From (2.10), (2.3) and (2.4), we have e 2 ¼ 1 uv b2 a2 ; A 2 1 3 e A 3 ¼ ðuvx ux vÞ þ au2 v2 þ b4 a3 : 4 8
e 1 ¼ a1 ; A
ð2:11Þ
Substituting (2.9) into the first and second expressions of (2.2) yields e j ¼ 2Bjþ1 2au A e jþ1 ; Bj;x þ 2b2 u A 2 e e jþ1 : Cj;x 2b v A j ¼ 2Cjþ1 þ 2av A
ð2:12Þ
Eqs. (2.10) and (2.12) can be written as the Lenard recursive form Kgj1 ¼ Jgj ;
ð2:13Þ
jP0
e j ÞT and two operators where gj1 ¼ ðCj ; Bj ; A 0 1 0 0 0 o 2b2 u K ¼ @ o 0 2b2 v A; J ¼ @ 2 u u v o
K and J are defined by (o ¼ ox ) 1 2 2au 0 2av A: v o
ð2:14Þ
The first several Lenard gradients are 0
g1
0 1 0 ¼ @ 0 A; 0
0
1 v g0 ¼ @ u A; a1
1 1 1 2 v auv B 2 x 2 C B C B 1 C 1 2 B g1 ¼ B ux au v C C; 2 2 B C @ 1 A uv b2 a2 2
0
1 1 1 1 3 2 2 3 1 2 2 2 v aðuv avðuv a b þ Þ þ u vÞ þ u v uv xx x x x B4 C 4 4 8 2 B C B1 C 1 1 3 1 2 2 C 2 2 3 2 B g2 ¼ B uxx aðu vÞx þ auðuvx ux vÞ þ a u v b u v C: 4 4 4 8 2 B C @ A 1 3 ðuvx ux vÞ þ au2 v2 þ b4 a3 4 8 Assume that the time dependence of v for the spectral problem (2.1) obeys the equation ! ðmÞ ðmÞ V11 V12 ðmÞ ðmÞ ¼ vtm ¼ V v; V ðmÞ ðmÞ V21 V11
ð2:15Þ
with ðmÞ
V11 ¼
m X
e j gðkÞ2 nðkÞmj ; A
j¼0
ðmÞ
V12 ¼
m X j¼0
Bj gðkÞnðkÞmj ;
ðmÞ
V21 ¼
m X
Cj gðkÞnðkÞmj :
ð2:16Þ
j¼0
Noticing gðkÞ2 ¼ anðkÞ þ b2 , the compatibility condition of (2.1) and (2.16) yields the zero-curvature equation, Utm VxðmÞ þ ½U ; V ðmÞ ¼ 0, which is equivalent to the hierarchy of nonlinear evolution equations ðutm ; vtm ÞT ¼ Xm ;
mP0
ð2:17Þ
with the vector field Xj ¼ PKgj1 ¼ PJgj ;
ð2:18Þ
314
X. Geng, H.H. Dai / Chaos, Solitons and Fractals 20 (2004) 311–321
where P is the projective map c ¼ ðcð1Þ ; cð2Þ ; cð3Þ ÞT ! ðcð1Þ ; cð2Þ ÞT . The first three equations in the hierarchy (2.17) are as follows (X0 ¼ 0) ut1 ¼ ux ; vt1 ¼ vx ; 1 ut2 ¼ uxx aðu2 vÞx 2bu2 v 4b4 a2 u ; 2 1 vt2 ¼ vxx þ aðuv2 Þx 2buv2 4b4 a2 v ; 2 1 3 3 8 ut3 ¼ uxxx aðu2 vÞxx þ aðu2 vx uvux Þx þ a2 ðu3 v2 Þx b2 u 2vux au2 v2 b4 a3 ; 4 2 4 3 1 3 3 8 2 4 3 2 2 2 2 3 2 2 vt3 ¼ vxxx þ aðuv Þxx þ aðuvvx v ux Þx þ a ðu v Þx b v 2uvx þ au v þ b a : 4 2 4 3
ð2:19Þ ð2:20Þ
ð2:21Þ
Eqs. (2.20) and (2.21) are the coupled mixed nonlinear Schr€ odinger equation and its higher-order equation, respectively. The coupled mixed nonlinear Schr€ odinger equation has possesses infinitely many symmetries, which constitute some infinite-dimensional Lie algebra [15]. We now define the generating function of fgk g: gk ¼
