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Relation among nonlinear evolution equations and their reductions
Chaos, Solitons and Fractals 27 (2006) 813–821 www.elsevier.com/locate/chaos
Relation among nonlinear evolution equations and their reductions Jinbing Chen *, Xianguo Geng Department of Mathematics, Zhengzhou University, Zhengzhou 450052, Henan, PR China
Abstract The LCZ soliton hierarchy is presented, and their generalized Hamiltonian structures are deduced. From the compatibility of soliton equations, it is shown that this soliton hierarchy is closely related to the Burger equation, the mKP equation and a new (2 + 1)-dimensional nonlinear evolution equation (NEE). Resorting to the nonlinearization of Lax pairs (NLP), all the resulting NEEs are reduced into integrable Hamiltonian systems of ordinary differential equations (ODEs). As a concrete application, the solutions for NEEs can be derived via solving the corresponding ODEs. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction Soliton equations are a remarkable class of NEEs having nice mathematical and physical properties in the nonlinear science. Thus the investigation for them is one of the most highlighted subjects in mathematical physics. In the past decades, many systematic methods were developed to study soliton equations such as the inverse scattering method, the Darboux transformation, the algebro-geometric method and the dressing method [1–6], etc. It is worthwhile to mention that the NLP [7–9] is also effective in studying soliton equations, and has been generalized to investigate multidimensional problems recently. The most important message relating to the progress of this manipulation is that the compatible solutions of soliton equations naturally give rise to solutions of (2 + 1)-dimensional NEEs [10,11]. Consequently many soliton equations were solved after this procedure [12–14]. Usually one considers multi-dimensional 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. Along with this idea, we devote to reduce NEEs into integrable ODEs such that solutions of NEEs can be achieved by solving ODEs. For this purpose, on application of the NLP, a family of finite-dimensional Hamiltonian systems (FDHSs) are obtained constituting the decomposition of the resulting NEEs, which are completely integrable in the Liouville sense. This article is organized as follows. In Section 2, the LCZ soliton hierarchy and two (2 + 1)-dimensional systems are deduced. In Section 3, the NLP with the Bargmann constraint is studied. In Section 4, the integrability of FDHSs is completed by means of the elliptic coordinates and quasi-Abel–Jacobi coordinates. In Section 5, the NEEs are reduced into FDHSs such that solutions for NEEs can be attained by solving integrable ODEs.
*
Corresponding author. E-mail address: [email protected] (J. Chen).
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.04.054
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J. Chen, X. Geng / Chaos, Solitons and Fractals 27 (2006) 813–821
2. The nonlinear evolution equations In this section, we deduce the soliton hierarchy, the Burger equation, the mKP equation, a new (2 + 1)-dimensional NEE, and further calculate the generalized Hamiltonian structures for the soliton hierarchy. Now let us begin with the LCZ eigenvalue problem [15,16], k þ v u þ v u1 ; ð2:1Þ ; u¼ ux ¼ U u; U ¼ uv kv u2 where k is a constant spectral parameter, and u, v are two potentials. Take into account the adjoint representation of (2.1), X aj bj þ cj j a bþc ¼ V x ¼ ½U ; V ; V ¼ k ; ð2:2Þ bj cj aj b c a jP0 which yields
1
aj ¼ 2o ðvbj ucj Þ;
bjþ1 cjþ1
¼
v 2uo1 v
12 o þ 2uo1 u
12 o 2vo1 v
v þ 2vo1 u
!
bj ; cj
ð2:3Þ
where o = o/ox, oo1 = o1o = 1. From (2.3) and initial values a0 = 2, b0 = c0 = 0, we have a1 ¼ 0;
b1 ¼ 2u;
a2 ¼ v2 u2 ;
c1 ¼ 2v;
b2 ¼ vx 2uv;
c2 ¼ ux 2v2 .
For any positive integral n, we introduce an auxiliary eigenvalue problem of (2.1), ! ðnÞ ðnÞ V 12 V 11 ðnÞ ðnÞ utn ¼ V u; V ¼ ; n P 1; ðnÞ ðnÞ V 21 V 11
ð2:4Þ
ð2:5Þ
where ðnÞ
V 11 ¼
n1 X
aj knj þ cn ;
ðnÞ
V 12 ¼
j¼0
n X ðbj þ cj Þknj ; j¼0
ðnÞ
V 21 ¼
n X ðbj cj Þknj . j¼0
Then the compatibility condition of (2.1) and (2.5), i.e., uxtn ¼ utn x , leads to the zero-curvature equation ðnÞ ¼ 0; U tn V ðnÞ x þ ½U ; V
ð2:6Þ
which is nothing but the LCZ soliton hierarchy ðu; vÞTtn ¼ Jgn1 ; where
gn ¼
ð2:7Þ
n P 1;
bnþ1 ; anþ1 cnþ1
J¼
o 0 . 0 o
Together with (2.3) and the Lenard recursive relation Kgj1 = Jgj (j P 0), we get ! 1 2 0 o ou 2 . K¼ 12 o2 uo ov vo Designating g2 = (2, 0)T and g1 = (0, 2)T, it is clear that ker J ¼ fq1 g1 þ q2 g2 j8q1 ; q2 2 Rg. On use of the Lenard recursive relation, gj (j P 0) can be uniquely established up to ker J. In particular, g0 ¼ ð2u; 2vÞT ;
g1 ¼ ðvx 2uv; 3v2 u2 ux ÞT .
ð2:8Þ
Here {gj} is the Lenard sequence, and K, J are the Lenard operator pairs. Now we list the first two nontrivial equations of (2.7) with y = t2 and t = t3, respectively, vy ¼ uxx þ 2uux 6vvx ; uy ¼ vxx 2ux v 2uvx ; 8 1 > < ut ¼ uxxx þ 3u2 ux 3ux v2 þ 3vvxx þ 3v2x 6uvvx ; 2 > : v ¼ 1v þ 3u v þ 3u2 v þ 3u v þ 6uu v 15v2 v . t xxx xx x x x x x 2
ð2:9Þ ð2:10Þ
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815
If simply taking u = v, (2.9) can be reduced as the well-known Burger equation uy ¼ uxx 4uux . Let r(x, y, t) = v(x, y, t) be the compatible solution of (2.9) and (2.10). Following (2.9), a lengthy calculation shows that 2 2 o1 x r y ¼ ux þ u 3r ;
1 2 o1 x r yy ¼ r xxx 8rr y 24r r x 4r x ox r y .
