Journal of the Physical Society of Japan Vol. 71, No. 11, November, 2002, pp. 2583–2586 #2002 The Physical Society of Japan
LETTERS
Conservation Laws of the Two-Dimensional Toda Lattice Hierarchy Da-jun Z HANG Department of Mathematics, Shanghai University, Shanghai 200436, People’s Republic of China (Received August 7, 2002)
A novel method of constructing the conservation laws of (1+2)-dimensional differential-difference systems is proposed. By introducing the generalized Riccati equation related to the pseudo-difference operator, we obtain the infinitely many conserved densities and the associated fluxes of the twodimensional Toda lattice hierarchy. Moreover, this method presents more forms of the conservation laws than the previous approach [K. Kajiwara and J. Satsuma: J. Math. Phys. 32 (1991) 506]. KEYWORDS: conservation law, two-dimensional Toda lattice hierarchy, generalized Riccati equation DOI: 10.1143/JPSJ.71.2583
1.
Introduction
As is well known, conservation laws (CLs) are important to integrable systems. There have been some successful methods developed for finding the infinitely many CLs.1–10) Recently, another method11) was proposed for constructing CLs for (1+1)-dimensional Lax integrable differentialdifference systems. The basis of this approach is a simple equality: ðn þ 1Þ ln ¼ ðE 1Þ ln ðnÞ; ð1:1Þ ðnÞ by which the CL can easily be obtained from the concerned Lax pair. Here, E is a shift operator. Furthermore, with the help of the Riccati equation related to the eigenvalue problem, an infinite number of CLs are consequently derived. The theory developed by Sato and coworkers12–14) is a powerful tool for studying high-dimensional systems and their related properties such as Lax pairs, symmetries, CLs and constraints.15–20) In ref. 19, the two-dimensional Toda lattice (2DTL) hierarchy has been discussed, starting from two pseudo-difference operators: L ¼ E þ u1 ðnÞ þ u2 ðnÞE1 þ u3 ðnÞE2 þ ;
ð1:2Þ
M ¼ v0 ðnÞE1 þ v1 ðnÞ þ v2 ðnÞE þ v3 ðnÞE2 þ ;
ð1:3Þ
and
where ui and vi are the functions of the variables n, x1 ; x2 ; . . ., y1 ; y2 ; . . . and only n is a discrete variable. Because of the definitions of ui and vi , there exist two forms of the CLs of the 2DTL hierarchy,19) the semidiscrete form @ ð1:4Þ F ¼ ðE 1ÞJ @tm and the continuous form @ @ F¼ J; @tm @ts
hierarchy in a new way which is different from KajiwaraSatsuma’s approach.19) The key element of this procedure is to introduce two solvable generalized Riccati equations which are related to pseudo-difference operators (1:2) and (1:3), respectively. We first construct the semidiscrete CLs according to formula (1:1). Then, with the help of the two Riccati equations, we further derive the infinitely many conserved densities and associated fluxes. We also consider the CLs between two continuous directions in this way. This method can be seen as a generalization of the method in ref. 11. To our knowledge, this is the first time that the infinite CLs of the 2TDL hierarchy have been considered through the generalized Riccati equation. This method is new and sufficiently flexible for application to the highdimensional differential-difference systems related to pseudo-difference operators. It not only allows us to obtain the explicit forms of the associated fluxes, but also presents more forms of the CLs which were not derived from the previous approach.19) The letter is organized as follows. In §2, we give a brief sketch of the 2DTL hierarchy. In §3, the CLs are considered. 2.
