Darboux transformation,infinite conservation laws,and exact solutions for the nonlocal Hirota equation with variable coefficients

This article presents the construction of a nonlocal Hirota equation with variable coefficients and its Darboux transformation.Using zero-seed solutions,1-soliton and 2-soliton solutions of the equation are constructed through the Darboux transformation,along with the expression for N-soliton solutions.Influence of coefficients that are taken as a function of time instead of a constant,i.e.,coefficient functionδ(t),on the solutions is investigated by choosing the coefficient functionδ(t),and the dynamics of the solutions are analyzed.This article utilizes the Lax pair to construct infinite conservation laws and extends it to nonlocal equations.The study of infinite conservation laws for nonlocal equations holds significant implications for the integrability of nonlocal equations.

A nonlinear discrete integrable coupling system and its infinite conservation laws

We construct a nonlinear integrable coupling of discrete soliton hierarchy, and establish the infinite conservation laws （CLs） for the nonlinear integrable coupling of the lattice hierarchy. As an explicit application of the method proposed in the paper, the infinite conservation laws of the nonlinear integrable coupling of the Volterra lattice hierarchy are presented.

Two integrable generalizations of WKI and FL equations: Positive and negative flows, and conservation laws

With the aid of Lenard recursion equations, an integrable hierarchy of nonlinear evolution equations associated with a 2 × 2 matrix spectral problem is proposed, in which the first nontrivial member in the positive flows can be reduced to a new generalization of the Wadati–Konno–Ichikawa(WKI) equation. Further, a new generalization of the Fokas–Lenells(FL) equation is derived from the negative flows. Resorting to these two Lax pairs and Riccati-type equations, the infinite conservation laws of these two corresponding equations are obtained.

Backlund transformation,infinite number of conservation laws and fission properties of an integro-differential model for ocean internal solitary waves

This paper presents an analytical investigation of the propagation of internal solitary waves in the ocean of finite depth.Using the multi-scale analysis and reduced perturbation methods,the integro-differential equation is derived,which is called the intermediate long wave(ILW)equation and can describe the amplitude of internal solitary waves.It can reduce to the Benjamin-Ono equation in the deep-water limit,and to the KdV equation in the shallow-water limit.Little attention has been paid to the features of integro-differential equations,especially for their conservation laws.Here,based on Hirota bilinear method,Backlund transformations in bilinear form of ILW equation are derived and infinite number of conservation laws are given.Finally,we analyze the fission phenomenon of internal solitary waves theoretically and verify it through numerical simulation.All of these have potential value for the further research on ocean internal solitary waves.

An extension of integrable equations related to AKNS and WKI spectral problems and their reductions

A novel hierarchy of integrable nonlinear evolution equations related to the combined Ablowitz–Kaup–Newell–Segur（AKNS） and Wadati–Konno–Ichikawa（WKI） spectral problems is proposed,from which the Lax pair for a corresponding negative flow and its infinite many conservation laws are obtained.Furthermore,a reduction of this hierarchy is discussed,by which a generalized sinh-Gordon equation is derived on the basis of its negative flow.