1 X
gk nðkÞk1 ;
ð2:22Þ
k¼0
which satisfies ðK nðkÞJ Þgk ¼ 0;
ð1Þ
ð2Þ
ð1Þ
ugk þ vgk þ ogk ¼ 0
with the aid of (2.13) and Jg0 ¼ 0. Assume that ðanðkÞ þ b2 Þcð3Þ gðkÞcð2Þ V ðu; v; kÞ ¼ r½c ¼ : gðkÞcð1Þ ðanðkÞ þ b2 Þcð3Þ
ð2:23Þ
ð2:24Þ
Then we have the fundamental identity Vx ½U ; V ¼ U ½P fðK nðkÞJ Þcg;
ð2:25Þ
where U is the differential of the map ðu; vÞT ! U ðu; vÞ. Therefore, ðK nðkÞJ Þc ¼ 0 implies o detr½c ¼ 0, from which we deduce det r½gk ¼ 1
ð2:26Þ
in view of ð3Þ
ðanðkÞ þ b2 Þgk ¼ 1 þ
1 X
Ak nðkÞk1 :
ð2:27Þ
k¼0
3. The Bargmann constraint and finite-dimensional integrable systems In this section, we shall discuss the nonlinearization of the spectral problem (2.1). Let us consider N copies of the spectral problem (2.1)
pj;x qj;x
¼
nðkj Þ gðkj Þu gðkj Þv nðkj Þ
pj ; qj
16j6N
with distinct eigenvalues k ¼ k1 ; . . . ; kN . A simple calculation gives gðkj Þq2j dkj =du ¼ rkj ¼ 2 dkj =dv gðkj Þpj
ð3:1Þ
X. Geng, H.H. Dai / Chaos, Solitons and Fractals 20 (2004) 311–321
315
up to a constant factor. We introduce the Bargmann constraint [4] N X Pg0 ¼ rkj ; j¼1
that is u ¼ hgðKÞp; pi;
v ¼ hgðKÞq; qi;
ð3:2Þ
where h; i stands for the canonical inner product in RN , K ¼ diagðk1 ; . . . ; kN Þ, p ¼ ðp1 ; . . . ; pN ÞT , q ¼ ðq1 ; . . . ; qN ÞT . Substituting (3.2) into (3.1), we arrive at a finite-dimensional Hamiltonian system oH ; px ¼ nðKÞp hgðKÞp; pigðKÞq ¼ oq ð3:3Þ oH qx ¼ hgðKÞq; qigðKÞp nðKÞq ¼ op with the Hamiltonian 1 H ¼ hnðKÞp; qi þ hgðKÞp; pihgðKÞq; qi: 2
ð3:4Þ
Let Gk ¼
N X j¼1
0 1 0 1 gðkÞ2 Qk ðgðKÞq; qÞ 0 e k a 1 B C r j 2 @0A¼ ; 2 @ gðkÞ Qk ðgðKÞp; pÞ A nðkÞ nðkj Þ gðkÞ2 gðkÞ 2 c0 1 þ Q ðgðKÞ p; qÞ
ð3:5Þ
k
where rekj ¼ ðrkj ; pj qj Þ , c0 ¼ hp; qi a is defined as T
1
Qk ð~ p; q~Þ ¼ hðnðkÞI nðKÞÞ1 p~; q~i ¼
is a conserved integral of the x-flow and the bilinear function Qk ð~ p; q~Þ on RN
N X j¼1
p~j q~j : nðkÞ nðkj Þ
Then a direct verification shows that ðK nðkj ÞJ Þ rekj ¼ 0, which implies ðK nðkÞJ ÞGk ¼ 0:
ð3:6Þ
Utilizing (2.24), we introduce 1 þ Qk ðgðKÞ2 p; qÞ gðkÞQk ðgðKÞp; pÞ Vk ¼ r½Gk ¼ ; gðkÞQk ðgðKÞq; qÞ 1 Qk ðgðKÞ2 p; qÞ
ð3:7Þ
whose the determinant, det Vk , is invariant under the action of the x-flow. Resorting to the identities gðkÞ2 ¼ anðkÞ þ b2 ;
Qk ðgðKÞ3 p~; q~Þ ¼ gðkÞ2 Qk ðgðKÞ~ p; q~Þ ahgðKÞ~ p; q~i;