Let us first recall briefly the 2DTL hierarchy according to refs. 19 and 20. Consider pseudo-difference operators (1:2) and (1:3). Let Bm ðnÞ ¼ ðLm Þþ and Cm ðnÞ ¼ ðM m Þ , where ð Þþ means that part of the shift operator containing only non-negative powers and ð Þ means that part containing only negative powers, then we have B1 ðnÞ ¼ E þ u1 ðnÞ;
E-mail:
[email protected]
ð2:1aÞ
B2 ðnÞ ¼ E2 þ ðu1 ðnÞ þ u1 ðn þ 1ÞÞE þ u2 ðnÞ þ u2 ðn þ 1Þ þ u1 ðnÞ2 ;
ð2:1bÞ
; ð1:5Þ
and C1 ðnÞ ¼ v0 ðnÞE1 ;
where tm and ts denote two different directions in fxi ; yi g. However, Kajiwara-Satsuma’s approach19) only allows us to obtain the infinitely many conserved densities but no explicit associated fluxes. In this letter, we reconsider the CLs of the 2DTL
The 2DTL hierarchy
ð2:2aÞ
C2 ðnÞ ¼ v0 ðnÞv0 ðn 1ÞE2 þ v0 ðnÞðv1 ðnÞ þ v1 ðn 1ÞÞE1 ; : Now, consider the linear problems
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ð2:2bÞ
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J. Phys. Soc. Jpn., Vol. 71, No. 11, November, 2002
LETTERS
LðnÞ ¼ ðnÞ;
ð2:3aÞ
ðnÞxm ¼ Bm ðnÞðnÞ;
ð2:3bÞ
ðnÞym ¼ Cm ðnÞðnÞ
ð2:3cÞ
D. ZHANG
where 1 X ðnÞ ¼ !j ðnÞ j ðn 1Þ j¼1
!ðnÞ ¼
ð3:6Þ
with
and
!1 ðnÞ ¼ v0 ðnÞ;
M ðnÞ ¼
ð2:4aÞ
ðnÞxm
1 ðnÞ; ¼ Bm ðnÞ ðnÞ;
ð2:4bÞ
!3 ðnÞ ¼ v0 ðnÞðv1 ðnÞ þ v2 ðnÞv0 ðn þ 1ÞÞ;
ðnÞym ¼ Cm ðnÞ ðnÞ:
ð2:4cÞ
:
ð2:5Þ
In the following, we construct the CLs of the 2DTL hierarchy in a new way, which differs from KajiwaraSatsuma’s approach.19) First, we differentiate eq. (1:1) directly with respect to xm and note that (2:3b) yields Bm ðnÞðnÞ ; ðln ðnÞÞxm ¼ ðE 1Þ ð3:8Þ ðnÞ
The related Lax equations, Lxm ¼ ½Bm ðnÞ; L ; Lym ¼ ½Cm ðnÞ; L ; Mxm ¼ ½Bm ðnÞ; M ; Mym ¼ ½Cm ðnÞ; M ; give an infinite number of nonlinear differential-difference equations for ui and vi , which are called the 2DTL hierarchy.19) The lowest equations in this hierarchy are
!2 ðnÞ ¼ v0 ðnÞv1 ðnÞ;
ð3:7Þ 2
which is simply the semidiscrete CL between the directions n and xm . In the same way, we have another three semidiscrete CLs: Cm ðnÞðnÞ ; ð3:9Þ ðln ðnÞÞym ¼ ðE 1Þ ðnÞ
u1 ðnÞx1 ¼ u2 ðn þ 1Þ u2 ðnÞ;
ð2:6Þ
u1 ðnÞy1 ¼ v0 ðnÞ v0 ðn þ 1Þ;
ð2:7Þ
v0 ðnÞx1 ¼ v0 ðnÞðu1 ðnÞ u1 ðn 1ÞÞ;
ð2:8Þ
v0 ðnÞy1 ¼ v0 ðnÞðv1 ðn 1Þ v1 ðnÞÞ:
ð2:9Þ
ðln !ðn þ 1ÞÞxm ¼ ðE 1Þ
Bm ðnÞ ðnÞ ; ðnÞ
ð3:10Þ
Eliminating u1 from eqs. (2:7) and (2:8), we can obtain the 2DTL equation19)
ðln !ðn þ 1ÞÞym ¼ ðE 1Þ
Cm ðnÞ ðnÞ : ðnÞ
ð3:11Þ
2
@ ln v0 ðnÞ ¼ 2v0 ðnÞ v0 ðn þ 1Þ v0 ðn 1Þ: @x1 @y1 3.
