Combined Wronskian solutions to the 2D Toda molecule equation

Transcription

1 Combned Wronskan solutons to the 2D Toda molecule equaton Wen-Xu Ma Department of Mathematcs and Statstcs, Unversty of South Florda, Tampa, FL , USA Abstract By combnng two peces of b-drectonal Wronskan solutons, molecule solutons n Wronskan form are presented for the fnte, sem-nfnte and nfnte blnear 2D Toda molecule equatons. In the cases of fnte and sem-nfnte lattces, separated-varable boundary condtons are mposed. The Jacob denttes for determnants are the key tool employed n the soluton formulatons. Key words: Toda lattce, Wronskan soluton, Solton equaton PACS: Ik, p 1 Introducton The study of solton equatons presents nterestng mathematcal theores to deal wth nonlnear equatons. Wronskan determnants, double Wronskan determnants and bdrectonal Wronskan determnants are used to construct exact solutons to solton equatons, among whch are the KdV equaton, the Boussnesq equaton, the KP equaton, the Toda lattce equaton and the 2D Toda lattce equaton see, e.g., 1-10). The Plücker relatons for determnants and the Jacob denttes for determnants are the key tools employed n formulatng exact solutons to solton equatons 1, 11. Generc mult-exponental wave solutons can be constructed by the multple expfuncton method 12. The approach generalzes the transformed ratonal functon method 13 and the Hrota perturbaton technque 1, and t s very powerful whle applyng computer algebra systems 12. The resultng multple wave solutons contan lnear combnaton solutons of exponental waves 14, 15 and resonant soltons 16. Emal: mawx@cas.usf.edu 1

2 Ths also shows that solton equatons can possess lnear superpostons among partcular solutons 14, 15, and thus possess lnear subspaces of solutons. Therefore, though solton equatons are nonlnear, they are good neghbors to lnear equatons. However, gven the complexty that nonlnear equatons brng, there s a need to develop more explct and systematc formulatons for generatng exact solutons. Ths paper s one of such exploratons. In ths paper, we would lke to formulate molecule b-drectonal Wronskan solutons for the 2D Toda molecule 2DTM) equaton n blnear form: 2 τ n x τ n τ n τ n x = τ n+1τ n 1 n three cases of the fnte lattce: 1 n N, the sem-nfnte lattce: 1 n <, and the nfnte lattce: < n <. For the frst two cases, we mpose the separatedvarable boundary condtons: τ 0 = φ 1 x)χ 1 y), τ N+1 = φ 2 x)χ 2 y); and τ 0 = φx)χy); respectvely, where all φ- and χ-functons are arbtrarly gven. The dfference among these three cases s that we smply don t requre any boundary condtons at n = ±. We wll show that combnng two peces of b-drectonal Wronskan solutons 17 yelds a requred molecule soluton. By molecule solutons, we mean a knd of determnant solutons whose determnants have orders dependng on the dscrete ndependent varable n. The Jacob denttes for determnants are the key tool employed n the soluton formulatons 2 B-drectonal Wronskans and the Jacob dentty We provde the defnton of the b-drectonal Wronskan determnant and dscuss the Jacob dentty for determnants for the reader s convenence and ease of reference. A b-drectonal Wronskan determnant s defned as follows. Defnton 2.1 A b-drectonal Wronskan determnant of order n assocated wth 2

3 Υ = Υx, y) s defned by 1 ) 1Υ = 1, n Υ Υ n 1 n 1 Υ x Υ 2 x Υ n x n 1 Υ. n 1 x n 1 Υ..... n x n 1 Υ 2n 2 x n 1 n 1 Υ. 2.1) The determnant n 2.1) s a Wronskan determnant n both horzontal and vertcal drectons. That s why t s called b-drectonal. Let us next state the Jacob dentty and gve a drect proof by usng the Laplace Expanson Theorem. Let n > 2 be an nteger, A = a, ) 1, n be a square matrx of order n and D denote the determnant of A, that s, So D s an nth-order determnant. D = deta) = a, 1, n. 2.2) The, ) mnor of A s defned as the n 1)th-order determnant obtaned by strkng out the th row and the th column of D, denoted by D. All such mnors are called frst mnors. The, ; k, l) mnor of A s defned as the n 2)thorder determnant obtaned by strkng out the th and th rows and the kth and lth, columns of D, denoted by D. All such mnors are called second mnors. k, l Now we can state the Jacob dentty 1, 18 as follows. Theorem 2.1 Let n > 3, A = a, ) 1, n and D = deta). For 1 n, we have, D D D D = D k, l D, 2.3), where D and D k, l are the, ) mnor and the, ; k, l) mnor of A, respectvely. Proof: By the propertes of determnants, wthout loss of generalty, we only need to verfy the Jacob dentty for = 1 and = 2. Let us denote the, ) cofactor of A by 3

