Continuous limits and integrability for a semidiscrete system Zuo-nong Zhu Department of Mathematics, Shanghai Jiao Tong University, P R China

Transcription

1 Continuous limits and integrability for a semidiscrete system Zuo-nong Zhu Department of Mathematics, Shanghai Jiao Tong University, P R China the 3th GCOE International Symposium, Tohoku University, February, 2011, Sendai, Japan

2 Outline Introduction A semidiscrete two-coupled mkdv system and its Lax pairs Darboux transformation and Exact solutions Continuous limits theories of the coupled semidiscrete mkdv Conclusions

3 Introduction A brief recall of the solitons Solitary wave is first discovered in water waves. 1834, John Scott Russell observed a large solitary wave in a shallow water channel in Scotland. This solitary wave is now known as an example of 1-soliton, and is described by a solution of the Korteweg-de Vries (KdV) equation. The KdV eq. is a one dimensional nonlinear PDE which approximately described the wave elevation (η = η(x, t)) above mean height (h).

4 The eq. derived by Korteweg-de Vries (1895) 1 η t + η x + 3 gh 2h ηη x + h2 2 (1 3 T )η xxx = 0 (1) where g is gravity, T is normalized surface tension. Consider a surface wave on water which is assumed to be irrotational and incompressible. The KdV equation is derived from the two-

5 dimensional Euler equation: ɛφ xx + φ zz = 0 [ φ z z=0 = 0 ] 1 ɛ φ z βη x φ x η t [ φ t (βφ2 x + β ] ɛ φ2 z) + η z=1+βη(x,t) = 0 z=1+βη(x,t) = 0 under the assumptions of weak nonlinearity(small amplitude) and weak dispersion(long wave). Russell s observation was finally confirmed in Mathematics.

6 1965, Zabusky and Kruskal found the Solitons. They consider u t + uu x + δ 2 u xxx = 0, u(x, 0) = cosπx, 0 x 2 (2) They choose δ = After a short time the wave steepens and almost produces a shock, but the dispersive term then becomes significant and some sort of local balance between nonlinearity and dispersion ensues. At later times the solution develops a train of eight waves, each like sech 2 functions, with the faster waves for ever catching-up and overtaking the slower waves. These nonlinear waves can interact strongly and then continue thereafter almost as if there had been no interaction at all.

7 Zabusky and Kruskal introduced the term solitons, to emphasize the particle-like character. Solitons are localized, stable, nonlinear waves. The interaction of two solitary waves is elastic, i.e., speeds and amplitudes of two solitary waves are preserved upon interaction. 1967, the method of inverse scattering transformation (IST) Gardner, Greene, Kuskal, Miura. u t + 6uu x + u xxx = 0 (3) u(x, 0) = φ(x) (4)

8 Soliton solutions of KdV (1)u(x, 0) = 2sech 2 x, solitary wave one-soliton u(x, t) = 2k 2 sech 2 k(x x 0 4k 2 t) (5) u max = 2k 2, speed = 2u max. (2)u(x, 0) = 6sech 2 x, two-solitons (3)u(x, 0) = N(N + 1)sech 2 x, N-soliton (4)u(x, 0) = 4sech 2 x, two solitons + a dispersive-wave...

9 KdV equation possesses rich mathematical structures including N-soliton solutions, Lax pair infinitely many conserved quantities Hamiltonian integrable...

10 As is well known, to construct explicit exact solutions for an integrable system is always an important topic. However, it is very difficult to construct exact solutions for the nonlinear systems. Many important and very useful approaches of obtaining the explicit exact solutions to nonlinear systems have been found. Inverse Scattering Transformations Darboux Transformations Hirota Direct Method...

11 Many processes in physics are described mathematically by differential equations(pdes, ODEs) that reflect the smoothness of natural processes. However, the assumption of a space-time continuum is no longer always adequate. We have seen that there exist a lot of discrete phenomena in physical realm, such as quantum mechanics, quantum gravity.

12 We have learned to appreciate the inherent discreteness of the physical phenomena at the atomic and subatomic level. Perhaps discrete things are more fundamental than continuous. To describe these discrete phenomena we need a mathematical machinery comprising differential-difference equations (continuous time and discrete space) and difference-difference equations (discrete time and discrete space).

