Hirota’s difference equations

A review of selected topics for Hirota’s bilinear difference equation (HBDE) is given. This famous three-dimensional difference equation is known to provide a canonical integrable discretization for most of the important types of soliton equations. Similar to continuous theory, HBDE is a member of an infinite hierarchy. The central point of our paper is a discrete version of the zero curvature condition explicitly written in the form of discrete Zakharov-Shabat equations for M-operators realized as difference or pseudo-difference operators. A unified approach to various types of M-operators and zero curvature representations is suggested. Different reductions of HBDE to two-dimensional equations are considered, with discrete counterparts of the KdV, sine-Gordon, Toda chain, relativistic Toda chain, and other examples.

[1]  G. Segal,et al.  Loop groups and equations of KdV type , 1985 .

[2]  N. Saitoh,et al.  Linearization of bilinear difference equations , 1987 .

[3]  Ryogo Hirota,et al.  Nonlinear Partial Difference Equations III; Discrete Sine-Gordon Equation , 1977 .

[4]  L. Faddeev,et al.  Quantum inverse scattering method on a spacetime lattice , 1992 .

[5]  A. Valleriani,et al.  DYNKIN TBA'S , 1992 .

[6]  P. Pearce,et al.  Conformal weights of RSOS lattice models and their fusion hierarchies , 1992 .

[7]  T. Miwa.  On Hirota's difference equations , 1982 .

[8]  M. Jimbo,et al.  TRANSFORMATION GROUPS FOR SOLITON EQUATIONS , 1982 .

[9]  Masaki Kashiwara,et al.  Operator Approach to the Kadomtsev-Petviashvili Equation —Transformation Groups for Soliton Equations III— , 1981 .

[10]  K. Takasaki,et al.  Toda lattice hierarchy , 1984 .

[11]  Sergei Petrovich Novikov,et al.  NON-LINEAR EQUATIONS OF KORTEWEG-DE VRIES TYPE, FINITE-ZONE LINEAR OPERATORS, AND ABELIAN VARIETIES , 1976 .

[12]  M. Jimbo,et al.  Solitons and Infinite Dimensional Lie Algebras , 1983 .

[13]  R. Hirota.  Discrete Two-Dimensional Toda Molecule Equation , 1987 .

[14]  Y. Suris.  A discrete-time relativistic Toda lattice , 1995, solv-int/9510007.

[15]  A. Zhedanov,et al.  Faces of Relativistic Toda Chain , 1997 .

[16]  A. Mironov,et al.  Towards unified theory of $2d$ gravity , 1992 .

[17]  A. Zamolodchikov.  On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering theories , 1991 .

[18]  Alexei Zhedanov,et al.  Discrete Darboux transformations, the discrete-time Toda lattice, and the Askey-Wilson polynomials , 1994 .

[19]  V. Matveev,et al.  Differential-difference evolution equations. II (Darboux transformation for the Toda lattice) , 1979 .

[20]  Ryogo Hirota,et al.  Nonlinear Partial Difference Equations. IV. Bäcklund Transformation for the Discrete-Time Toda Equation , 1978 .

[21]  FUNCTIONAL RELATIONS IN SOLVABLE LATTICE MODELS I: FUNCTIONAL RELATIONS AND REPRESENTATION THEORY , 1993, hep-th/9309137.

[22]  Dressing method, Darboux transformation and generalized restricted flows for the KdV hierarchy , 1993 .

[23]  Mikio Sato.  Soliton Equations as Dynamical Systems on Infinite Dimensional Grassmann Manifold , 1983 .

[24]  A. Orlov,et al.  Non-local integrable equations as reductions of the Toda hierarchy , 1991 .

[25]  Leon A. Takhtajan,et al.  Hamiltonian methods in the theory of solitons , 1987 .

[26]  Yasuhiro Ohta,et al.  Casorati and Discrete Gram Type Determinant Representations of Solutions to the Discrete KP Hierarchy , 1993 .

[27]  U. Pinkall,et al.  The discrete quantum pendulum , 1993 .

[28]  Ryogo Hirota,et al.  Nonlinear Partial Difference Equations. : I. A Difference Analogue of the Korteweg-de Vries Equation , 1977 .

[29]  Ryogo Hirota,et al.  Nonlinear Partial Difference Equations. V. Nonlinear Equations Reducible to Linear Equations , 1979 .

[30]  R. Hirota.  Discrete Analogue of a Generalized Toda Equation , 1981 .

[31]  V. Matveev,et al.  Darboux transformation and explicit solutions of the Kadomtcev-Petviaschvily equation, depending on functional parameters , 1979 .

[32]  M. Kontsevich.  Intersection theory on the moduli space of curves , 1991 .

[33]  Simon Ruijsenaars,et al.  Relativistic Toda systems , 1989 .