4 C, : C, = 1) + D. 2.4) We partton the matrx A nto four blocks as follows: A1 A a1,1 A = 2 a 1,2, A A 3 A 1 =. 2.5) 4 a 2,1 a 2,2 By the Laplace Expanson Theorem, we have C 1,1 C 1,2 0 C 2,1 C 2,2 A1 A 2 C 3,1 C 3,2 1 0 A 3 A C n,1 C n,2 0 1 = D 0 0 D A 2 0 A 4. Takng determnants on both sdes leads to DC 1,1 C 2,2 C 1,2 C 2,1 ) = D 1, 2 1, 2 D ) If D 0, ths gves the desred Jacob dentty, upon usng 2.4). If D = 0, we take another matrx A = A + εi n, where I n s the nth-order dentty matrx, and then takng the lmt ε 0 of the resultng dentty 2.6) assocated wth A yelds the desred Jacob dentty. Note that we have used a fact that f ε s small enough, the matrx A s nvertble. 3 Combned b-drectonal Wronskan solutons Let us now start to construct combned b-drectonal Wronskan solutons to the 2D Toda molecule 2DTM) equaton 2 τ n x τ n τ n τ n x = τ n+1τ n 1, 3.1) whch s equvalent to D x D y τ n τ n = 2τ n+1 τ n 1, 3.2) where D x and D y are Hrota s dfferental operators 1, 19. We wll present the soluton formulatons n the fnte, sem-nfnte and nfnte cases separately. 4

5 3.1 Fnte lattce We consder the fnte 2DTM equaton 2 τ n x τ n τ n τ n x = τ n+1τ n 1, 1 n N, 3.3) wth the followng separated-varable boundary condtons: τ 0 = φ 1 x)χ 1 y), τ N+1 = φ 2 x)χ 2 y), 3.4) where φ and χ, = 1, 2, are four arbtrarly gven functons of the ndcated varables. We apply the Jacob denttes for determnants to guarantee a class of combned molecule b-drectonal Wronskan solutons to ths boundary problem. Set N = N 1 + N 2 + 4, where N 1 and N 2 are non-negatve ntegers. Let us combne two peces of b-drectonal Wronskan determnant functons to ntroduce τ n as follows: τ n = 1 ) 1Φx, y), 0 n N 1, 1, N1 n+1 τ N1 +1 = 1, τ N1 +2 = 0, τ N1 +3 = 0, τ N1 +4 = 1, 3.5) τ n = 1 ) 1Ψx, y), N n N , n N+N 2 One pece s defned over the N 1 lattce ponts: 1 n N 1, and the other pece, over the N 2 lattce ponts: N N = N n N. In between, set τ n as ether zero or one. We wll prove that ths combned Wronskan determnant functon solves the 2DTM equaton 3.3) wth the boundary condtons 3.4). Note that the two nvolved determnants n the presented soluton formulaton are b-drectonal Wronskan determnants and ther orders depend on the dscrete ndependent varable n. Therefore, 3.5) presents combned molecule b-drectonal Wronskan solutons. Solvng the 2DTM equaton: Let us frst prove that τ n defned by 3.5) solves the 2DTM equaton 3.3) when 1 n N 1. For brevty, we assume that Φ, = ) Φx, y),, ) If n = N 1, the 2DTM equaton 3.3) becomes Φ 1,1 Φ 0,0 Φ 1,0 Φ 0,1 = Φ 0,0 Φ 0,1 Φ 1,0 Φ 1,1, 3.7) 5