13 Discrete (full discrete, or semidiscrete) KdV equation, discrete mkdv equation, discrete nonlinear Schrödinger equation,... These discrete equations not only are integrable discrete versions for the corresponding continuous systems, but also have wide applications. The discussion of continuous limits of discrete systems is one of the important topics in soliton theory. The relations between discrete KdV-type and KdV-type equations have been outlined. (Schwarz, 1982, Adv. in Math; Toda and Wadati, 1973, J Phys Soc Japan; Kupershmidt, 1985; Morosi and Pizzocchero, 1996, Commun. Math.Phys; Gieseker, 1996, Commun.Math.Phys; Zeng and Wojciechowski, 1995, J Phys A).

14 In the work of Morosi and Pizzocchero, the KdV theory including the infinitely many commuting vector fields, the conserved functions, the Lax pairs and the bi-hamiltonian structure is recovered systematically through the continuous limit of the corresponding discrete KdV system. Recently, the coupled KdV-type systems (continuous and discrete) have been extensively studied. Let us recall some works for the coupled mkdv. Iwao and Hirota, (1997, J. Phys. Soc. Jpn), and Tsuchida and Wadati (1998, J. Phys. Soc. Jpn.), discussed a coupled mkdv system, M 1 u i t + 6 C jk u j u k u i x + 3 u i = 0, i = 0, 1,.., M 1(6) x3 j,k=0

15 or an equivalent form, M 1 u i t 6 ɛ j u 2 u i j x + 3 u i x 3 = 0, ɛ j = ±1, i = 0, 1,..M 1 j=0 Iwao and Hirota obtained soliton solutions to equation (6) by the direct method. Tsuchida and Wadati shown that its initial value problem can be solved by the inverse scattering method (ISM). They constructed the N-soliton solutions of the coupled mkdv by ISM. For example, the self-focusing type of the coupled mkdv equation M 1 u i t + 6 u 2 u i j x + 3 u i = 0, i = 0, 1,.., M 1 (8) x3 j=0 (7)

16 with M = 2m possesses an explicit 1-soliton solution: Q (m) (x, t) = 2iη 1 sech[2η 1 x 8η 1 (η1 2 3ξ2 1 )t x 0] 2 2m 1 l=0 c (l) 1 2 [ C 1 (0)e 2iξ 1x 8iξ 1 (ξ 2 1 3η2 1 )t + C 2 (0)e 2iξ 1x+8iξ 1 (ξ 2 1 3η2 1 )t ] (9) The semidiscrete version of the coupled mkdv (6) is given by a (i) M 1 n = 1 + C t jk a (j) n a (k) n (a (i) n+1 a(i) n 1 ), i = 0, 1,..., M 1 j,k=0 The semidiscrete system yields coupled mkdv (6) through the con- (10) 1 2

17 tinuous limits a (i) n = δu i (x, τ), x = (n + 2t)δ, τ = 1 3 δ3 t Tsuchida, Ujino, and Wadati (1998, J Math Phys) studied its initial value problem. The general N-soliton solutions to equation (10) are obtained by the ISM. The self-focusing type of the semidiscrete coupled mkdv equation a (i) M 1 n = 1 + n t j=0 a (j)2 (a (i) n+1 a(i) n 1 ), i = 0, 1,..., M 1 (11)

18 with M = 2m has the 1-soliton solution a (i) n (t) = sech[2nw + 2(sinh2W cos2θ)t + φ 0 ]sinh2w 2 2m 1 l=0 1 (0) 2 c (l) 1 2 [ c (i) 1 (0)e2i(nθ+(cosh2W sin2θ)t) + c (i) 1 (0)e 2i(nθ+(cosh2W sin2θ)t) ], i = 0, 1,..., 2m 1. (12)

19 Q: What is the relation between the solutions (9) and (12)? In the special case θ = 0, the solution (12) yields the solution (9) in the continuous limit. However, in the general, it is very difficult to check whether the solutions of (10) reduce to the ones of (6) or not. To the best of our knowledge, the continuous limit theory (not only on the level of equation, but also on the level of the other integrabilities) of the semidiscrete coupled mkdv equation has not been established.