ð2:11Þ
Substituting (2.11) into (2.10)2 implies the following (2 + 1)-dimensional NEEs, 1 3 rt ¼ ðrxx 8r3 Þx ðo1 ryy 4rx o1 x r y Þ; 8 8 x 1 rt ¼ rxxx þ 12r2 rx þ 3rry þ 3rx o1 x ry . 2
ð2:12Þ ð2:13Þ
It is remarkable to see that (2.12) is the mKP equation [17] and (2.13) is a new (2 + 1)-dimensional system, which are closely connected with two soliton equations (2.9) and (2.10). In what follows, we are ready to calculate the generalized Hamiltonian structure of (2.7). It is easy from (2.1) and (2.2) to calculate that oU oU oU tr V ¼ 2a; tr V ¼ 2b; tr V ¼ 2a 2c. ð2:14Þ ok ou ov Noticing the trace identity [18], T d d o ; ð2aÞ ¼ ks ks ð2b; 2a 2cÞT ; du dv ok where s is a constant to be fixed. After a simple calculation we know that s = 0 and T d d ; Hn ¼ gn ; n 2 Zþ ; du dv where Hn ¼ anþ1 =n. Therefore (2.7) adopts the following generalized Hamiltonian form: T dHn1 dHn1 ; . ðu; vÞTtn ¼ J du dv In particular, the Hamiltonian form of (2.9) reads H1 ¼ v2 u2 . In order to proceed our work, let us introduce a generating function of {gk}: 1 X gk kk1 ; gk ¼ g1 þ k¼0
it is easy to check that (K kJ)gk = 0. Denoting g ¼ ðgð1Þ ; gð2Þ ÞT ¼ V(n) can be rewritten as a concise form, V
ðnÞ
,rðu; v; kÞ½g ¼
ð2Þ 12 gð1Þ þ kgð2Þ x vg 1 ð2Þ ð1Þ ðg þ g Þx ðu vÞgð2Þ 2
Pn1
ð2:15Þ
n1j , j¼0 gj1 k
1 ðgð2Þ gð1Þ Þx ðu þ vÞgð2Þ 2 1 ð1Þ g þ vgð2Þ kgð2Þ 2 x
an immediate consequence is that
! .
ð2:16Þ
A straightforward calculation delivers V x ½U ; V ¼ U ½ðK kJ Þg; ð2:17Þ where U ðnÞ ¼ ded e¼0 U ðu þ en1 ; v þ en2 Þ. Therefore, (K kJ)g = 0 implies that o det r[g] = 0 [18]. Whence, we get from (2.15) and (2.16) that det r½gk ¼ 4k2 .
ð2:18Þ
3. The nonlinearization of Lax pairs Let k1, k2, . . ., kN be N distinct eigenvalues and u = (pj, qj)T. We take N copies of the eigenvalue problem (2.1), ! ! pj pj kj þ v u þ v ; 1 6 j 6 N. ¼ ð3:1Þ qj qj uv kj v x
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J. Chen, X. Geng / Chaos, Solitons and Fractals 27 (2006) 813–821
A simple calculation gives rise to ðK kj J Þrkj ¼ 0;
rkj ¼
T T dkj dkj ; ¼ q2j p2j ; p2j þ 2pj qj þ q2j . du dv
Let us focus on the Bargmann constraint [7–9] N X rkj ; g0 ¼
ð3:2Þ
ð3:3Þ
j¼1
which yields 1 u ¼ ðhp; pi hq; qiÞ; 2
1 v ¼ ðhp; pi þ 2hp; qi þ hq; qiÞ; 2
ð3:4Þ
where p = (p1, . . ., pN)T, q = (q1, . . ., qN)T, and hÆ, Æi stands for the standard inner product in RN . On insertion of the expression (3.4) into (3.1), we obtain the following FDHS: 8 1 oH 0 > > < px ¼ Kp þ 2 ðhp; pi þ 2hp; qi þ hq; qiÞp þ ðhp; pi þ hp; qiÞq ¼ oq ; ð3:5Þ > 1 oH 0 > : qx ¼ Kq ðhp; pi þ 2hp; qi þ hq; qiÞq ðhp; qi þ hq; qiÞp ¼ ; 2 op where K = diag(k1, . . ., kN), and 1 H 0 ¼ hKp; qi ðhp; pi þ hp; qiÞðhp; qi þ hq; qiÞ. 2
ð3:6Þ
Only for convenience, we introduce a bilinear function Qk(n, g) which plays an important role in our consideration, N 1 X X nj gj ¼ km1 hKm n; gi. Qk ðn; gÞ ¼ hðkI KÞ1 n; gi ¼ k kj m¼0 j¼1 Designating Gk ¼ g1 þ
N X Qk ðq; qÞ Qk ðp; pÞ rkj ¼ ; k kj 2 þ Qk ðp; pÞ þ 2Qk ðp; qÞ þ Qk ðq; qÞ j¼1
it is not difficult to check that (K kJ)Gk = 0. Substituting (3.7) into (2.16) leads to 2ðQk ðKp; pÞ þ hp; pi þ hp; qiÞ 2k þ 2Qk ðKp; qÞ ; Vk ¼ 2ðQk ðKq; qÞ þ hp; qi þ hq; qiÞ 2k 2Qk ðKp; qÞ
ð3:7Þ
ð3:8Þ
in view of Qk(Kn, g) = kQk(n, g) hn, gi. Assuming that Fk = det Vk, it is easy to see that Fk is invariant under the action of x-flow. Hence the generating function of integrals of motion for (3.5) can be written as F k ¼ 4k2 þ 4ðhp; pi þ hp; qiÞðhp; qi þ hq; qiÞ 8kQk ðKp; qÞ þ 4ðhp; pi þ hp; qiÞQk ðKq; qÞ Qk ðKp; pÞ Qk ðKp; qÞ þ 4ðhp; qi þ hq; qiÞQk ðKp; pÞ þ Qk ðKp; qÞ Qk ðKq; qÞ 1 X F m km ; ¼ 4k2 þ
ð3:9Þ
m¼0
where F 0 ¼ 8hKp; qi þ 4ðhp; pi þ hp; qiÞðhp; qi þ hq; qiÞ ¼ 8H 0 ; F 1 ¼ 8hK2 p; qi þ 4ðhp; pi þ hp; qiÞhKq; qi þ 4ðhp; qi þ hq; qiÞhKp; pi; F mþ1 ¼ 8hKmþ2 p; qi þ 4ðhp; pi þ hp; qiÞhKmþ1 q; qi þ 4ðhp; qi þ hq; qiÞhKmþ1 p; pi m1 jþ1 mj X hK p; pi hK p; qi þ jþ1 ; m P 1. p; qi hKmj q; qi j¼0 hK
ð3:10Þ
Assuming that Fk is the Hamiltonian function of the symplectic space ðR2N ; dp ^ dqÞ, we remind that the Poisson bracket of two functions F and G in the symplectic space ðR2N ; dp ^ dqÞ is defined as [20] ! N X oF oG oF oG oF oG oF oG ¼ ; ; . fF ; Gg ¼ oqj opj opj oqj oq op op oq j¼1
J. Chen, X. Geng / Chaos, Solitons and Fractals 27 (2006) 813–821
Denoting the conjugate variable of Fk by sk, a direct calculation provides a canonical equation oF k =oqk pk d pk ¼ Irk F k ¼ ¼ W ðk; kk Þ ; dsk qk oF k =opk qk where I¼
0 1
1 ; 0
K 4l Vkþ4 W ðk; lÞ ¼ kl V 21 k
V 12 k K
817
ð3:11Þ
! ð3:12Þ
;
with K = hp, pi + 2hp, qi + hq, qi + Qk(Kp, p) + Qk(Kq, q). Proposition 1. Along with the sk-flow, the Lax matrix Vl satisfies Lax equation dV l ¼ ½W ðk; lÞ; V l ; dsk
ð3:13Þ
and fF l ; F k g ¼ 0;
k; l 2 C;
fF j ; F k g ¼ 0;
j; k ¼ 1; 2; . . .
