Bilinear Equations and B"acklund Transformation for Generalized Ultradiscrete Soliton Solution

Bilinea r Equati ons and B¨ ac klund T ransfor mati on for Gener alized Ultra discrete Solit on Sol ution Hidetomo Nagai 1 , Daisuk e T ak ahashi 2 F acult y of Scienc e and Engineering, W aseda Univ ersit y , 3-4 -1, Okubo, Shinjuku-ku, T oky o 169-8555, Japan Abstract Ultradiscrete soliton equations and B¨ ac klund transformation for a general ized soliton solu- tion are presented. The equations include the ultradiscrete KdV eq uation or t he ultradiscrete T o da equation in a sp ecial case. W e also exp ress the solution by the u ltradiscrete p ermanent, whic h is defined b y ultradiscretizi ng the signature-free determinant, that is, the permanent. Moreo v er, we discuss a relation b etw ee n B¨ acklund transformations for d iscrete an d ultradis- crete KdV equations. 1 In tro duction Soliton equation ha s explicit N -soliton solutions a nd an infinite num ber of conserved quantities generally . In the b eginning of the development o f the soliton theo ry , co nt inuous or semi-dis c rete soliton equa tions w ere studied mainly . F or example, the Kor teweg-de V ries (KdV) e quation is a contin uous soliton equation of PDE type, and the T o da equation is a semi-discre te so lito n equation with contin uous a nd dis crete indep enden t v ar iables. There are tw o t yp es o f soliton so lutions to the bilinear equatio ns derived from these equations. One is expres sed b y a sum o f a finite num ber of exp onential functions, whic h was firs t prop osed by Hirota[1, 2]. W e ca ll this type of expr ession Type I. The other is expr essed b y W ro nski determinant [3, 4]. W e call this type of expr ession Type II. After the discov ery of v a rious contin uo us o r semi-discrete soliton equations, discrete soliton equations o f which indep e nden t v ar iables are a ll discre te w ere pro p os ed[5, 6]. Discrete soliton equation is a ls o transfor med into the bilinear equation a nd has multi-soliton solutions. It has also t wo t y pes of express ions, Type I and I I, where the determina n t of Type I I is generally the Casorati determinant for discr ete solito n equations . Discretization pro ces s is completed if dep endent a nd indep endent v ariables ar e all discretized. In the 1 990s, T o kihiro et a l. prop osed the ultr a discretization method to discretize dep endent v ariables [7]. The key formula in the metho d is lim ε → +0 ε log( e a/ε + e b/ε ) = max( a, b ) . (1) Usual addition, m ultiplicatio n and division f or the rea l v alues in the or iginal discrete equation are r eplaced with max op eration, addition and subtraction resp ectively by this metho d. Due to these replacements, dep endent v ariables can b e discrete in the ultradiscrete equation if we use appropria te constan ts and initial v alue s . Ma n y ultradiscr ete soliton equations or cellular automata hav e b een pro po sed and the integrabilit y is shown even for the dig itized e quations[8, 9]. How ever, the o per ation in the ultradiscrete e quation c o rresp onding to the subtraction in the discrete e q uation is not well-defined. Thu s we c a n not ultradiscretize a discrete equation auto- matically . This obstructio n is called a ‘nega tiv e problem’[10, 11]. Th us the a b ove so liton solution 1 e-mail hdnagai@aoni.w aseda.jp 2 e-mail daisuke t@w aseda.jp 1 of T yp e I I c a n no t b e ultra discretized dir ectly since the antisymmetry is crucia l for the determi- nant. O n the o ther hand, the so lution of Type I can b e ultradiscretized genera lly choo sing the appropria te parameters included in the solution. The imbalance b etw een the tw o types of expression for the ultradiscr ete soliton solution is partially solved. One of the authors (T ak ahashi) and Hirota prop osed the ultradiscrete ana logue of determinant solution for the ultradiscrete KdV (uKdV) equation[1 