20 Aiming to get more insight on the relation between semidiscrete coupled mkdv and the coupled mkdv equation, we will propose a new two-coupled semidiscrete mkdv, and show that the continuous limit theories of this semidiscrete coupled mkdv including the Lax pairs, the Darboux transformation, soliton solutions, and the conservation laws yield the corresponding results of the two-coupled mkdv equation. Under the reduction, the continuous limit theories of the semidiscrete coupled mkdv yield the ones of the mkdv.

21 A new semidiscrete two-coupled mkdv system and its Lax pairs we propose the following semidiscrete two-coupled mkdv system: a n,t = (1 + a 2 n + b 2 n)[a n+2 a n 2 + 2a n 1 2a n+1 + 2b n (a n+1 a n 1 )(b n+1 + b n 1 ) +a n (a 2 n+1 a 2 n 1 + b 2 n 1 b 2 n+1) + a n+2 (a 2 n+1 + b 2 n+1) a n 2 (a 2 n 1 + b 2 n 1)], b n,t = (1 + a 2 n + b 2 n)[b n+2 b n 2 + 2b n 1 2b n+1 + 2a n (b n+1 b n 1 )(a n+1 + a n 1 ) +b n (b 2 n+1 b 2 n 1 + a 2 n 1 a 2 n+1) + b n+2 (a 2 n+1 + b 2 n+1) b n 2 (a 2 n 1 + b 2 n 1)]. (13) We can show that this last system converges to the two-coupled mkdv system u τ = 6(u 2 + v 2 )u x + u xxx, v τ = 6(u 2 + v 2 )v x + v xxx, (14)

22 under the transformations a n = δu(x, τ), b n = δv(x, τ); x = nδ, τ = 2δ 3 t. (15) From the construction of the matrix Ablowitz-Ladik equation hierarchy (Gordoa, Pickering, Zhu, 2010, J. Math. Phys.), we obtain the following Lax pair for the semidiscrete coupled mkdv equation (13): N n = where λi U n U n λ 1 I U n = a n b n dψ n Eψ n = N n ψ n, dt = M nψ n (16), M n = A(λ, λ 1, a n, b n ) B(λ, λ 1, a n, b n ) B(λ 1, λ, a n, b n ) A(λ 1, λ, a n, b n ) (17) b n a n, A(λ, λ 1, a n, b n ) = A 11 A 12 A 12 A 11,

23 B(λ, λ 1, a n, b n ) = B 11 B 12 B 12 B 11 A 11 = λ4 4 3λ λ 2 (a n a n 1 + b n b n 1 ) + λ 2 (2 a n a n 1 b n b n 1 ) + 11, A 12 = (λ 2 + λ 2 )(a n b n 1 b n a n 1 ) + 12 with B 11 = B 11 (a n, b n ) = λ 3 a n + λ 3 a n 1 + λ 13 + λ 1 14, B 12 = B 11 (b n, a n ),

24 11 = (a n a n 1 + b n b n 1 ) 2 (a n b n 1 b n a n 1 ) 2 2a n a n 1 2b n b n 1 +a n a n 2 + b n b n 2 + a n+1 a n 1 + b n+1 b n 1 + (a 2 n 1 + b 2 n 1)(a n a n 2 + b n b n 2 ) +(a 2 n + b 2 n)(a n+1 a n 1 + b n+1 b n 1 ) 3 2, 12 = 2(a n a n 1 + b n b n 1 )(a n b n 1 b n a n 1 ) 2a n b n 1 + 2b n a n 1 +a n b n 2 b n a n 2 + a n+1 b n 1 b n+1 a n 1 + (a 2 n 1 + b 2 n 1)(a n b n 2 b n a n 2 ) +(a 2 n + b 2 n)(a n+1 b n 1 b n+1 a n 1 ) 13 = a n (a n a n 1 + b n b n 1 2) b n (b n a n 1 a n b n 1 ) + a n+1 (a 2 n + b 2 n + 1) 14 = a n 1 (a n a n 1 + b n b n 1 2) b n 1 (a n b n 1 b n a n 1 ) + a n 2 (a 2 n 1 + b 2 n 1 + 1)

25 Darboux transformation of the coupled semidiscrete mkdv The original idea of Darboux transformation: Darboux found that a solution of the Schrödinger equation L(u)φ = λφ, L(u) = φ xx u can be transformed into a solution φ = ( x φ 0x φ 0 )φ of L(ũ) φ = λ φ by means of eigenfunction φ 0 satisfying L(u)φ 0 = λ 0 φ 0 ũ = u + 2(lnφ 0 ) xx

26 Darboux s idea has been applied to construct exact solutions to many integrable systems, including KdV equation, mkdv equation, AKNS system, et al.