ð3:14Þ
Proof. In terms of the following identities: Qk ðKl n; gÞ ¼ kQk ðKl1 n; gÞ þ hKl1 n; gi; 1 ðQ ðn; gÞ Ql ðn; gÞÞ; hðlI KÞ1 ðkI KÞ1 n; gi ¼ lk k after some direct but tedious calculation, we know that (3.13) holds. For (3.14), we can readily work out in analogy to the treatment in [19]. h
4. Elliptic coordinates and the Liouville integrability To prove the complete integrability of FDHSs, the functional independence indicating a sufficient number of integrals of motion is essential for the Liouville sense. Following (3.8), we define 21 11 2 F k ¼ V 12 k V k ðV k Þ ¼
4bðkÞ 4RðkÞ ¼ 2 ; aðkÞ a ðkÞ
ð4:1Þ
mðkÞ ; aðkÞ nðkÞ ; ¼ 2ðhp; qi þ hq; qi þ Qk ðKq; qÞÞ ¼ 2ðhp; qi þ hq; qiÞ aðkÞ
V 12 k ¼ 2ðhp; pi þ hp; qi þ Qk ðKp; pÞÞ ¼ 2ðhp; pi þ hp; qiÞ
ð4:2Þ
V 21 k
ð4:3Þ
where aðkÞ ¼
N Y
ðk kk Þ;
bðkÞ ¼
k¼1
nðkÞ ¼
N Y
N þ2 Y
ðk kN þk Þ;
mðkÞ ¼
k¼1
ðk mk Þ;
N Y
ðk lk Þ;
k¼1
RðkÞ ¼ aðkÞbðkÞ ¼
k¼1
2N þ2 Y
ð4:4Þ
ðk kk Þ.
k¼1
Here {lk} and {mk} are called the elliptic coordinates. Comparing the coefficients with respect to kN1 of both sides of (4.2) and (4.3) with the aid of (4.4) gives a relation between the canonical variables p and q and the elliptic coordinates, N X ðkj lj Þ ¼ j¼1
hKp; pi ; hp; pi þ hp; qi
N X ðkj mj Þ ¼ j¼1
hKq; qi . hp; qi þ hq; qi
ð4:5Þ
On use of the Lax equation (3.13), we get dV 12 l 12 11 ¼ 2 W 11 ðk; lÞV 12 l W ðk; lÞV l ; dsk
dV 21 l 11 21 ¼ 2 W 21 ðk; lÞV 11 l W ðk; lÞV l . dsk
ð4:6Þ
818
J. Chen, X. Geng / Chaos, Solitons and Fractals 27 (2006) 813–821
Setting k = lk and k = mk, respectively, it follows from (4.1)–(4.3) that pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi 2 Rðlk Þ 2 Rðmk Þ 11 ; V . ¼ ¼ V 11 lk mk aðlk Þ aðmk Þ
ð4:7Þ
Resorting to (3.13), (4.2), (4.3), (4.6) and (4.7), it is easy to calculate that 16
1 dl kmðkÞ pffiffiffiffiffiffiffiffiffiffiffiffi k ¼ ; Rðlk Þ dsk aðkÞðk lk Þm0 ðlk Þ
1 dm knðkÞ pffiffiffiffiffiffiffiffiffiffiffi k ¼ . aðkÞðk mk Þn0 ðmk Þ 16 Rðmk Þ dsk
ð4:8Þ
With the help of the Lagrange interpolation formula, we arrive at N X j¼1
lN j dl kN þ1j pkffiffiffiffiffiffiffiffiffiffiffiffi k ¼ ; aðkÞ 16 Rðlk Þ dsk
N X j¼1
mN j dmk kN þ1j pk ffiffiffiffiffiffiffiffiffiffiffi . ¼ aðkÞ 16 Rðmk Þ dsk
ð4:9Þ
These formulae naturally lead to the consideration of the Riemann surface C of hyperelliptic curve given by the affine equation n2 = R(k), which is genus of N. It is known that ~j ¼ x
kN j dk pffiffiffiffiffiffiffiffiffiffi ; 16 RðkÞ
ð4:10Þ
1 6 j 6 N;
are N linearly independent homomorphic differentials on C. Assuming that P ðkÞ ¼ ðk; n ¼ on C, let us introduce the quasi-Abel–Jacobi coordinates N Z P ðlk Þ N Z P ðmk Þ X X ~ ~ ¼ ~ ~ j; ; w ¼ 1 6 j 6 N. x x / j j j k¼1
P0
k¼1
pffiffiffiffiffiffiffiffiffiffi RðkÞÞ and a fixed point P0
ð4:11Þ
P0
Therefore, (4.9) is of the form ~ dw kN jþ1 j . ¼ dsk aðkÞ
~ d/ kN jþ1 j ; ¼ dsk aðkÞ
ð4:12Þ
After these preparations, we attain a proposition indicating the functional independence of the resulting integrals of motion. Proposition 2. F0, F1, . . ., FN1 determined by (3.10) are functionally independent. Proof. Designating the conjugate variable of Fk by sk, from the definition of Poisson bracket, we have 1 1 ~ ~ X X d/ d/ j j k ~ ; F kg ¼ ~ ; F k gkk ¼ ¼ f/ f/ k ; j j dsk ds k k¼0 k¼0
1 6 j 6 N.
On the other hand, 1 X kN 1 ¼ ¼ Ak kk ; 1 1 aðkÞ ð1 k1 k Þ ð1 kN k Þ k¼0
where
0 A0 ¼ 1;
A1 ¼ s1 ;
1
X C 1B Ak ¼ B si Aj C sk þ A; k@ iþj¼k;
k P 2;
i;jP1
kk1
kkN
and a supplementary definition Ak = 0 (k P 1). Together with (4.12), a direct computation with sk ¼ þ þ ~ =dsk ¼ Akþ1j . Whence, we have gives rise to d/ j 0 1 1 A1 A2 AN 1 B 1 A1 AN 2 C C ! B B ~ ~ .. C .. .. d/ d/ B C ¼B ;...; ð4:13Þ . C; . . B C ds0 dsN 1 .. B C @ 0 . A1 A 1
J. Chen, X. Geng / Chaos, Solitons and Fractals 27 (2006) 813–821
819
~ ¼ ð/ ~ ;...;/ ~ ÞT . Supposing that PN 1 c dF k ¼ 0, we derive where / 1 N k¼0 k 0¼
N 1 X
~ Þ¼ ck x2 ðI dF k ; I d/ j
k¼0
N 1 X
~ ; F kg ¼ ck f/ j
k¼0
N 1 X
ck
k¼0
~ d/ j ; dsk
1 6 j 6 N.
Because the coefficient determinant of (4.13) is equal to 1, it means that c0 = = cN1 = 0. This completes the proof. h ~ is sufficient in verifying the above proposition as well. As a result of Actually the quasi-Abel–Jacobi coordinate w j Proposition 2, we obtain Corollary 1. The finite-dimensional Hamiltonian system dp oF k ¼ ; dsk oq
dq oF k ¼ ; dsk op
ð4:14Þ
k P 0;
is completely integrable in the Liouville sense. 5. Reductions to the nonlinear evolution equations In this section, all the resulting NEEs are reduced into integrable FDHSs of ODEs. For this purpose we introduce a set of polynomial integrals {Hm} for (3.5), m 1 1 1 1X H 1 ¼ F 1; H mþ2 ¼ F mþ2 H j H mj ; m P 0; ð5:1Þ H 0 ¼ F 0; 8 8 8 2 j¼0 which is in agreement with F k ¼ 4ðk þ H k Þ2 ;
ð5:2Þ
where Hk ¼
1 X
H m km1 .