the generating function of integrals for the finite-dimensional Hamiltonian system (3.3) can be written as 1 þ Qk ðgðKÞ2 p; qÞ gðkÞQk ðgðKÞp; pÞ ; Fk ¼ det Vk ¼ gðkÞQk ðgðKÞq; qÞ 1 Qk ðgðKÞ2 p; qÞ
ð3:8Þ
ð3:9Þ
that is
Q ðgðKÞ3 p; pÞ Qk ðgðKÞ2 p; qÞ Fk ¼ 1 2Qk ðgðKÞ2 p; qÞ þ ahgðKÞp; piQk ðgðKÞq; qÞ þ k Qk ðgðKÞ2 p; qÞ Qk ðgðKÞq; qÞ 1 X ¼ 1 þ nðkÞm1 Fm ;
ð3:10Þ
m¼0
where F0 ¼ 2hgðKÞ2 p; qi þ ahgðKÞp; pihgðKÞq; qi ¼ 2aH 2b2 hp; qi; 3 j 2 mj1 m1 X p; qi hgðKÞ nðKÞ p; pi hgðKÞ nðKÞ 2 m m Fm ¼ 2hgðKÞ nðKÞ p; qi þ ahgðKÞp; pihgðKÞnðKÞ q; qi þ : 2 j hgðKÞnðKÞmj1 q; qi j¼0 hgðKÞ nðKÞ p; qi
ð3:11Þ
316
X. Geng, H.H. Dai / Chaos, Solitons and Fractals 20 (2004) 311–321
In what follows, we prove that the involutivity of fFm g, that is fFm ; Fl g ¼ 0 for m; l P 0. Here the Poisson bracket of smooth functions in the symplectic space ðR2N ; dp ^ dqÞ is defined as N X of og of og of og of og ¼ ff ; gg ¼ ; ; : oqj opj opj oqj oq op op oq j¼1 By using (3.9), a direct calculation shows that fF1 ; Fk g ¼ 0;
81; k 2 C
ð3:12Þ
with the help of equality E D ðnð1ÞI nðKÞÞ1 ðnðkÞI nðKÞÞ1 gðKÞl p~; q~ ¼
h i 1 Qk ðgðKÞl p~; q~Þ Q1 ðgðKÞl p~; q~Þ : nð1Þ nðkÞ
ð3:13Þ
Therefore, we have the following fact. Proposition 3.1. The functions Fm defined by (3.11) are in involution in pairs, fFm ; Fl g ¼ 0;
8m; l P 0:
ð3:14Þ
Theorem 3.1. The finite-dimensional Hamiltonian systems p sm ¼
oFm ; oq
q sm ¼
oFm ; op
mP0
ð3:15Þ
are completely integrable in the Liouville sense. Proof. By Proposition 3.1, we need only prove that the conserved integrals fFmþl g; 0 6 l 6 N 1, are functionally independent. In fact, from (3.11) we have oFmþl ¼ 2gðKÞ2 nðKÞmþl q: op p¼0 Therefore, we arrive oFm oFmþ1 op1 op1 oF m oFmþ1 op2 op2 oF m oFmþ1 opN opN
at
oFmþN 1 op1 N oFmþN 1 Y Y N gðkj Þ2 nðkj Þm qj ðnðkj Þ nðki ÞÞ 6 0: op2 ¼ ð2Þ 1 6 i
ð3:16Þ
Then the N 1-forms dFmþl , 0 6 l 6 N 1, are linearly independent over some region of R2N . Therefore, Fm ; Fmþ1 ; . . . ; FmþN 1 are functionally independent. The proof is completed. h View the generating function Fk as a Hamiltonian in the symplectic space (R2N ; dp ^ dq), whose canonical equation is pj;sk ¼ qj;sk
oFk 2gðkÞ2 gðkj Þqj Qk ðgðKÞp; pÞ þ 2gðkj Þ2 pj þ 2gðkj Þ2 pj Qk ðgðKÞ2 p; qÞ ; ¼ oqj nðkÞ nðkj Þ
oFk 2gðkÞ2 gðkj Þpj Qk ðgðKÞq; qÞ 2gðkj Þ2 qj 2gðkj Þ2 qj Qk ðgðKÞ2 p; qÞ : ¼ ¼ opj nðkÞ nðkj Þ
ð3:17Þ
Through tedious calculations, we arrive at ðusk ; vsk ÞT ¼ 2gðkÞ2 PJGk ; in view of (3.17) and (3.2).
