Explicit solutions for a hierarchy of differential-difference equations

A new discrete isospectral problem and the corresponding hierarchy of nonlinear differential-difference equations are proposed. On the basis of the theory of algebraic curves, the continuous flow and discrete flow related to the hierarchy of differential-difference equations are straightened out using the Abel-Jacobi coordinates. The meromorphic function and the Baker-Akhiezer function are introduced on the hyperelliptic curve. Quasi-periodic solutions of the corresponding hierarchy of differential-difference equations are constructed with the help of the asymptotic properties and the algebro-geometric characters of the meromorphic function, the Baker-Akhiezer function and the hyperelliptic curve.

[1]  E. Belokolos,et al.  Algebro-geometric approach to nonlinear integrable equations , 1994 .

[2]  Boris Dubrovin,et al.  Theta functions and non-linear equations , 1981 .

[3]  Xianguo Geng,et al.  Algebro–geometric constructions of the discrete Ablowitz–Ladik flows and applications , 2003 .

[4]  Xianguo Geng,et al.  Decomposition of the Discrete Ablowitz–Ladik Hierarchy , 2007 .

[5]  Shunichi Tanaka,et al.  Periodic Multi-Soliton Solutions of Korteweg-de Vries Equation and Toda Lattice , 1976 .

[6]  Shunichi Tanaka,et al.  Analogue of Inverse Scattering Theory for the Discrete Hill's Equation and Exact Solutions for the Periodic Toda Lattice , 1976 .

[7]  Solomon J. Alber,et al.  On finite-zone solutions of relativistic Toda lattices , 1989 .

[8]  Vadym Vekslerchik,et al.  Finite-genus solutions for the Ablowitz-Ladik hierarchy , 1999, solv-int/9903004.

[9]  Xianguo Geng,et al.  Algebro-geometric solution of the 2+1 dimensional Burgers equation with a discrete variable , 2002 .

[10]  M. Ablowitz,et al.  The Periodic Cubic Schrõdinger Equation , 1981 .

[11]  A. Chowdhury,et al.  On the quasi-periodic solutions to the discrete nonlinear Schrodinger equation , 1987 .

[12]  G. Teschl,et al.  Algebro-Geometric Quasi-Periodic Finite-Gap Solutions of the Toda and Kac-van Moerbeke Hierarchies , 1995 .

[13]  Joe W. Harris,et al.  Principles of Algebraic Geometry , 1978 .

[14]  Helge Holden,et al.  Algebro-Geometric Solutions of the Baxter–Szegő Difference Equation , 2005 .

[15]  N. Bogolyubov,et al.  Discrete periodic problem for the modified nonlinear Korteweg-de Vries equation , 1981 .

[16]  Fritz Gesztesy,et al.  An Alternative Approach to Algebro-Geometric Solutions of the AKNS Hierarchy , 1997, solv-int/9707009.

[17]  C. David Levermore,et al.  Finite genus solutions to the Ablowitz‐Ladik equations , 2010 .

[18]  S. Novikov,et al.  Theory of Solitons: The Inverse Scattering Method , 1984 .

[19]  A. Roy Chowdhury,et al.  The quasiperiodic solutions to the discrete nonlinear Schrödinger equation , 1987 .

[20]  Helge Holden,et al.  Soliton Equations and Their Algebro-Geometric Solutions: The AKNS Hierarchy , 2003 .

[21]  D. Mumford.  Tata Lectures on Theta I , 1982 .