ð5:3Þ
m¼0
Proposition 3. The polynomial integral {Hm} is completely integrable in the Liouville sense, 1. fH k ; H l g ¼ 0;
k; l 2 C;
fH i ; H j g ¼ 0;
i; j ¼ 0; 1; 2; . . .
ð5:4Þ
2. H0, H1, . . ., HN1 described by (5.1) are functionally independent. Proof. Taking advantage of the identity (5.2), a simple calculation provides fH k ; H l g ¼
16
1 pffiffiffiffiffiffiffiffiffiffiffi fF k ; F l g ¼ 0. F kF l
Substituting (5.3) into (5.4)1, 0 0 1 1 dF 0 B B C B dF 1 C B B C 1B B dF 2 C B B C¼ B B . C 8B B . C B @ . A @ dF N 1
it is shown that (5.4)2 holds immediately. Noticing (5.1), 1 10 0 0 0 dH 0 B C 1 0 0C CB dH 1 C B C .. .. .. C C CB . . . CB dH 2 C; CB . C .. CB . C . 0 A@ . A dH N 1 1
which means that dH0, dH1, . . ., dHN1 are linearly independent in view of Proposition 2. Then H0, H1, . . ., HN1 are functionally independent as well. h
820
J. Chen, X. Geng / Chaos, Solitons and Fractals 27 (2006) 813–821
Remark. Due to the Liouville integrability of H0, H1, . . ., HN1, two Hamiltonian systems ðR2N ; dp ^ dq; H 0 Þ and ðR2N ; dp ^ dq; H k Þ are compatible and their flows naturally commute [20]. Acting with J1K upon (3.3) k 1 times delivers N X
kk1 ckþ1 g2 ; j rkj ¼ gk1 þ c3 gk3 þ þ ckþ1 g1 þ ~
ð5:5Þ
k P 2;
j¼1
where cj and ~cj are constants of integration. Together with (5.5), (3.7) can be rewritten as Gk ¼ g1 þ ¼ g1 þ
N 1 N X X X rkj ¼ g1 þ kk kk1 j rkj k kj j¼1 k¼1 j¼1 1 X
¼ ck g k þ
kk ðgk1 þ c3 gk3 þ þ ckþ1 g1 þ ~ckþ1 g2 Þ
k¼1 1 X
ð5:6Þ
kk ~ckþ1 g2 ;
k¼1
where ck ¼ 1 þ
P1
k2 . k¼0 ckþ3 k
The combination of (2.16) and (3.8) gives rise to # 1 X V k ¼ rðkÞ½Gk ¼ rðkÞ ck gk þ kk ~ckþ1 g2 ¼ rðkÞ½ck gk ; "
ð5:7Þ
k¼1
F k ¼ 4c2k k2 .
ð5:8Þ
Therefore (5.2) and (5.8) result in k ck = k + Hk. Denoting variables of Hk and Hk by tk and tk, respectively, and noting that V(1) = 2U, we have d 1 d ¼ . dtk 4kck dsk
ð5:9Þ
Now we are in position to reveal the relation between NEEs and ODEs. On one hand, du ¼ 4kðhp; pi þ 2hp; qi þ hq; qiÞðQk ðp; pÞ þ 2Qk ðp; qÞ þ Qk ðq; qÞÞ þ 8kðQk ðKp; pÞ þ Qk ðKq; qÞÞ; dsk dv ¼ 4kðhp; pi hq; qiÞðQk ðp; pÞ þ 2Qk ðp; qÞ þ Qk ðq; qÞÞ þ 8kðQk ðKp; pÞ Qk ðKq; qÞÞ; dsk
ð5:10Þ ð5:11Þ
and ðu; vÞTsk ¼ 4kJGk .
ð5:12Þ
On the other hand, 1 X d u 1 d u 1 ¼ ¼ 4kJGk ¼ Jgk1 kk . dtk v 4kck dsk v 4kck k¼1
ð5:13Þ
Summing up these results, we encapsulate the following main results. Theorem 1. Let (p(x, tn), q(x, tn))T be a compatible solution of H0 and Hn1, then 1 uðx; tn Þ ¼ ðhp; pi hq; qiÞ; 2
1 vðx; tn Þ ¼ ðhp; pi þ 2hp; qi þ hq; qiÞ; 2
n P 2;
is a solution of the nth LCZ soliton Eq. (2.7). Theorem 2. Let (p(x, y, t), q(x, y, t))T be a compatible solution of H0, H1 and H2, i.e., p p p ¼ IrH 0 ; ¼ IrH 1 ; ¼ IrH 2 ; q y q t q x then
ð5:14Þ
J. Chen, X. Geng / Chaos, Solitons and Fractals 27 (2006) 813–821
r ¼ 12ðhpðx; y; tÞ; pðx; y; tÞi þ 2hpðx; y; tÞ; qðx; y; tÞi þ hqðx; y; tÞ; qðx; y; tÞiÞ;
821
ð5:15Þ
is a solution of (2 + 1)-dimensional equations (2.12) and (2.13). t
Proof. Let hjj be the solution operator of initial value problems of the Hamiltonian system ðR2N ; dp ^ dq; H j Þ. Following Proposition 3, it is apparent that ht3 and hy2 are two commuting operators [20], i.e.,
p0 p0 p x y t x t ¼ h0 h3 hy2 . ¼ h0 h2 h3 q0 q0 q Consequently it is known that (5.15) is the compatible solution of (2.9) and (2.10), and then (5.15) solves (2.12) and (2.13) simultaneously. h
Acknowledgements This work was supported by the National Natural Science Foundation of China (No. 10471132) and the Special Foundation for Chinese Major State Basic Research Project ‘‘Nonlinear Science’’. The author (J.B. Chen) would like to express his sincere thanks to Professor Wadati for his helpful suggestions.