ð3:18Þ
X. Geng, H.H. Dai / Chaos, Solitons and Fractals 20 (2004) 311–321
317
4. Decomposition of soliton equations In order to decompose the soliton equations (2.17) into the Hamiltonian systems of ordinary differential equations, we define a new set of integrals fHk g recursively by F1 ¼ 2aH2 þ 2b2 H1 a2 H12 ;
F0 ¼ 2aH1 ;
F2 ¼ 2aH3 þ 2b2 H2 2a2 H1 H2 2ab2 H12 ; Fm ¼ 2aHmþ1 þ 2b2 Hm a2
m X
Hj Hmj 2ab2
j¼0
m1 X
Hj Hm1j b4
j¼0
m2 X
ð4:1Þ Hj Hm2j ;
m P 3;
j¼0
which is put in the equivalent form Fk ¼ ð1 gðkÞ2 Hk Þ2
ð4:2Þ
with the aid of the generating function Hk ¼
1 X
Hk nðkÞk1 :
ð4:3Þ
k¼1
The involutivity of fHk g is based on the equality fH1 ; Hk g ¼
1 gðkÞ2 gð1Þ2
pffiffiffiffiffiffiffiffiffi fF1 ; Fk g ¼ 0: Fk F1
The constraint (3.2) can be written as N X
rekj ¼ g0 þ d0
ð4:4Þ
j¼1
with the constant vector d0 ¼ ð0; 0; c0 ÞT . Operating with J 1 K upon (4.4) k times and noting ker J ¼ f.g0 j 8.ðconstantÞg give N X
nðkj Þk rekj ¼ gk þ c1 gk1 þ c2 gk2 þ þ ck g0 þ ð1Þk ak b2k d0 ;
kP1
ð4:5Þ
j¼1
in view of ðK nðkj ÞJ Þ rekj ¼ 0 and (2.13), where cj ’s are constants of integration. Resorting to (3.5) and (4.5), we arrive at ( ) N 1 N X X X ad0 rekj k1 k e kþ1 k 2k ¼ nðkÞ nðkj Þ r kj þ ð 1Þ a b d0 Gk ¼ nðkÞ nðkj Þ anðkÞ þ b2 j¼1 k¼0 j¼1 ¼
1 X
nðkÞk1 fgk þ c1 gk1 þ c2 gk2 þ þ ck g0 g ¼ ck gk
ð4:6Þ
k¼0
with ck ¼ 1 þ
1 X
ck nðkÞk :
ð4:7Þ
k¼1
From (2.26) and (3.7), we have Vk ¼ rðkÞ½ck gk ¼ ck rðkÞ½gk ;
ð4:8Þ
Fk ¼ c2k ;
ð4:9Þ
which implies by comparing (4.2) and (4.9) that ck ¼ 1 gðkÞ2 Hk ; c1 ¼ aH1 ;
ck ¼ aHk b2 Hk1 ;
ð4:10Þ k P 2:
ð4:11Þ
318
X. Geng, H.H. Dai / Chaos, Solitons and Fractals 20 (2004) 311–321
Denote the variables of the Hk -flow and Hk -flow by tk and tk , respectively. From (4.2), we have d 1 d ¼ : dtk 2gðkÞ2 ck dsk
ð4:12Þ
Using (3.18), (4.6), (2.23) and (2.18), we obtain (X0 ¼ 0) 1 X 1 1 utk usk ¼ ¼ PJGk ¼ PJgk ¼ Xk nðkÞk1 : 2 vtk v ck sk 2gðkÞ ck k¼1
ð4:13Þ
Therefore, we obtain the assertion. Theorem 4.1. Let ðpðx; tk Þ; qðx; tk ÞÞT be a compatible solution of the H -flow and Hk -flow. Then uðx; tk Þ ¼ hgðKÞpðx; tk Þ; pðx; tk Þi;
vðx; tk Þ ¼ hgðKÞqðx; tk Þ; qðx; tk Þi
ð4:14Þ
solves the kth soliton equation ðutk ; vtk ÞT ¼ Xk ðu; vÞ:
ð4:15Þ