ð2:10Þ
The CLs of the 2DTL hierarchy
Let us consider the generalized Riccati equations associated with pseudo-difference operators (1:2) and (1:3). If we set ðnÞ ðnÞ ¼ ; ð3:1Þ ðn þ 1Þ
it is easy to rewrite the above CLs in the following explicit forms: ! 1 1 X X ð0Þ j j ðnÞ ¼ ðE 1Þ JjðmÞ ðnÞj ; ð3:13Þ j¼1
we have, from (2:3a), that ðnÞ ¼ 1 þ u1 ðnÞðnÞ þ u2 ðnÞðn 1ÞðnÞ þ u3 ðnÞðn 2Þðn 1ÞðnÞ þ
1 X
ð3:2Þ
j¼1
jð0Þ ðnÞj
1 X
1 X
2 ðnÞ ¼ u1 ðnÞ;
3 ðnÞ ¼ u1 ðnÞ2 þ u2 ðnÞ; : ð3:4Þ
Similarly, we have another generalized Riccati equation related to (1:3) and (2:4a): 1 !ðnÞ ¼ v0 ðnÞ þ v1 ðnÞ!ðnÞ þ v2 ðnÞ!ðnÞ!ðn þ 1Þ þ ; ð3:5Þ
¼ ðE 1Þ !
j
ð0Þ j ðnÞ
¼ ðE 1Þ xm
¼ ðE 1Þ ym
KjðmÞ ðnÞj ;
ð3:14Þ
1 X
j PðmÞ j ðnÞ ;
ð3:15Þ
j QðmÞ j ðnÞ :
ð3:16Þ
j¼0
! j
ð0Þ j ðnÞ
1 X j¼1
ym
j¼0
j¼0
and
j¼1
xm
!
j¼1
which we call the generalized Riccati equation. This equation can be solved by setting 1 X ðnÞ ¼ j ðnÞj ð3:3Þ
1 ðnÞ ¼ 1;
Then, by employing expansions (3:3), (3:6) and some simple formulas such as 1 1 X X 1 k 1 lnð1 qÞ ¼ q; ¼ qk ; ð3:12Þ k 1 q k¼0 k¼1
1 X j¼0
That is to say, the infinitely many semidiscrete CLs can be described as jð0Þ ðnÞxm ¼ ðE 1ÞJjðmÞ ðnÞ;
j ¼ 1; 2; . . . ;
ð3:17Þ
jð0Þ ðnÞym ¼ ðE 1ÞKjðmÞ ðnÞ;
j ¼ 1; 2; . . . ;
ð3:18Þ
ðmÞ
ð0Þ j ðnÞxm ¼ ðE 1ÞPj ðnÞ;
j ¼ 0; 1; . . . ;
ð3:19Þ
ðmÞ
ð0Þ j ðnÞym ¼ ðE 1ÞQj ðnÞ;
j ¼ 0; 1; . . . :
ð3:20Þ
J. Phys. Soc. Jpn., Vol. 71, No. 11, November, 2002
LETTERS
Some of the conserved densities jð0Þ ðnÞ, ð0Þ j ðnÞ and the ðmÞ associated fluxes JjðmÞ ðnÞ, KjðmÞ ðnÞ, PðmÞ j ðnÞ, Qj ðnÞ are listed here:
1ð0Þ ðnÞ ¼ u1 ðnÞ;
ð3:21aÞ
1 2ð0Þ ðnÞ ¼ u1 ðnÞ2 u2 ðnÞ; 2 1 3ð0Þ ðnÞ ¼ u1 ðnÞ3 u1 ðnÞu2 ðnÞ 3 u1 ðn 1Þu2 ðnÞ u3 ðnÞ;
ð3:21cÞ ð3:22aÞ
ð0Þ 1 ðnÞ ¼ v1 ðn þ 1Þ;
ð3:22bÞ
1 v1 ðn þ 1Þ2 þ v0 ðn þ 2Þv2 ðn þ 1Þ; 2