References [1] Novikov SP, Manakov SV, Pitaevskii LP, Zakharov VE. Theory of solitons, the inverse scattering methods. New York: Consultants Bureau; 1984. [2] Newell AC. Solitons in mathematics and physics. Philadelphia, PA: SIAM; 1985. [3] Ablowitz MJ, Segur H. Solitons and the inverse scattering transform. Philadelphia, PA: SIAM; 1981. [4] Matveev VB, Salle MA. Darboux transformations and solitons. Berlin: Springer; 1991. [5] Belokolos ED, Bobenko AI, Enolskii VZ, Its AR, Matveev VB. Algebro-geometric approach to nonlinear integrable equations. Berlin: Springer; 1994. [6] Zakharov E, Shabat AB. Funct Anal Appl 1974;8:226. [7] Cao CW. Sci China A 1990;33:528. [8] Cao CW, Geng XG. J Phys A: Math Gen 1990;23:4117. [9] Cao CW, Geng XG. Nonlinear physics. Berlin: Springer; 1990. p. 68. [10] Cheng Y, Li YS. Phys Lett A 1991;157:22. [11] Cheng Y, Li YS. J Phys A: Math Gen 1992;25:419. [12] Zhou RG. J Math Phys 1997;38:2535. [13] Cao CW, Wu YT, Geng XG. J Math Phys 1999;40:3948. [14] Geng XG, Wu YT, Cao CW. J Phys A: Math Gen 1999;32:3733. [15] Li YS, Chen DY. Acta Math Sin 1986;29:127. [16] Li YS, Chen DY, Zeng YB. Proc 1983 Beijing Symp on Diff Geom and Diff Eqs. Beijing: Science Press; 1986. p. 359. [17] Konopelchenko BG, Dubrovsky VG. Phys Lett A 1984;102:15. [18] Tu GZ. J Math Phys 1989;30:330. [19] Chen JB. Chaos, Solitons & Fractals 2004;19:905. [20] Arnold VI. Mathematical methods of classical mechanics. Berlin: Springer; 1978.
Relation among nonlinear evolution equations and their reductions Jinbing Chen *, Xianguo Geng Department of Mathematics, Zhengzhou University, Zhengzhou 450052, Henan, PR China
Abstract The LCZ soliton hierarchy is presented, and their generalized Hamiltonian structures are deduced. From the compatibility of soliton equations, it is shown that this soliton hierarchy is closely related to the Burger equation, the mKP equation and a new (2 + 1)-dimensional nonlinear evolution equation (NEE). Resorting to the nonlinearization of Lax pairs (NLP), all the resulting NEEs are reduced into integrable Hamiltonian systems of ordinary differential equations (ODEs). As a concrete application, the solutions for NEEs can be derived via solving the corresponding ODEs. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction Soliton equations are a remarkable class of NEEs having nice mathematical and physical properties in the nonlinear science. Thus the investigation for them is one of the most highlighted subjects in mathematical physics. In the past decades, many systematic methods were developed to study soliton equations such as the inverse scattering method, the Darboux transformation, the algebro-geometric method and the dressing method [1–6], etc. It is worthwhile to mention that the NLP [7–9] is also effective in studying soliton equations, and has been generalized to investigate multidimensional problems recently. The most important message relating to the progress of this manipulation is that the compatible solutions of soliton equations naturally give rise to solutions of (2 + 1)-dimensional NEEs [10,11]. Consequently many soliton equations were solved after this procedure [12–14]. Usually one considers multi-dimensional 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. Along with this idea, we devote to reduce NEEs into integrable ODEs such that solutions of NEEs can be achieved by solving ODEs. For this purpose, on application of the NLP, a family of finite-dimensional Hamiltonian systems (FDHSs) are obtained constituting the decomposition of the resulting NEEs, which are completely integrable in the Liouville sense. This article is organized as follows. In Section 2, the LCZ soliton hierarchy and two (2 + 1)-dimensional systems are deduced. In Section 3, the NLP with the Bargmann constraint is studied. In Section 4, the integrability of FDHSs is completed by means of the elliptic coordinates and quasi-Abel–Jacobi coordinates. In Section 5, the NEEs are reduced into FDHSs such that solutions for NEEs can be attained by solving integrable ODEs.
*
Corresponding author. E-mail address: [email protected] (J. Chen).
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.04.054
814
J. Chen, X. Geng / Chaos, Solitons and Fractals 27 (2006) 813–821
2. The nonlinear evolution equations In this section, we deduce the soliton hierarchy, the Burger equation, the mKP equation, a new (2 + 1)-dimensional NEE, and further calculate the generalized Hamiltonian structures for the soliton hierarchy. Now let us begin with the LCZ eigenvalue problem [15,16], k þ v u þ v u1 ; ð2:1Þ ; u¼ ux ¼ U u; U ¼ uv kv u2 where k is a constant spectral parameter, and u, v are two potentials. Take into account the adjoint representation of (2.1), X aj bj þ cj j a bþc ¼ V x ¼ ½U ; V ; V ¼ k ; ð2:2Þ bj cj aj b c a jP0 which yields
1
aj ¼ 2o ðvbj ucj Þ;
bjþ1 cjþ1
¼
v 2uo1 v
12 o þ 2uo1 u
12 o 2vo1 v
v þ 2vo1 u
!
bj ; cj
ð2:3Þ
where o = o/ox, oo1 = o1o = 1. From (2.3) and initial values a0 = 2, b0 = c0 = 0, we have a1 ¼ 0;
b1 ¼ 2u;
a2 ¼ v2 u2 ;
c1 ¼ 2v;
b2 ¼ vx 2uv;
c2 ¼ ux 2v2 .
For any positive integral n, we introduce an auxiliary eigenvalue problem of (2.1), ! ðnÞ ðnÞ V 12 V 11 ðnÞ ðnÞ utn ¼ V u; V ¼ ; n P 1; ðnÞ ðnÞ V 21 V 11
ð2:4Þ
ð2:5Þ
where ðnÞ
V 11 ¼
n1 X
aj knj þ cn ;
ðnÞ
V 12 ¼
j¼0
n X ðbj þ cj Þknj ; j¼0
ðnÞ
V 21 ¼
n X ðbj cj Þknj . j¼0
Then the compatibility condition of (2.1) and (2.5), i.e., uxtn ¼ utn x , leads to the zero-curvature equation ðnÞ ¼ 0; U tn V ðnÞ x þ ½U ; V
ð2:6Þ
which is nothing but the LCZ soliton hierarchy ðu; vÞTtn ¼ Jgn1 ; where
gn ¼
ð2:7Þ
n P 1;
bnþ1 ; anþ1 cnþ1
J¼
o 0 . 0 o
Together with (2.3) and the Lenard recursive relation Kgj1 = Jgj (j P 0), we get ! 1 2 0 o ou 2 . K¼ 12 o2 uo ov vo Designating g2 = (2, 0)T and g1 = (0, 2)T, it is clear that ker J ¼ fq1 g1 þ q2 g2 j8q1 ; q2 2 Rg. On use of the Lenard recursive relation, gj (j P 0) can be uniquely established up to ker J. In particular, g0 ¼ ð2u; 2vÞT ;
g1 ¼ ðvx 2uv; 3v2 u2 ux ÞT .
ð2:8Þ
Here {gj} is the Lenard sequence, and K, J are the Lenard operator pairs. Now we list the first two nontrivial equations of (2.7) with y = t2 and t = t3, respectively, vy ¼ uxx þ 2uux 6vvx ; uy ¼ vxx 2ux v 2uvx ; 8 1 > < ut ¼ uxxx þ 3u2 ux 3ux v2 þ 3vvxx þ 3v2x 6uvvx ; 2 > : v ¼ 1v þ 3u v þ 3u2 v þ 3u v þ 6uu v 15v2 v . t xxx xx x x x x x 2
ð2:9Þ ð2:10Þ
J. Chen, X. Geng / Chaos, Solitons and Fractals 27 (2006) 813–821
815
If simply taking u = v, (2.9) can be reduced as the well-known Burger equation uy ¼ uxx 4uux . Let r(x, y, t) = v(x, y, t) be the compatible solution of (2.9) and (2.10). Following (2.9), a lengthy calculation shows that 2 2 o1 x r y ¼ ux þ u 3r ;
1 2 o1 x r yy ¼ r xxx 8rr y 24r r x 4r x ox r y .