27 We obtain the following results: Theorem: The solutions (a n, b n ) of semidiscrete two-coupled mkdv equation (8) are mapped into new solutions (ã n, b n ) under the Darboux transformation (ψ n, a n, b n ) ( ψ n, ã n, b n ), where ã n, b n, ψ n are given by ã n = 1 (12) [(T 2 n+1 T (11) n+1 )b n + T (13) n+1 T (14) n+1 ] (18) bn = 1 (11) [(T 2 n+1 T (12) n+1 )a n + T (14) n+1 + T (13) n+1 ] (19) ψ n = T n ψ n (20)

28 and the matrix T n is written as: T n (λ) = where T (13) n λ + λ 1T n (11) λ + λ 1T n (12) T n (13) T n (14) λ λ 1T n (12) λ + λ 1T n (11) T n (14) T n (13) 1 (11) + λt T n (13) T n (14) T n (14) T n (13) T n (11) = T n (12) λ 1 λ λt (12) n = β n σ n = (λ 1 1 λ 3 1 )γ n, T n (14) σ n n λ 1 1 λ (12) + λt n (11) + λt = (λ 1 1 λ 3 1 )φ n σ n n (21)

29 β n = (λ 1 ψ (1) n (λ 1 )) 2 + (λ 1 ψ (2) n (λ 1 )) 2 + (ψ (3) n (λ 1 )) 2 + (ψ (4) n (λ 1 )) 2, γ n = ψ (1) n (λ 1 )(ψ (3) n (λ 1 ) ψ (4) n (λ 1 )) ψ (2) n (λ 1 )(ψ (3) n (λ 1 ) + ψ (4) n (λ 1 )), φ n = ψ (1) n (λ 1 )(ψ (3) n (λ 1 ) + ψ (4) n (λ 1 )) + ψ (2) n (λ 1 )(ψ (3) n (λ 1 ) ψ (4) n (λ 1 )), σ n = (ψ (1) n (λ 1 )) 2 + (ψ (2) n (λ 1 )) 2 + (λ 1 ψ (3) n (λ 1 )) 2 + (λ 1 ψ (4) n (λ 1 )) 2. where ψ n (λ) = (ψ (1) n (λ), ψ (2) n (λ), ψ (3) n (λ), ψ (4) n (λ)) T is the eigenfunction of the spectral problem (16) corresponding to the solutions a n and b n.

30 Continuous limits theories of the coupled semidiscrete mkdv system (13) We will show that continuous limit theories of the semidiscrete coupled mkdv (13) including the Lax pairs, the Darboux transformation, soliton solutions, and conservation laws can yield the corresponding results of the coupled mkdv (14). We first give the Darboux transformation of the coupled mkdv system (14). The linear eigenvalue problem of (14) is written as (Zhang etal, 2008, J Phys. A) φ x = U 0 φ, φ τ = V 0 φ (22)

31 where U 0 = U = u zi U U zi v v u, V 01 (z) =, V 0 = V 11 V 12 V 12 V 11 V 01(z) V 02 (z) V 02 ( z) V 01 ( z), V 02 (z) = (23) V 13 V 14 V 14 V 13 V 11 = 4z 3 + 2(u 2 + v 2 )z, V 13 = 4uz 2 + 2u x z + 2u 3 + u xx + 2v 2 u, V 12 = 2vu x 2uv x, We obtain the following result:, V 14 = 4vz 2 + 2v x z + 2v 3 + v xx + 2u 2 v Theorem: The solutions (u, v) of the coupled mkdv equation (14) are mapped into new solutions (ũ, ṽ) under the Darboux

32 transformation (φ, u, v) ( φ, ũ, ṽ), where ũ, ṽ, φ are given by ũ = v + D (13) D (14), ṽ = u + D (13) + D (14) (24) φ = Dφ, (25) where z + D (11) z + D (12) D (13) D (14) z D (12) z + D (11) D (14) D (13) D(z) = κ D (13) D (14) z + D (11) z + D (12) D (14) D (13) z D (12) z + D (11) (26) where κ is an arbitrary constant, and