6 whch s obvously true. Let 1 n N 1 1. We ntroduce three knds of determnants: D 1 = 1 ) 1Φx, y) = τ n 1, 3.8) 1, N1 n+2 D 1 D 1, k, l = the determnant obtaned by strkng out the th row and th column of D 1, = the determnant obtaned by strkng out the th and th rows and the kth and lth columns of D 1, 3.9) 3.10) whch are the determnant, and a frst mnor and a second mnor of a correspondng matrx, respectvely. Usng ths determnant notaton, we can easly compute N1 n + 2 N1 n + 1, N 1 n + 2 τ n = D 1, τ n+1 = D 1, N 1 n + 2 N 1 n + 1, N 1 n + 2 τ n N1 x = D n + 1 τ n N1 1, N 1 n + 2 = D n + 2 1, N 1 n τ n N1 x = D n N 1 n + 1 Now t follows that for each 1 n N 1 1, the 2DTM equaton 3.3) s equvalent to N1 n + 1 N1 n + 2 N1 n + 1 N1 n + 2 D 1 D 1 D 1 D 1 N 1 n + 1 N 1 n + 2 N 1 n + 2 N 1 n + 1 N1 n + 1, N 1 n + 2 = D 1 D 1. N 1 n + 1, N 1 n + 2 These are smply the Jacob denttes for determnants. Therefore, τ n defned by 3.5) solves 3.3) when 1 n N 1. When n = N 1 +, 1 4, t s drect to check that the 2DTM equaton 3.3) holds. Let us now smlarly prove that τ n defned by 3.5) solves the 2DTM equaton 3.3) when N n N. Assume for brevty that ) Ψx, Ψ, = y),, ) If n = N 1 + 5, the 2DTM equaton 3.3) reduces to Ψ 0,0 Ψ 0,1 Ψ 1,1 Ψ 0,0 Ψ 1,0 Ψ 0,1 = Ψ 1,0 Ψ 1,1, 3.12) 6

7 whch s clearly rght. Let N n N. To apply the Jacob denttes for determnants, we ntroduce three knds of determnants: D 2 = 1 ) 1Ψx, y) = τ n+1, 3.13) 1, n N+N2 +1 D 2 D 2, k, l = the determnant obtaned by strkng out the th row and th column of D 2, = the determnant obtaned by strkng out the th and th rows and the kth and lth columns of D 2, 3.14) 3.15) whch are the determnant, and a frst mnor and a second mnor of a correspondng matrx, respectvely. In terms of ths determnant notaton, we can easly obtan n N + N2 + 1 n N + N2, n N + N τ n = D 2, τ n 1 = D 2, n N + N n N + N 2, n N + N τ n x = D n N + N 2 τ n n N + N2 2, n N + N = D + 1 2, n N + N 2 2 τ n n N + N2 x = D 2. n N + N 2 Then t follows from these formulas that for each N n N, the 2DTM equaton 3.3) s equvalent to n N + N2 n N + N2 + 1 D 2 D 2 n N + N 2 n N + N n N + N 2 n N + N2 + 1 D 2 D 2 n N + N n N + N 2 n N + N2, n N + N = D 2 D 2. n N + N 2, n N + N These are exactly the Jacob denttes for determnants. Therefore, τ n defned by 3.5) solves the 2DTM equaton 3.3) when N n N. Satsfyng the boundary condtons: To satsfy two boundary condtons n 3.4), we requre that Φx, y) = N 1 +1 =1 u x)v y), Ψx, y) = N 2 +1 where all functons u, v, r and s are to be determned. 7 =1 r x)s y), 3.16)