ð2:11Þ
Substituting (2.11) into (2.10)2 implies the following (2 + 1)-dimensional NEEs, 1 3 rt ¼ ðrxx 8r3 Þx ðo1 ryy 4rx o1 x r y Þ; 8 8 x 1 rt ¼ rxxx þ 12r2 rx þ 3rry þ 3rx o1 x ry . 2
ð2:12Þ ð2:13Þ
It is remarkable to see that (2.12) is the mKP equation [17] and (2.13) is a new (2 + 1)-dimensional system, which are closely connected with two soliton equations (2.9) and (2.10). In what follows, we are ready to calculate the generalized Hamiltonian structure of (2.7). It is easy from (2.1) and (2.2) to calculate that oU oU oU tr V ¼ 2a; tr V ¼ 2b; tr V ¼ 2a 2c. ð2:14Þ ok ou ov Noticing the trace identity [18], T d d o ; ð2aÞ ¼ ks ks ð2b; 2a 2cÞT ; du dv ok where s is a constant to be fixed. After a simple calculation we know that s = 0 and T d d ; Hn ¼ gn ; n 2 Zþ ; du dv where Hn ¼ anþ1 =n. Therefore (2.7) adopts the following generalized Hamiltonian form: T dHn1 dHn1 ; . ðu; vÞTtn ¼ J du dv In particular, the Hamiltonian form of (2.9) reads H1 ¼ v2 u2 . In order to proceed our work, let us introduce a generating function of {gk}: 1 X gk kk1 ; gk ¼ g1 þ k¼0
it is easy to check that (K kJ)gk = 0. Denoting g ¼ ðgð1Þ ; gð2Þ ÞT ¼ V(n) can be rewritten as a concise form, V
ðnÞ
,rðu; v; kÞ½g ¼
ð2Þ 12 gð1Þ þ kgð2Þ x vg 1 ð2Þ ð1Þ ðg þ g Þx ðu vÞgð2Þ 2
Pn1
ð2:15Þ
n1j , j¼0 gj1 k
1 ðgð2Þ gð1Þ Þx ðu þ vÞgð2Þ 2 1 ð1Þ g þ vgð2Þ kgð2Þ 2 x
an immediate consequence is that
! .
ð2:16Þ
A straightforward calculation delivers V x ½U ; V ¼ U ½ðK kJ Þg; ð2:17Þ where U ðnÞ ¼ ded e¼0 U ðu þ en1 ; v þ en2 Þ. Therefore, (K kJ)g = 0 implies that o det r[g] = 0 [18]. Whence, we get from (2.15) and (2.16) that det r½gk ¼ 4k2 .
ð2:18Þ
3. The nonlinearization of Lax pairs Let k1, k2, . . ., kN be N distinct eigenvalues and u = (pj, qj)T. We take N copies of the eigenvalue problem (2.1), ! ! pj pj kj þ v u þ v ; 1 6 j 6 N. ¼ ð3:1Þ qj qj uv kj v x
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J. Chen, X. Geng / Chaos, Solitons and Fractals 27 (2006) 813–821
A simple calculation gives rise to ðK kj J Þrkj ¼ 0;
rkj ¼
T T dkj dkj ; ¼ q2j p2j ; p2j þ 2pj qj þ q2j . du dv
Let us focus on the Bargmann constraint [7–9] N X rkj ; g0 ¼
ð3:2Þ
ð3:3Þ
j¼1
which yields 1 u ¼ ðhp; pi hq; qiÞ; 2
1 v ¼ ðhp; pi þ 2hp; qi þ hq; qiÞ; 2
ð3:4Þ
where p = (p1, . . ., pN)T, q = (q1, . . ., qN)T, and hÆ, Æi stands for the standard inner product in RN . On insertion of the expression (3.4) into (3.1), we obtain the following FDHS: 8 1 oH 0 > > < px ¼ Kp þ 2 ðhp; pi þ 2hp; qi þ hq; qiÞp þ ðhp; pi þ hp; qiÞq ¼ oq ; ð3:5Þ > 1 oH 0 > : qx ¼ Kq ðhp; pi þ 2hp; qi þ hq; qiÞq ðhp; qi þ hq; qiÞp ¼ ; 2 op where K = diag(k1, . . ., kN), and 1 H 0 ¼ hKp; qi ðhp; pi þ hp; qiÞðhp; qi þ hq; qiÞ. 2
ð3:6Þ
Only for convenience, we introduce a bilinear function Qk(n, g) which plays an important role in our consideration, N 1 X X nj gj ¼ km1 hKm n; gi. Qk ðn; gÞ ¼ hðkI KÞ1 n; gi ¼ k kj m¼0 j¼1 Designating Gk ¼ g1 þ
N X Qk ðq; qÞ Qk ðp; pÞ rkj ¼ ; k kj 2 þ Qk ðp; pÞ þ 2Qk ðp; qÞ þ Qk ðq; qÞ j¼1
it is not difficult to check that (K kJ)Gk = 0. Substituting (3.7) into (2.16) leads to 2ðQk ðKp; pÞ þ hp; pi þ hp; qiÞ 2k þ 2Qk ðKp; qÞ ; Vk ¼ 2ðQk ðKq; qÞ þ hp; qi þ hq; qiÞ 2k 2Qk ðKp; qÞ
ð3:7Þ
ð3:8Þ
in view of Qk(Kn, g) = kQk(n, g) hn, gi. Assuming that Fk = det Vk, it is easy to see that Fk is invariant under the action of x-flow. Hence the generating function of integrals of motion for (3.5) can be written as F k ¼ 4k2 þ 4ðhp; pi þ hp; qiÞðhp; qi þ hq; qiÞ 8kQk ðKp; qÞ þ 4ðhp; pi þ hp; qiÞQk ðKq; qÞ Qk ðKp; pÞ Qk ðKp; qÞ þ 4ðhp; qi þ hq; qiÞQk ðKp; pÞ þ Qk ðKp; qÞ Qk ðKq; qÞ 1 X F m km ; ¼ 4k2 þ
ð3:9Þ
m¼0
where F 0 ¼ 8hKp; qi þ 4ðhp; pi þ hp; qiÞðhp; qi þ hq; qiÞ ¼ 8H 0 ; F 1 ¼ 8hK2 p; qi þ 4ðhp; pi þ hp; qiÞhKq; qi þ 4ðhp; qi þ hq; qiÞhKp; pi; F mþ1 ¼ 8hKmþ2 p; qi þ 4ðhp; pi þ hp; qiÞhKmþ1 q; qi þ 4ðhp; qi þ hq; qiÞhKmþ1 p; pi m1 jþ1 mj X hK p; pi hK p; qi þ jþ1 ; m P 1. p; qi hKmj q; qi j¼0 hK
ð3:10Þ
Assuming that Fk is the Hamiltonian function of the symplectic space ðR2N ; dp ^ dqÞ, we remind that the Poisson bracket of two functions F and G in the symplectic space ðR2N ; dp ^ dqÞ is defined as [20] ! N X oF oG oF oG oF oG oF oG ¼ ; ; . fF ; Gg ¼ oqj opj opj oqj oq op op oq j¼1
J. Chen, X. Geng / Chaos, Solitons and Fractals 27 (2006) 813–821
Denoting the conjugate variable of Fk by sk, a direct calculation provides a canonical equation oF k =oqk pk d pk ¼ Irk F k ¼ ¼ W ðk; kk Þ ; dsk qk oF k =opk qk where I¼
0 1
1 ; 0
K 4l Vkþ4 W ðk; lÞ ¼ kl V 21 k
V 12 k K
817
ð3:11Þ
! ð3:12Þ
;
with K = hp, pi + 2hp, qi + hq, qi + Qk(Kp, p) + Qk(Kq, q). Proposition 1. Along with the sk-flow, the Lax matrix Vl satisfies Lax equation dV l ¼ ½W ðk; lÞ; V l ; dsk
ð3:13Þ
and fF l ; F k g ¼ 0;
k; l 2 C;
fF j ; F k g ¼ 0;
j; k ¼ 1; 2; . . .