J1ð1Þ ðnÞ ¼ u2 ðnÞ; J2ð1Þ ðnÞ ¼ u1 ðn 1Þu2 ðnÞ u3 ðnÞ; J1ð2Þ ðnÞ ¼ u2 ðnÞðu1 ðn 1Þ þ u1 ðnÞÞ u3 ðnÞ u3 ðn þ 1Þ;
ð3:21bÞ
ð0Þ 0 ðnÞ ¼ ln v0 ðn þ 1Þ;
ð0Þ 2 ðnÞ ¼
D. Z HANG
Cm ðnÞðnÞ ðnÞ Bm ðnÞðnÞ ðnÞ
Bm ðnÞ ðnÞ ðnÞ Cm ðnÞ ðnÞ ðnÞ Bm ðnÞ ðnÞ ðnÞ
¼
ys
¼
ys
¼
xs
¼
ys
¼
ys
Cs ðnÞðnÞ ðnÞ Cs ðnÞðnÞ ðnÞ
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;
Bs ðnÞ ðnÞ ðnÞ Cs ðnÞ ðnÞ ðnÞ Cs ðnÞ ðnÞ ðnÞ
ðs 6¼ mÞ;
ð3:32Þ
ym
;
ð3:33Þ
xm
;
ðs 6¼ mÞ;
ð3:34Þ
;
ðs 6¼ mÞ;
ð3:35Þ
xm
ym
:
ð3:36Þ
xm
The above six CLs suggest the following infinitely many CLs:
ð3:22cÞ
JjðmÞ ðnÞxs ¼ JjðsÞ ðnÞxm ;
ðs 6¼ mÞ;
j ¼ 1; 2; . . . ;
ð3:37Þ
ð3:23aÞ
KjðmÞ ðnÞys ¼ KjðsÞ ðnÞym ;
ðs 6¼ mÞ;
j ¼ 1; 2; . . . ;
ð3:38Þ
ð3:23bÞ
JjðmÞ ðnÞys
¼
KjðsÞ ðnÞxm ;
j ¼ 1; 2; . . . ;
ð3:39Þ
¼
PðsÞ j ðnÞxm ;
ðs 6¼ mÞ;
j ¼ 0; 1; . . . ;
ð3:40Þ
¼
QðsÞ j ðnÞym ;
ðs 6¼ mÞ;
j ¼ 0; 1; . . . ;
ð3:41Þ
¼
QðsÞ j ðnÞxm ;
j ¼ 0; 1; . . . ;
PðmÞ j ðnÞxs ð3:24Þ
K1ð1Þ ðnÞ ¼ v0 ðnÞ;
ð3:25aÞ
K2ð1Þ ðnÞ ¼ v0 ðnÞu1 ðn 1Þ;
ð3:25bÞ
K3ð1Þ ðnÞ ¼ v0 ðnÞðu1 ðn 1Þ2 þ u2 ðn 1ÞÞ;
ð3:25cÞ
QðmÞ j ðnÞys PðmÞ j ðnÞys
ð3:42Þ
ðv1 ðn 1Þ þ v1 ðnÞÞ;
ð3:26bÞ
Pð1Þ 0 ðnÞ ¼ u1 ðnÞ;
ð3:27aÞ
Pð1Þ 1 ðnÞ ¼ v0 ðn þ 1Þ;
ð3:27bÞ
ðmÞ where JjðmÞ ðnÞ, KjðmÞ ðnÞ, PðmÞ j ðnÞ and Qj ðnÞ are partly given by eqs. (3:23)–(3:30). ð1Þ ð1Þ Of course, if we express jð0Þ ðnÞ, ð0Þ j ðnÞ, Jj ðnÞ, Kj ðnÞ, ð1Þ ð1Þ Pj ðnÞ and Qj ðnÞ in terms of v0 ðnÞ using Lax equations (2.5), we obtain the infinitely many CLs of 2DTL equation (2:10). In addition, it is not difficult to give the CLs of the reduction equations of the 2DTL hierarchy.19,20) We also find that there are two different forms of the CLs between the same pair of distinct directions, for example, (3:8) and (3:10), (3:33) and (3:36). Therefore, one of the remaining problems is to determine whether or not these CLs are the same.
Pð1Þ 2 ðnÞ ¼ v0 ðn þ 1Þv1 ðn þ 1Þ;
ð3:27cÞ
4.