8 where Let us frst compute τ 0 as follows: τ 0 = 1 ) k 1Φx, y) 1,k N ) N k = u x)v y) =1 N 1 +1 ) 1u ) k 1v = x) y) x =1 1 ) U N1 +1 = x u x) 1 1,k N ,k N 1 +1 = detu N1 +1V N1 +1) = detu N1 +1) detv N1 +1), 3.17) 1, N 1 +1 k 1 ), V N1 +1 = x v y) k 1 1,k N ) We can now take φ 1 x) = detu N1 +1), χ 1 y) = detv N1 +1). 3.19) For two gven functons φ 1 x) and χ 1 y), we fx N 1 functons among u and v, 1 N 1 + 1, and then the condtons n 3.19) present two lnear ordnary dfferental equatons on the unfxed functons, let us say u k and v k, respectvely. The exstence theory of lnear dfferental equatons guarantees that we have solutons for u k and v k. Therefore, the frst boundary condton n 3.4) can be satsfed. where Smlarly, t can be shown that 1 ) R N2 +1 = x r x) 1 τ N+1 = detr N2 +1) dets N2 +1), 3.20) 1, N 2 +1 By the same reason, we can acheve k 1 ), S N2 +1 = x s y) k 1 1,k N ) φ 2 x) = detr N2 +1), χ 2 y) = dets N2 +1). 3.22) Therefore, the second boundary condton n 3.4) can be satsfed, too. To conclude, τ n defned by 3.5) and 3.16) solves the 2DTM equaton 3.3) and satsfes the boundary condtons n 3.4). 8

9 3.2 Sem-nfnte lattce There are two sem-nfnte lattce equatons: one s wth < n K and the other s wth L n <, where K, L Z are arbtrarly fxed. Note that the 2DTM equaton s nvarant under the reflecton n n and the translaton n n+m wth any gven m Z. Thus we only need to consder the followng sem-nfnte 2DTM equaton wth one separated-varable boundary condton at n = 0: 2 τ n x τ n τ n τ n x = τ n+1τ n 1, 1 n <, τ 0 = φx)χy), where φ and χ are two arbtrarly gven functons of the ndcated varables. 3.23) In 3.5), settng M = N 1 0 and lettng N, we obtan the requred combned molecule b-drectonal Wronskan soluton: τ n = 1 ) 1Φx, y), 0 n M, 1, M n+1 τ M+1 = 1, τ M+2 = 0, τ M+3 = 0, τ M+4 = 1, 3.24) τ n = 1 ) 1Ψx, y), M + 5 n <, 1, n M 4 where Ψx, y) s arbtrary but Φx, y) s defned by Φx, y) = M+1 =1 u x)v y), 3.25) whch satsfes 1 x u x) 1 = φx), 1, M+1 k 1 x v y) k 1 = χy). 3.26) 1,k M+1 As shown before, there s no problem for exstence of those functons u s and v s. 3.3 Infnte lattce The nfnte 2DTM equaton s 2 τ n x τ n τ n τ n x = τ n+1τ n 1, < n <. 3.27) 9

10 Smlarly, by extendng two boundares 0 and N to and, respectvely, we can obtan a class of combned molecule b-drectonal Wronskan solutons: τ n = 1 ) 1Φx, y), < n M, 1, M n+1 τ M+1 = 1, τ M+2 = 0, τ M+3 = 0, τ M+4 = 1, 3.28) τ n = 1 ) 1Ψx, y), M + 5 n <, 1, n M 4 where M Z, Φx, y) and Ψx, y) are all arbtrary. 4 Concludng remarks The combned molecule b-drectonal Wronskan solutons have been presented for the fnte, sem-nfnte and nfnte blnear 2D Toda molecule 2DTM) equatons. In the frst two cases, separated-varable boundary condtons were mposed. The Jacob denttes for determnants are the key tool employed. The success s to combne two peces of molecule b-drectonal Wronskan solutons n formulatng the solutons. Between the two peces of molecule b-drectonal Wronskan solutons, we defned τ n as ether zero or one to move from one pece to the other pece followng the 2DTM equatons. It s known that the fnte 2DTM equaton 3.3) has double Wronskan solutons whch satsfy the boundary condtons 20: τ 0 = φx), τ N+1 = χy), 4.1) where φ and χ are arbtrary functons of the ndcated varables. Our constructon tells that there exst combned molecule b-drectonal Wronskan solutons to the fnte 2DTM equaton 3.3) whch satsfy the above boundary condtons. These solutons correspond to the case of χ 1 y) = 1 and φ 2 x) = 1 n our formulaton of solutons for 3.3). Smlarly, we can get combned molecule b-drectonal Wronskan solutons to the fnte 2DTM equaton 3.3) whch satsfy the followng boundary condtons: τ 0 = φx), τ N+1 = ψx)χy), 4.2) where φ, ψ and χ are arbtrary functons of the ndcated varables. Moreover, forcng one of the boundary condtons n 4.1) to be constant there s no problem for exstence of such double Wronskan solutons, based on the prevous dscusson on the separatedvarable boundary condtons usng the exstence theory of solutons of lnear dfferental equatons), the same dea n our constructon can be used to connect the correspondng 10

