Two kinds of discrete integrable hierarchies of evolution equations and some algebraic-geometric solutions

Yufeng Zhang,Xiangzhi Zhang

doi:10.1186/s13662-017-1124-3

Yufeng Zhang, Xiangzhi Zhang

Open Access

https://doi.org/10.1186/s13662-017-1124-3

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Abstract

With the help of a loop algebra we first present a (1+1)-dimensional discrete integrable hierarchy with a Hamiltonian structure and generate a (2+1)-dimensional discrete integrable hierarchy, respectively. Then we obtain a new differential-difference integrable system with three-potential functions, whose algebraic-geometric solution is derived from the theory of algebraic curves, where we construct the new elliptic coordinates to straighten out the continuous and discrete flows by introducing the Abel maps as well as the Riemann-Jacobi inversion theorem.

Highlights

Integrable nonlinear lattice systems have important applications and rich mathematical structures in mathematical physics, statistical physics, and quantum physics
It is interesting how to generate integrable nonlinear lattice systems associated with mathematics and physics by various methods
Suris [ ] once derived a new lattice equation related to the relativistic Toda lattice hierarchy via a highly non-trivial Bäcklund transformation

Summary

Introduction

Integrable nonlinear lattice systems have important applications and rich mathematical structures in mathematical physics, statistical physics, and quantum physics. Tu Guizhang [ ] applied a compatibility condition of spectral problems and some Lie algebras to propose a powerful method for generating integrable differential-difference hierarchies and the corresponding Hamiltonian structures. Geng et al [ – ] further improved the method so as to conveniently investigate algebraic-geometric solutions of differential and difference equations by introducing a new matrix consisting of fundamental solutions of spectral problems which satisfy discrete zero-curvature equations. The ( + )-dimensional discrete integrable hierarchy obtained in the paper can be reduced to a new ( + )-dimensional integrable nonlinear difference system with three-potential functions, and the ( + )-dimensional discrete integrable hierarchy presented in the paper is obtained by a non-isospectral Lax pair based on the loop algebra and a zero-curvature equation.

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