Proof. We need only verify that fH ; Hk g ¼ 0. A direct calculation gives fhp; qi; Fk g ¼ 0, which leads to fhp; qi; Fj g ¼ 0 and fhp; qi; Hk g ¼ 0. This implies fH ; Hk g ¼ 0 because of (3.14) and F0 ¼ 2aH 2b2 hp; qi. Therefore, there exists the compatible solution of the H -flow and Hk -flow [16]. h
5. Straightening out of the flows Noting (3.8) and (3.9), we prove easily that Fk ¼ Vk12 Vk21 þ ðVk11 Þ2
h i RðnÞ ¼ 1 þ 2Qk ðgðKÞ2 p; qÞ Qk ðgðKÞp; pÞ ahgðKÞq; qi þ Qk ðgðKÞ3 q; qÞ þ Qk ðgðKÞ2 p; qÞ2 ¼ ; aðnÞ2
ð5:1Þ
mðnÞ ; aðnÞ nðnÞ ; ¼ gðkÞQk ðgðKÞq; qÞ ¼ gðkÞhgðKÞq; qi aðnÞ
ð5:2Þ
Vk12 ¼ gðkÞQk ðgðKÞp; pÞ ¼ gðkÞhgðKÞp; pi Vk21
where n ¼ nðkÞ, and N Y ½n nðkk Þ; aðnÞ ¼
bðnÞ ¼
k¼1
N Y
½n nðkkþN Þ;
k¼1
RðnÞ ¼ aðnÞbðnÞ ¼
2N Y
½n nðkj Þ;
ð5:3Þ
j¼1
mðnÞ ¼
N 1 Y
½n nðlj Þ;
nðnÞ ¼
j¼1
N 1 Y
½n nðmj Þ:
j¼1
From (5.3), we obtain N 1 X hgðKÞnðKÞp; pi ¼ c1 nðlj Þ; hgðKÞp; pi j¼1
N 1 X hgðKÞnðKÞq; qi ¼ c1 nðmj Þ; hgðKÞq; qi j¼1
1 hgðKÞ2 p; qi ¼ auv þ c2 ; 2 where c1 ¼
N X j¼1
nðkj Þ;
N 1 1X c2 ¼ c1 nðkjþN Þ: 2 2 j¼1
ð5:4Þ
ð5:5Þ
X. Geng, H.H. Dai / Chaos, Solitons and Fractals 20 (2004) 311–321
319
Resorting to (3.2) and (3.3), we have ux ¼ 2hgðKÞnðKÞp; pi 2uhgðKÞ2 p; qi;
ð5:6Þ
vx ¼ hgðKÞnðKÞq; qi þ 2vhgðKÞ2 p; qi; which, together with (5.4) and (5.5), implies o lnu ¼ 2 o lnv ¼ 2
N 1 X
nðlj Þ þ auv þ 2ðc1 c2 Þ;
j¼1 N 1 X
nðmj Þ auv 2ðc1 c2 Þ;
j¼1 N 1 X
o lnuv ¼ 2
nðmj Þ
j¼1
N 1 X
ð5:7Þ
! nðlj Þ :
j¼1
From (3.7), (3.17) and (4.12), we obtain i dVl12 4gðlÞ h gðkÞVl11 Vk12 gðlÞVk11 Vl12 ; ¼ nðlÞ nðkÞ dsk 21 i dVl 4gðlÞ h gðlÞVk11 Vl21 gðkÞVl11 Vk21 : ¼ nðlÞ nðkÞ dsk Using (5.1), we have pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rðnðlk ÞÞ Vl11 ; ¼ k aðnðlk ÞÞ
Vm11 ¼ k
ð5:8Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rðnðmk ÞÞ : aðnðmk ÞÞ
ð5:9Þ
Substituting l ¼ lk ; mk into the first expression and the second expression of (5.8), respectively, yields 1 dnðlk Þ 4gðkÞ2 mðnðkÞÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ¼ aðnðkÞÞ½nðkÞ nðlk Þm0 ðnðlk ÞÞ Rðnðlk ÞÞ dsk 1 dnðmk Þ 4gðkÞ2 nðnðkÞÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ¼ aðnðkÞÞ½nðkÞ nðlk Þn0 ðnðlk ÞÞ Rðnðmk ÞÞ dsk
ð5:10Þ
After some calculations we obtain by the interpolation formula for polynomials that N 1 X nðlk ÞN 1j dnðlk Þ 4gðkÞ2 nðkÞN 1j pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ¼ ; aðnðkÞÞ Rðnðlk ÞÞ dsk k¼1 N 1 X nðmk ÞN 1j dnðmk Þ 4gðkÞ2 nðkÞN 1j pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ¼ aðnðkÞÞ Rðnðmk ÞÞ dsk k¼1
