Explicit Solutions and Conservation Laws for a New Integrable Lattice Hierarchy

Abstract

An integrable lattice hierarchy is derived on the basis of a new matrix spectral problem. Then, some properties of this hierarchy are shown, such as the Liouville integrability, the bi-Hamiltonian structure, and infinitely many conservation laws. After that, the Darboux transformation of the first integrable lattice equation in this hierarchy is constructed. Eventually, the explicitly exact solutions of the integrable lattice equation are investigated via graphs.

Algebro-geometric Constructions of a Hierarchy of Integrable Semi-discrete Equations

“…Though using algebro-geometric method to solve soliton equations is systematically provided in the case of 2 × 2 matrix spectral problem [21][22][23][24], inferring the solutions is challenging because of concerning the knowledge of the discrete variables and algebraic curve. For this paper, we will discuss the following integrable hierarchy of semi-discrete equations which has been studied explicit solutions by Darboux transformation [25], where u n and v n are two potentials and t is the time variable. As an application, the main aim of this paper is to use algebraic geometric methods to solve the hierarchy (1).…”

Liouville integrability and explicit solutions for a Frobenius‐coupled integrable lattice hierarchy

“…In Yang et al, 14 a new integrable lattice hierarchy has been derived. The first nonlinear equation of this hierarchy is $$$$ \left\{\begin{array}{l}{u}_{n,t}={u}_n\left({u}_n{v}_{n-1}-{u}_{n+1}{v}_n\right),\\ {}{v}_{n,t}={v}_n\left({u}_n-{u}_{n+1}\right)+{v}_n\left({u}_n{v}_{n-1}-{u}_{n+1}{v}_n\right).\end{array}\right.$$…”

“…In Yang et al, 14 a new integrable lattice hierarchy has been derived. The first nonlinear equation of this hierarchy is…”