K1ð2Þ ðnÞ ¼ v0 ðnÞðv1 ðn 1Þ þ v1 ðnÞÞ;
ð3:26aÞ
K2ð2Þ ðnÞ ¼ v0 ðnÞv0 ðn 1Þ þ v0 ðnÞu1 ðn 1Þ
2 Pð2Þ 0 ðnÞ ¼ u2 ðnÞ þ u2 ðn þ 1Þ þ u1 ðnÞ ;
ð3:28aÞ
Pð2Þ 1 ðnÞ ¼ v0 ðn þ 1Þðu1 ðnÞ þ u1 ðn þ 1ÞÞ;
ð3:28bÞ
Pð2Þ 2 ðnÞ ¼ v0 ðn þ 1Þ½v0 ðn þ 2Þ þ v1 ðn þ 1Þðu1 ðnÞ þ u1 ðn þ 1ÞÞ ;
ð3:28cÞ
Qð1Þ 0 ðnÞ ¼ v1 ðnÞ;
ð3:29aÞ
Qð1Þ 1 ðnÞ ¼ v0 ðn þ 1Þv2 ðnÞ;
ð3:29bÞ
2 Qð2Þ 0 ðnÞ ¼ v0 ðnÞv2 ðn 1Þ v0 ðn þ 1Þv2 ðnÞ v1 ðnÞ :
ð3:30Þ @ @ ln ðnÞ @xs ð @xm Þ
@ @ ln ðnÞ @xm ð @xs Þ,
¼ it is easy to Second, in light of find the following continuous CL of the 2DTL hierarchy: Bm ðnÞðnÞ Bs ðnÞðnÞ ¼ ; ðs 6¼ mÞ: ð3:31Þ ðnÞ ðnÞ xs xm Similarly, we also have the other CLs:
Conclusions
We have described a simple way of finding the CLs for the 2DTL hierarchy and obtained the CLs between two arbitrary directions. Some CLs are the same as those derived through Kajiwara-Satsuma’s approach.19) However, some CLs and some quantities (fKjðmÞ ðnÞg, fPðmÞ j ðnÞg) are new because Bm ðnÞ is difficult to express using the powers of the pseudo-difference operator M, and Cm ðnÞ is difficult to express using the powers of L in the previous approach.19) With the help of the generalized Riccati equations, we can not only obtain the infinitely many conserved densities but also obtain the infinitely many associated fluxes, which has not been done previously. This method is novel and sufficiently flexible for application to the high-dimensional differential-difference systems. Acknowledgments This project is supported by the National Natural Science Foundation of China.
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J. Phys. Soc. Jpn., Vol. 71, No. 11, November, 2002
LETTERS
1) M. Wadati, H. Sanuki and K. Konno: Prog. Theor. Phys. 53 (1975) 419. 2) Y. Matsuno: J. Phys. Soc. Jpn. 59 (1990) 3093. 3) K. Konno, H. Sanuki and Y. H. Ichikawa: Prog. Theor. Phys. 52 (1974) 886. 4) T. Tsuchida and M. Wadati: J. Phys. Soc. Jpn. 67 (1998) 1175. 5) V. Zakharov and A. Shabat: Sov. Phys. JETP 34 (1972) 62. 6) M. Wadati: Prog. Theor. Phys. Suppl. No. 59 (1976) 36. 7) M. Wadati and M. Watanabe: Prog. Theor. Phys. 57 (1977) 808. 8) T. Tsuchida, H. Ujino and M. Wadati: J. Math. Phys. 39 (1998) 4785. 9) T. Tsuchida, H. Ujino and M. Wadati: J. Phys. A 32 (1999) 2239. 10) M. J. Ablowitz and J. F. Ladik: J. Math. Phys. 17 (1976) 1011. 11) D. J. Zhang and D. Y. Chen: Chaos, Solitons & Fractals 14 (2002) 573. 12) M. Sato: RIMS Kokyuroku Kyoto Univ. 439 (1981) 30.
D. ZHANG
13) M. Sato and Y. Sato: in Nonlinear Partial Differential Equations in Applied Science, ed. H. Fujita, P. D. Lax and G. Strang (Kinokuniya/ North-Holland, Tokyo, 1983) p. 259. 14) Y. Ohta, J. Satsuma, D. Takahashi and T. Tokihiro: Prog. Theor. Phys. Suppl. No. 94 (1988) 210. 15) K. Kajiwara, J. Matsukidaira and J. Satsuma: Phys. Lett. A 146 (1990) 115. 16) J. Matsukidaira, J. Satsuma and W. Strampp: J. Math. Phys. 31 (1990) 1426. 17) J. Sidorenko and W. Strampp: J. Math. Phys. 34 (1993) 1429. 18) Y. Cheng: J. Math. Phys. 33 (1992) 3774. 19) K. Kajiwara and J. Satsuma: J. Math. Phys. 32 (1991) 506. 20) B. Konopelchenko and W. Strampp: J. Math. Phys. 33 (1992) 3676.