1 6 j 6 N 1:
ð5:11Þ
Here nðlk Þ, nðmk Þ are called the elliptic coordinate of the finite-dimensional Hamiltonian system (3.3). Let us consider the hyperelliptic curve C defined by the affine equation, f2 ¼ RðnÞ, (n ¼ nðkÞ), with genus N 1 and the usual holomorphic differentials nN 1j dn ~ j ¼ pffiffiffiffiffiffiffiffiffiffi ; 1 6 j 6 N 1: x ð5:12Þ RðnÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Denote qðnðlk ÞÞ ¼ n ¼ nðlk Þ; f ¼ Rðnðlk ÞÞ , qðnðmk ÞÞ ¼ n ¼ nðmk Þ; f ¼ Rðnðmk ÞÞ 2 C. Let P0 2 C be fixed. We introduce the quasi-Abel–Jacobi coordinates by N 1 Z qðnðlk ÞÞ N 1 Z qðnðmk ÞÞ X X ~ j ; w~j ¼ ~ j ; 1 6 j 6 N 1: ð5:13Þ /~j ¼ x x k¼1
P0
k¼1
P0
Then (5.11) can be put in the form d/~j 4gðkÞ2 nðkÞN 1j ¼ ; dsk aðnðkÞÞ
dw~j 4gðkÞ2 nðkÞN 1j ¼ ; dsk aðnðkÞÞ
1 6 j 6 N 1:
ð5:14Þ
320
X. Geng, H.H. Dai / Chaos, Solitons and Fractals 20 (2004) 311–321
Let a1 ; b1 ; . . . ; aN 1 ; bN 1 be the canonical basis of cycles on the hyperelliptic curve C, and C ¼ ðAjk Þ1 ðN 1ÞðN 1Þ ;
Ajk ¼
Z
ð5:15Þ
~j: x ak
Define the normalized holomorphic differential by xs ¼
N 1 X
~j; Csj x
~: x ¼ ðx1 ; . . . ; xN 1 ÞT ¼ C x
ð5:16Þ
xs ¼ Bsk :
ð5:17Þ
j¼1
Then we have [17,18] Z
xs ¼ dsk ; ak
Z bk
The Abel map AðP Þ and the Abel–Jacobi coordinates are defined as Z P
X X nk AðPk Þ: x; A nk Pk ¼ AðP Þ ¼
ð5:18Þ
P0 N 1 X
/¼A
! qðnðlk ÞÞ
k¼1 N 1 X
w¼A
¼ ¼
k¼1
N 1 Z X k¼1
qðnðlk ÞÞ
qðnðmk ÞÞ
k
ð5:19Þ xj ¼ C w~:
P0
b ðz1 Þ ¼ Let Sk ¼ nðk1 Þ þ þ nðk2N Þ and R k
xj ¼ C /~;
P0
k¼1
! qðnðmk ÞÞ
N 1 Z X
Q2N
j¼1 ð1
nðkj Þz1 Þ. Then the coefficients in (see [7])
1 X 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Xk zk 1 b R ðz Þ k¼0
ð5:20Þ
are determined recursively by 1 K1 ¼ S1 ; 2 0 1 X 1 BSk þ Si Xj C Xk ¼ @ A: iþj¼k 2k
X0 ¼ 1;
ð5:21Þ
i;j P 1
From (4.10), (4.2) and (5.1) we get pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aðnðkÞÞck ¼ RðnðkÞÞ:
ð5:22Þ
Let Ck be the kth column vector of the matrix C. By (5.19), (5.14), (4.12) and (5.22) we have d/ d/~ 4gðkÞ2 nðkÞN 1 ðC1 nðkÞ1 þ þ CN 1 nðkÞN þ1 Þ; ¼C ¼ aðnðkÞÞ dsk dsk d/ 2nðkÞN 1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðC1 nðkÞ1 þ þ CN 1 nðkÞN þ1 Þ ¼ dtk RðnðkÞÞ ¼2
1 X
2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðC1 nðkÞ1 þ þ CN 1 nðkÞN þ1 Þ b ðnðkÞ1 Þ nðkÞ R
Xk nðkÞk1 ðC1 nðkÞ1 þ þ CN 1 nðkÞN þ1 Þ ¼
k¼0
1 X