33 D (11) = D (12) = z 1 (φ (1) (z 1 )) 2 + (φ (2) (z 1 )) 2 (φ (3) (z 1 )) 2 (φ (4) (z 1 )) 2 θ D (13) = 2z 1 φ (1) (z 1 )(φ (3) (z 1 ) φ (4) (z 1 )) φ (2) (z 1 )(φ (3) (z 1 ) + φ (4) (z 1 )) θ D (14) = 2z 1 φ (1) (z 1 )(φ (3) (z 1 ) + φ (4) (z 1 )) + φ (2) (z 1 )(φ (3) (z 1 ) φ (4) (z 1 )) θ with θ = (φ (1) (z 1 )) 2 + (φ (2) (z 1 )) 2 + (φ (3) (z 1 )) 2 + (φ (4) (z 1 )) 2 Here φ(z 1 ) = (φ (1) (z 1 ), φ (2) (z 1 ), φ (3) (z 1 ), φ (4) (z 1 )) T is an eigenfunction of the spectral problem (16) with z = z 1 corresponding to the solution u, v.

34 Remark The Darboux transformation obtained here is different from the one given by zhang etal. Let us establish the continuous limit theories for the semidiscrete coupled mkdv. Set ψ n (i) = δφ (i) (x, τ), ψ(i) n = δ 2 φ(i) (x, τ), λ = e zδ = 1 + δz δ2 z δ3 z 3 + o(δ 3 ), κ = 2. After a detailed calculations, we have Eψ n N n ψ n = (φ x Uφ)δ 2 + o(δ 2 ), (27) dψ n dt M nψ n = 2(φ τ V φ)δ 4 + o(δ 4 ). (28)

35 We thus obtain theorem The Lax pair of the coupled mkdv equation (14) is reconstructed by the continuous limit of the Lax pair of the coupled semidiscrete mkdv equation (13). Noting that T n (13) = 2D (13) δ + o(δ), T n (14) = 2D (14) δ + o(δ), T (11) n = T (12) n = A 1 + A 2 δ + o(δ) B 1 + B 2 δ + o(δ) = A 1 A 2B 1 A 1 B 2 δ + o(δ) = 1 + 2D (11) δ + o(δ), B 1 B 2 1 where A 1 = B 1 = θ, A 2 = 2z 1 [(φ (1) (z 1 )) 2 + (φ (2) (z 1 )) 2 ],

36 B 2 = 2z 1 [(φ (3) (z 1 )) 2 + (φ (4) (z 1 )) 2 ], we have ã n 1 2 ψ n T n (λ)ψ n = δ 2 ( φ D(z)φ) + o(δ 2 ). (12) (11) [(T n+1 T n+1 )b n+t (13) n+1 bn 1 (11) (12) [(T 2 n+1 T n+1 )a n+t (13) n+1 Therefore, we have T (14) n+1 ] = δ(ũ v D(13) +D (14) )+o(δ) +T (14) n+1 ] = δ(ṽ+u D(13) D (14) )+o(δ) theorem Continuous limits of the Darboux transformation and explicit solutions of the coupled semidiscrete mkdv (13) yield the corresponding results of the coupled mkdv (14). Remark The Darboux transformation, nonlinear superposition principle and the solutions for a semidiscrete mkdv system were

37 derived by Adler and Postnikov (2008, J. Phys. A: Math. Theor). However, their continuous limits do not lead to the corresponding results of the coupled mkdv system. Example 1 We will show that continuous limit of the 1-soliton solution to semidiscrete coupled mkdv equation (13) yields the 1- soliton solution to coupled mkdv equation (14). We take a n = b n = 0 as the seed solution. Solving the spectral equation corresponding to the seed gives ψ n (λ 1 ) = (ψ n (1) (λ 1 ), ψ n (2) (λ 1 ), ψ n (3) (λ 1 ), ψ n (4) (λ 1 )) T where ψ (1) n (λ 1 ) = c 11 e ξ 1(λ 1 ), ψ (2) n (λ 1 ) = c 21 e ξ 1(λ 1 ), ψ (3) n (λ 1 ) = c 31 e ξ 1(λ 1 1 ), ψ (4) n (λ 1 ) = c 41 e ξ 1(λ 1 1 )