ð3:14Þ
Proof. In terms of the following identities: Qk ðKl n; gÞ ¼ kQk ðKl1 n; gÞ þ hKl1 n; gi; 1 ðQ ðn; gÞ Ql ðn; gÞÞ; hðlI KÞ1 ðkI KÞ1 n; gi ¼ lk k after some direct but tedious calculation, we know that (3.13) holds. For (3.14), we can readily work out in analogy to the treatment in [19]. h
4. Elliptic coordinates and the Liouville integrability To prove the complete integrability of FDHSs, the functional independence indicating a sufficient number of integrals of motion is essential for the Liouville sense. Following (3.8), we define 21 11 2 F k ¼ V 12 k V k ðV k Þ ¼
4bðkÞ 4RðkÞ ¼ 2 ; aðkÞ a ðkÞ
ð4:1Þ
mðkÞ ; aðkÞ nðkÞ ; ¼ 2ðhp; qi þ hq; qi þ Qk ðKq; qÞÞ ¼ 2ðhp; qi þ hq; qiÞ aðkÞ
V 12 k ¼ 2ðhp; pi þ hp; qi þ Qk ðKp; pÞÞ ¼ 2ðhp; pi þ hp; qiÞ
ð4:2Þ
V 21 k
ð4:3Þ
where aðkÞ ¼
N Y
ðk kk Þ;
bðkÞ ¼
k¼1
nðkÞ ¼
N Y
N þ2 Y
ðk kN þk Þ;
mðkÞ ¼
k¼1
ðk mk Þ;
N Y
ðk lk Þ;
k¼1
RðkÞ ¼ aðkÞbðkÞ ¼
k¼1
2N þ2 Y
ð4:4Þ
ðk kk Þ.
k¼1
Here {lk} and {mk} are called the elliptic coordinates. Comparing the coefficients with respect to kN1 of both sides of (4.2) and (4.3) with the aid of (4.4) gives a relation between the canonical variables p and q and the elliptic coordinates, N X ðkj lj Þ ¼ j¼1
hKp; pi ; hp; pi þ hp; qi
N X ðkj mj Þ ¼ j¼1
hKq; qi . hp; qi þ hq; qi
ð4:5Þ
On use of the Lax equation (3.13), we get dV 12 l 12 11 ¼ 2 W 11 ðk; lÞV 12 l W ðk; lÞV l ; dsk
dV 21 l 11 21 ¼ 2 W 21 ðk; lÞV 11 l W ðk; lÞV l . dsk
ð4:6Þ
818
J. Chen, X. Geng / Chaos, Solitons and Fractals 27 (2006) 813–821
Setting k = lk and k = mk, respectively, it follows from (4.1)–(4.3) that pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi 2 Rðlk Þ 2 Rðmk Þ 11 ; V . ¼ ¼ V 11 lk mk aðlk Þ aðmk Þ
ð4:7Þ
Resorting to (3.13), (4.2), (4.3), (4.6) and (4.7), it is easy to calculate that 16
1 dl kmðkÞ pffiffiffiffiffiffiffiffiffiffiffiffi k ¼ ; Rðlk Þ dsk aðkÞðk lk Þm0 ðlk Þ
1 dm knðkÞ pffiffiffiffiffiffiffiffiffiffiffi k ¼ . aðkÞðk mk Þn0 ðmk Þ 16 Rðmk Þ dsk
ð4:8Þ
With the help of the Lagrange interpolation formula, we arrive at N X j¼1
lN j dl kN þ1j pkffiffiffiffiffiffiffiffiffiffiffiffi k ¼ ; aðkÞ 16 Rðlk Þ dsk
N X j¼1
mN j dmk kN þ1j pk ffiffiffiffiffiffiffiffiffiffiffi . ¼ aðkÞ 16 Rðmk Þ dsk
ð4:9Þ
These formulae naturally lead to the consideration of the Riemann surface C of hyperelliptic curve given by the affine equation n2 = R(k), which is genus of N. It is known that ~j ¼ x
kN j dk pffiffiffiffiffiffiffiffiffiffi ; 16 RðkÞ
ð4:10Þ
1 6 j 6 N;
are N linearly independent homomorphic differentials on C. Assuming that P ðkÞ ¼ ðk; n ¼ on C, let us introduce the quasi-Abel–Jacobi coordinates N Z P ðlk Þ N Z P ðmk Þ X X ~ ~ ¼ ~ ~ j; ; w ¼ 1 6 j 6 N. x x / j j j k¼1
P0
k¼1
pffiffiffiffiffiffiffiffiffiffi RðkÞÞ and a fixed point P0
ð4:11Þ
P0
Therefore, (4.9) is of the form ~ dw kN jþ1 j . ¼ dsk aðkÞ
~ d/ kN jþ1 j ; ¼ dsk aðkÞ
ð4:12Þ
After these preparations, we attain a proposition indicating the functional independence of the resulting integrals of motion. Proposition 2. F0, F1, . . ., FN1 determined by (3.10) are functionally independent. Proof. Designating the conjugate variable of Fk by sk, from the definition of Poisson bracket, we have 1 1 ~ ~ X X d/ d/ j j k ~ ; F kg ¼ ~ ; F k gkk ¼ ¼ f/ f/ k ; j j dsk ds k k¼0 k¼0
1 6 j 6 N.
On the other hand, 1 X kN 1 ¼ ¼ Ak kk ; 1 1 aðkÞ ð1 k1 k Þ ð1 kN k Þ k¼0
where
0 A0 ¼ 1;
A1 ¼ s1 ;
1
X C 1B Ak ¼ B si Aj C sk þ A; k@ iþj¼k;
k P 2;
i;jP1
kk1
kkN
and a supplementary definition Ak = 0 (k P 1). Together with (4.12), a direct computation with sk ¼ þ þ ~ =dsk ¼ Akþ1j . Whence, we have gives rise to d/ j 0 1 1 A1 A2 AN 1 B 1 A1 AN 2 C C ! B B ~ ~ .. C .. .. d/ d/ B C ¼B ;...; ð4:13Þ . C; . . B C ds0 dsN 1 .. B C @ 0 . A1 A 1
J. Chen, X. Geng / Chaos, Solitons and Fractals 27 (2006) 813–821
819
~ ¼ ð/ ~ ;...;/ ~ ÞT . Supposing that PN 1 c dF k ¼ 0, we derive where / 1 N k¼0 k 0¼
N 1 X
~ Þ¼ ck x2 ðI dF k ; I d/ j
k¼0
N 1 X
~ ; F kg ¼ ck f/ j
k¼0
N 1 X
ck
k¼0
~ d/ j ; dsk
1 6 j 6 N.