e k nðkÞk1 ; X
ð5:23Þ
k¼1
with the constants (Ck ¼ 0, k P N ) e k ¼ 2ðXk1 C1 þ Xk2 C2 þ þ X0 Ck Þ; X
k P 1:
ð5:24Þ
X. Geng, H.H. Dai / Chaos, Solitons and Fractals 20 (2004) 311–321
321
Theorem 5.1 (Straightening out of the flow) 1 d/ X e k nðkÞk1 ; X ¼ dtk k¼1
d/ e ¼ Xk ; dtk
dw e k; ¼ X dtk
1 X dw e k nðkÞk1 ; X ¼ dtk k¼1
ð5:25Þ ð5:26Þ
k P 1:
Proof. In a way similar to the calculation of (5.24), we can derive the second expression of (5.25). Comparing the coefficients of kk1 in both sides of (5.25) yields (5.26). As an application of Theorem 5.1, we obtain the compatible solutions of flows Xk in Abel–Jacobi coordinates: e 1x þ X e k tk ; / ¼ /0 þ X
e 1x X e k tk ; w ¼ w0 X
kP2
ð5:27Þ
which can be inverted into explicit solutions in the original coordinates with the help of the Jacobi inversion technique [7,10,17]. h
Acknowledgements Project 10071075 was supported by National Natural Science Foundation of China. This work was supported by the Special Funds for Chinese Major State Basic Research Project ‘‘Nonlinear Science’’ and a grant from the Research Grants Council of the HKSAR, China (Project no. 9040644).
References [1] Ablowitz MJ, Segur H. Solitons and the inverse scattering transform. Philadelphia: SIAM; 1981. [2] Novikov SP, Manakov SV, Pitaevskii LP, Zakharov VE. Theory of solitons, the inverse scattering methods. New York: Consultants Bureau; 1984. [3] Wadati M, Konno K, Ichikawa YH. J Phys Soc Japan 1979;46:1965. [4] Cao CW, Geng XG. In: Nonlinear physics, research reports in physics. Berlin: Springer; 1990. p. 68–78. [5] Cao CW. Sci China A 1990;33:528. [6] Cao CW, Geng XG. J Phys A: Math Gen 1990;23:4117. [7] Cao CW, Wu YT, Geng XG. J Math Phys 1999;40:3948. [8] Dai HH, Geng XG. J Phys Soc Japan 1999;68:2878. [9] Cao CW, Geng XG, Wu YT. J Phys A: Math Gen 1999;32:8059. [10] Geng XG, Cao CW, Dai HH. J Phys A: Math Gen 2001;34:989. [11] Zeng YB. J Phys A: Math Gen 2000;33:L115. [12] Zhou RG. J Math Phys 1997;38:2535. [13] Geng XG. Physica A 2003;317:270. [14] Qiao ZJ. Rev Math Phys 2001;13:545. [15] Geng XG. Ann Diff Eqs 1989;5:397. [16] Arnold VI. Mathematical methods of classical mechanics. Berlin: Springer; 1978. [17] Griffiths P, Harris J. Principles of algebraic geometry. New York: Wiley; 1994. [18] Mumford D. Tata lectures on theta II. Boston: Birkh€auser; 1984.