38 with Noting that ξ 1 (λ 1 ) = n ln λ 1 + [ 1 4 (λ4 1 3λ 4 1 ) + 2λ ]t. ξ 1 (λ 1 ) = nz 1 δ + [ 1 4 (e4z 1δ 3e 4z 1δ ) + 2e 2z 1δ 3 2 ]t = z 1 nδ + 8z 3 1 δ3 t + o(δ 3 ) = z 1 x + 4z 3 1 τ + o(δ3 ) ξ 1 (λ 1 1 ) = nz 1δ + [ 1 4 (e 4z 1δ 3e 4z 1δ ) + 2e 2z 1δ 3 2 ]t we have = z 1 nδ 8z 3 1 δ3 t + o(δ 3 ) = z 1 x 4z 3 1 τ + o(δ3 ) ψ (i) n (λ 1 ) = φ (i) (z 1 ) + o(δ 3 ), i = 1, 2, 3, 4 (29)

39 where φ (1) (z 1 ) = c 11 e ζ 1, φ (2) (z 1 ) = c 21 e ζ 1, φ (3) (z 1 ) = c 31 e ζ 1, φ (4) (z 1 ) = c 41 e ζ 1 with ζ 1 = z 1 x + 4z1 3τ. Remark φ(z 1 ) = (φ (1) (z 1 ), φ (2) (z 1 ), φ (3) (z 1 ), φ (4) (z 1 )) T is just the eigenfunction for the coupled mkdv equation corresponding to the seed solution u = v = 0. By using the Darboux transformation, we obtain 1-soliton solution

40 to semidiscrete two-coupled mkdv a n = (λ3 1 λ 1 1 )(c 11c 41 + c 21 c 31 ) 2(c c2 41 )eη 1 b n = (λ3 1 λ 1 1 )(c 21c 41 c 11 c 31 ) 2(c c2 41 )eη 1 sech(ξ 1 (λ 1 ) ξ 1 (λ 1 1 ) + η 1), sech(ξ 1 (λ 1 ) ξ 1 (λ 1 1 ) + η 1), where e 2η 1 = λ2 1 (c2 11 +c2 21 ) c 2. Setting λ 1 = e z1δ gives 31 +c2 41 a n = δu + o(δ), b n = δv + o(δ), (30) where u = 2z 1(c 11 c 41 + c 21 c 31 ) (c c2 41 )eε 1 v = 2z 1(c 21 c 41 c 11 c 31 ) (c c2 41 )eε 1 sech(2ζ 1 + ε 1 ), sech(2ζ 1 + ε 1 )

41 with e 2ε 1 = c2 11 +c2 21 c 2. This shows that the soliton solutions of the 31 +c2 41 coupled semidiscrete mkdv yield the ones of the coupled mkdv equation through the continuous limits: a n lim δ 0 δ = u, lim b n = v. (31) δ 0 δ It is well known that the conservation laws are an important integrability feature. Finally we will show that continuous limits of the conservation laws of the coupled semidiscrete mkdv equation (13) yield the ones of the coupled mkdv (14). Using a similar method of constructing conservation laws (Tsuchida, Ujino and Wadati, 1998, J. Math. Phys.), we can conclude that semidiscrete coupled mkdv

42 (13) possesses infinitely many conservation laws: (L (i) n ) t = (E 1)F (i) n, i 1. (32) The first three conserved densities are given by L (1) n = 2 log(1 + a 2 n + b 2 n) L (2) n = 2(a n+1 a n + b n+1 b n ) L (3) n = 2[2a n a n+1 b n b n+1 + a n a n+2 + b n b n+2 + (a n a n+2 +b n b n+2 )(a 2 n+1 + b2 n+1 )] + (a2 n+1 b2 n+1 )(a2 n b 2 n) The corresponding flux functions are very complicated. Here we

43 only write the first two fluxes. F (1) n = 4[(a n+1 a n 1 + b n+1 b n 1 )(a 2 n + b 2 n) + (a 2 n 1 b2 n 1 )(a2 n b 2 n) +(a n a n 2 + b n b n 2 )(a 2 n 1 + b2 n 1 ) + 4a na n 1 b n b n 1 + a n+1 a n 1 +b n+1 b n 1 + a n a n 2 + b n b n 2 2a n a n 1 2b n b n 1 ]