0]. One of the authors (Na- gai) pro pos ed the simila r type of so lution for the ultra discrete T oda (uT oda ) equation[1 1]. This analogue is calle d an ‘ultradis c r ete p ermanent’ (UP) defined by max[ a ij ] ≡ max π X 1 ≤ i ≤ N a iπ i , (2) where [ a ij ] denotes an arbitr a ry N × N ma trix and π = { π 1 , π 2 , . . . , π N } a n a rbitrary p ermutation of 1, 2, . . . , N . The N -solito n solution to the ultra discrete bilinea r equation of uKdV eq uation is expressed by the following tw o forms , f n i = max µ j =0 , 1  X 1 ≤ j ≤ N µ j s j ( n, i ) − X 1 ≤ j 1, − p N +1 − q N +1 is equiv alent to − p N +1 + q N +1 − 2 by the dis p er s ion relatio n. In the case of p N +1 ≤ 1, we can prove ˜ f n +1 i − 1 + ˜ g n i − p N +1 + q N +1 − 2 < ˜ f n +1 i − 1 + ˜ g n i − p N +1 − q N +1 ≤ ˜ f n − 1 i − 1 + ˜ g n +2 i − p N +1 + q N +1 (60) after the pr o o f in the previous section. Thus (52) and (59) are equiv alent each other. 5 Concluding Remarks In this article, we consider the g eneralized so liton solution of tw o types a nd give the ultradiscr ete soliton equatio ns and the B¨ acklund transforma tion. The equa tions ar e equiv alent to the uKdV and the uT o da equation in a sp ecial case. The B¨ a c klund tra nsformation holds under the conditio n (28). The disp ersion rela tions (10) and (12) satisfy the co ndition. This means that the B¨ acklund transformatio ns for the uKdV a nd the uT o da equations a re a ls o obtained as a s p ecia l ca se. F ur- thermore, we discuss the ultradiscretiza tio n of the B¨ acklund transforma tions for the discrete KdV equation assuming that determinant is replace d with UP . Although determina n t c a nnot be ultra- discretized directly in general, their counterpart gives the B¨ acklund tra ns formations for the uKdV equation. References [1] R. Hirota: Exact Solution o f the Mo dified Korteweg-de V ries Equation for Multiple Co llisions of Solitons, J. P h ys. So c. J a pan, 33 (1972) 145 6 –1458 . [2] R. Hir ota: Exact Solution of the Sine-Gor don Equation for Multiple Collisions o f Solitons, J . Phys. So c. Japan, 33 (197 2) 14 59–14 6 3. [3] J. Satsuma: A W ro nskian Represent ation of N -Soliton solutions of Nonlinear Ev olution Equa- tions. J. Phys. So c. Ja pa n, 46 (1979 ) 359–3 6 0. 10 [4] N. C. F reeman and J. J . C. Nimmo: Soliton solutions of the Korteweg-de V reis and K adomtsev- Petviash vili equations : the W r o nskian technique, Phys. Lett. 95A (198 3 ) 1. [5] R. Hirota: Nonlinear Partial Difference Equations . I. A Difference Analogue of the Ko rteweg- de V ries Equatio n, J. Phys. So c. Japan, 43 (197 7) 14 24–14 33. [6] R. Hirota : Nonlinear Partial Difference Equations. IV. B¨ acklund T r ansformation for the Discrete-Time T oda Equa tion, J. Phys. So c . J apan, 45 (197 8) 321– 332. [7] T. T okihir o, D. T ak ahashi, J. Matsuk idaira and J. Satsuma: F rom Soliton E quations to Int egrable Cellular Automata thro ugh a Limiting Pro cedure, P hys. Rev . Lett. 76 (19 96) 3247– 3250. [8] J. Matsukidaira, J. Satsuma, D. T ak ahashi, T. T okihiro and M. T orii: T o da-type cellular automaton and its N -solito n solution, Phys. Lett. A 22 5 (199 7) 287 –295. [9] S. Tsujimoto and R. Hirota: Ultradiscr ete K dV E q uation, J. Ph ys. So c. Japan, 67 (1998) 1809– 1810. [10] D. T ak aha shi, R. Hirota: Ultra dis crete Soliton Solutio n of Permanen t Type, J. Phys. Soc. Japan, 76 (2 007) 104 007–1 04012 . [11] H. Nagai, A New Expre s sion of Soliton Solution to the Ultradiscrete T o da Eq ua tion, J. P hys. A: Ma th. Theo r. 41 (200 8), 23520 4(12pp). [12] N. Shinzaw a and R. Hirota: The B¨ acklund tra ns formation equations for the ultradiscre te K P equation, J. Phys. A: Math. Gen. 36 (200 3) 44 67–46 75. [13] S. Iso jima, S. Kub o, M. Murata and J. Satsuma: Discrete and ultra discrete B¨ a c klund trans- formation for KdV equation, J. P h ys. A :Math. Theo r. 41 (200 8) 0252 0 5(8pp). 11