Because the coefficient determinant of (4.13) is equal to 1, it means that c0 = = cN1 = 0. This completes the proof. h ~ is sufficient in verifying the above proposition as well. As a result of Actually the quasi-Abel–Jacobi coordinate w j Proposition 2, we obtain Corollary 1. The finite-dimensional Hamiltonian system dp oF k ¼ ; dsk oq
dq oF k ¼ ; dsk op
ð4:14Þ
k P 0;
is completely integrable in the Liouville sense. 5. Reductions to the nonlinear evolution equations In this section, all the resulting NEEs are reduced into integrable FDHSs of ODEs. For this purpose we introduce a set of polynomial integrals {Hm} for (3.5), m 1 1 1 1X H 1 ¼ F 1; H mþ2 ¼ F mþ2 H j H mj ; m P 0; ð5:1Þ H 0 ¼ F 0; 8 8 8 2 j¼0 which is in agreement with F k ¼ 4ðk þ H k Þ2 ;
ð5:2Þ
where Hk ¼
1 X
H m km1 .
ð5:3Þ
m¼0
Proposition 3. The polynomial integral {Hm} is completely integrable in the Liouville sense, 1. fH k ; H l g ¼ 0;
k; l 2 C;
fH i ; H j g ¼ 0;
i; j ¼ 0; 1; 2; . . .
ð5:4Þ
2. H0, H1, . . ., HN1 described by (5.1) are functionally independent. Proof. Taking advantage of the identity (5.2), a simple calculation provides fH k ; H l g ¼
16
1 pffiffiffiffiffiffiffiffiffiffiffi fF k ; F l g ¼ 0. F kF l
Substituting (5.3) into (5.4)1, 0 0 1 1 dF 0 B B C B dF 1 C B B C 1B B dF 2 C B B C¼ B B . C 8B B . C B @ . A @ dF N 1
it is shown that (5.4)2 holds immediately. Noticing (5.1), 1 10 0 0 0 dH 0 B C 1 0 0C CB dH 1 C B C .. .. .. C C CB . . . CB dH 2 C; CB . C .. CB . C . 0 A@ . A dH N 1 1
which means that dH0, dH1, . . ., dHN1 are linearly independent in view of Proposition 2. Then H0, H1, . . ., HN1 are functionally independent as well. h
820
J. Chen, X. Geng / Chaos, Solitons and Fractals 27 (2006) 813–821
Remark. Due to the Liouville integrability of H0, H1, . . ., HN1, two Hamiltonian systems ðR2N ; dp ^ dq; H 0 Þ and ðR2N ; dp ^ dq; H k Þ are compatible and their flows naturally commute [20]. Acting with J1K upon (3.3) k 1 times delivers N X
kk1 ckþ1 g2 ; j rkj ¼ gk1 þ c3 gk3 þ þ ckþ1 g1 þ ~
ð5:5Þ
k P 2;
j¼1
where cj and ~cj are constants of integration. Together with (5.5), (3.7) can be rewritten as Gk ¼ g1 þ ¼ g1 þ
N 1 N X X X rkj ¼ g1 þ kk kk1 j rkj k kj j¼1 k¼1 j¼1 1 X
¼ ck g k þ
kk ðgk1 þ c3 gk3 þ þ ckþ1 g1 þ ~ckþ1 g2 Þ
k¼1 1 X
ð5:6Þ
kk ~ckþ1 g2 ;
k¼1
where ck ¼ 1 þ
P1
k2 . k¼0 ckþ3 k
The combination of (2.16) and (3.8) gives rise to # 1 X V k ¼ rðkÞ½Gk ¼ rðkÞ ck gk þ kk ~ckþ1 g2 ¼ rðkÞ½ck gk ; "
ð5:7Þ
k¼1
F k ¼ 4c2k k2 .
ð5:8Þ
Therefore (5.2) and (5.8) result in k ck = k + Hk. Denoting variables of Hk and Hk by tk and tk, respectively, and noting that V(1) = 2U, we have d 1 d ¼ . dtk 4kck dsk
ð5:9Þ
Now we are in position to reveal the relation between NEEs and ODEs. On one hand, du ¼ 4kðhp; pi þ 2hp; qi þ hq; qiÞðQk ðp; pÞ þ 2Qk ðp; qÞ þ Qk ðq; qÞÞ þ 8kðQk ðKp; pÞ þ Qk ðKq; qÞÞ; dsk dv ¼ 4kðhp; pi hq; qiÞðQk ðp; pÞ þ 2Qk ðp; qÞ þ Qk ðq; qÞÞ þ 8kðQk ðKp; pÞ Qk ðKq; qÞÞ; dsk
ð5:10Þ ð5:11Þ
and ðu; vÞTsk ¼ 4kJGk .
ð5:12Þ
On the other hand, 1 X d u 1 d u 1 ¼ ¼ 4kJGk ¼ Jgk1 kk . dtk v 4kck dsk v 4kck k¼1
ð5:13Þ
Summing up these results, we encapsulate the following main results. Theorem 1. Let (p(x, tn), q(x, tn))T be a compatible solution of H0 and Hn1, then 1 uðx; tn Þ ¼ ðhp; pi hq; qiÞ; 2
1 vðx; tn Þ ¼ ðhp; pi þ 2hp; qi þ hq; qiÞ; 2
n P 2;
is a solution of the nth LCZ soliton Eq. (2.7). Theorem 2. Let (p(x, y, t), q(x, y, t))T be a compatible solution of H0, H1 and H2, i.e., p p p ¼ IrH 0 ; ¼ IrH 1 ; ¼ IrH 2 ; q y q t q x then
ð5:14Þ
J. Chen, X. Geng / Chaos, Solitons and Fractals 27 (2006) 813–821
r ¼ 12ðhpðx; y; tÞ; pðx; y; tÞi þ 2hpðx; y; tÞ; qðx; y; tÞi þ hqðx; y; tÞ; qðx; y; tÞiÞ;
821
ð5:15Þ
is a solution of (2 + 1)-dimensional equations (2.12) and (2.13). t
Proof. Let hjj be the solution operator of initial value problems of the Hamiltonian system ðR2N ; dp ^ dq; H j Þ. Following Proposition 3, it is apparent that ht3 and hy2 are two commuting operators [20], i.e.,
p0 p0 p x y t x t ¼ h0 h3 hy2 . ¼ h0 h2 h3 q0 q0 q Consequently it is known that (5.15) is the compatible solution of (2.9) and (2.10), and then (5.15) solves (2.12) and (2.13) simultaneously. h
Acknowledgements This work was supported by the National Natural Science Foundation of China (No. 10471132) and the Special Foundation for Chinese Major State Basic Research Project ‘‘Nonlinear Science’’. The author (J.B. Chen) would like to express his sincere thanks to Professor Wadati for his helpful suggestions.
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