44 F (2) n = (1 + a 2 n + b 2 n){4a n 1 b n (a n b n 1 + a n+1 b n+1 ) +4a n+1 b n 1 (a n b n+1 + a n 1 b n ) 4a n+1 a n 1 4b n+1 b n 1 +2[b n 1 b n+2 + b n+1 b n 2 + b n b n 1 + a n+2 a n 1 +b n+1 b n + a n+1 a n 2 + a n a n 1 + a n a n+1 + (a n a n+1 b n b n+1 )(a 2 n 1 b2 n 1 ) + (a n+1a n 2 + b n+1 b n 2 )(a 2 n 1 + b2 n 1 ) +(a n 1 a n+2 + b n 1 b n+2 )(a 2 n+1 + b2 n+1 ) +(a n a n 1 b n b n 1 )(a 2 n+1 b2 n+1 )]} 4a2 n 4b 2 n On the other hand, the coupled mkdv equation (14) has the following infinite many conservation laws: (ω i ) τ = (J i ) x, i 1. (33)

45 The explicit forms of the first three conserved densities and fluxes are given by ω 1 = (u 2 + v 2 ), ω 2 = 1 4 (u2 + v 2 ) x, ω 3 = 1 4 [uu xx + vv xx + (u 2 + v 2 ) 2 ]; J 1 = (3u 4 + 3v 4 u 2 x vx 2 + 2uu xx + 2vv xx + 6u 2 v 2 ), J 2 = 3(v 3 v x + u 3 u x + u 2 vv x + v 2 uu x ) (uu xxx + vv xxx ), J 3 = 1 4 [u2 xx + uu xxxx u x u xxx + 4u 6 + 6(uu x ) 2 +10u 3 u xx + 12u 2 v 4 2(u x v) u 2 vv xx + vxx 2 + vv xxxx v x v xxx +4v 6 + 6(vv x ) v 3 v xx + 12u 4 v 2 2(uv x ) uu xx v uu x vv x ]. Thus, the conservation laws for the both semidiscrete coupled mkdv

46 and coupled mkdv have been obtained. However, we find that the conservation laws of the semidiscrete coupled mkdv do not yield the ones of the coupled mkdv in the continuous limit. It would be interesting to recover the conservation laws of the coupled mkdv as a continuous limit. To this end, we form the following linear combinations for the conserved densities and fluxes: σ n (1) = L (1) n, σ n (2) = L (2) n L (1) n, σ n (3) = L (3) n 2L (2) n + L (1) n, σ n (4) = L (4) n 5L (3) n + 7L (2) n 3L (1) n ; B (1) n = F (1) n, B (2) n = F (2) n F (1) n, B (3) n = F (3) n 2F (2) n + F (1) n, B (4) n = F (4) n 5F (3) n + 7F (2) n 3F (1) n. (34) We then have new conservation laws of the semidiscrete coupled

47 mkdv equation Further, we can show that (σ (i) n ) t = (E 1)B (i) n, i = 1, 2, 3, 4. (35) (σ (i) n ) t (E 1)B (i) n = 4 i [(ω i ) τ (J i ) x ]δ 4+i +o(δ 4+i ), i = 1, 2, 3, 4. (36) This implies that the continuous limit of the new conservation laws of the semidiscrete coupled mkdv equation yields the conservation laws of the coupled mkdv equation. But, it is not clear how to recover the other conservation laws of the coupled mkdv equation from the ones of the semidiscrete coupled mkdv equation through continuous limits.

48 Remark Under the reduction b n = 0, we obtain the continuous limit theory of the semidiscrete mkdv equation, a n,t = (1 + a 2 n)[a n+2 a n 2 + 2a n 1 2a n+1 +(a n + a n+2 )a 2 n+1 (a n + a n 2 )a 2 n 1 ] The theory of the mkdv including the Lax pairs, the Darboux transformation, the soliton solutions, and the conservation laws is recovered through the continuous limit of the corresponding theory of the semidiscrete mkdv equation.

49 Conclusions We have recalled some facts for solitons. We have stated the significance of investigation for the discrete integrable systems. We have proposed a new semidiscrete two-coupled mkdv equation, and constructed its Darboux transformation, soliton solutions, and the conservation laws. We have shown that the theories of the coupled semidiscrete mkdv system including the Lax pair, the Darboux transformation, soliton solutions, and the conservation laws yield the corresponding theories of the coupled mkdv through the continuous limit.

50 THANK YOU