11 double Wronskan soluton wth a molecule b-drectonal Wronskan soluton to form a new soluton to the fnte, sem-nfnte or nfnte 2DTM equatons. But ths knd of solutons s not molecule. Acknowledgements: The work was supported n part by the State Admnstraton of Foregn Experts Affars of Chna, the Natonal Natural Scence Foundaton of Chna Nos , and ), Chunhu Plan of the Mnstry of Educaton of Chna, Zheang Innovaton Proect Grant No. T200905), the Natural Scence Foundaton of Shangha and the Shangha Leadng Academc Dscplne Proect No. J50101). References 1 R. Hrota, The Drect Method n Solton Theory, Cambrdge Unversty Press 2004). 2 J. Hetarnta, Hrota s blnear method and solton solutons, Phys. AUC 15 part 1) 2005) W. X. Ma, Complexton solutons to the Korteweg-de Vres equaton, Phys. Lett. A ) W. X. Ma, K. Maruno, Complexton solutons of the Toda lattce equaton, Physca A ) W. X. Ma, Solton, poston and negaton solutons to a Schrödnger self-consstent source equaton, J. Phys. Soc. Jpn ) W. X. Ma, Complexton solutons of the Korteweg-de Vres equaton wth self-consstent sources, Chaos, Soltons & Fractals ) W. X. Ma, Y. You, Solvng the Korteweg-de Vres equaton by ts blnear form: Wronskan solutons, Trans. Amer. Math. Soc ) C. X. L, W. X. Ma, X. J. Lu, Y. B. Zeng, Wronskan solutons of the Boussnesq equaton soltons, negatons, postons and complextons, Inverse Problems ) W. X. Ma, C. X. L, J. S. He, A second Wronskan formulaton of the Boussnesq equaton, Nonlnear Anal ) W. X. Ma, An applcaton of the Casoratan technque to the 2D Toda lattce equaton, Mod. Phys. Lett. B ) W. X. Ma, A. Abdelabbar, M. G. Asaad, Wronskan and Gramman solutons to a 3+1)-dmensonal generalzed KP equaton, Appl. Math. Comput ) W. X. Ma, T. W. Huang, Y. Zhang, A multple exp-functon method for nonlnear dfferental equatons and ts applcaton, Phys. Scrpta ) , 8 pp. 11

12 13 W. X. Ma, J.-H. Lee, A transformed ratonal functon method and exact solutons to the 3+1 dmensonal Jmb0-Mwa equaton, Chaos, Soltons and Fractals ) W. X. Ma, E. G. Fan, Lnear superposton prncple applyng to Hrota blnear equatons, Comput. Math. Appl ) W. X. Ma, Y. Zhang, Y. N. Tang, J. Y. Tu, prernt 2011). 16 R. Hrota, M. Ito, Resonance of soltons n one dmenson, J. Phys. Soc. Jpn ) A. N. Leznov, M. V. Savelev, Theory of group representatons and ntegraton of nonlnear systems x a,z z = expkx) a, Physca D ) T. Takag, Lecture n Algebra Tokyo, Kyortsu, 1965). 19 R. Hrota, A new form of Bäcklund transformatons and ts relaton to the nverse scatterng problem, Progr. Theoret. Phys ) R. Hrota, Y. Ohta, J. Satsuma, Wronskan structures of solutons for solton equatons, Progr. Theoret. Phys. Suppl. No )