Zuo-nong Zhu

Some new results on the Blaszak–Marciniak lattice: Bäcklund transformation and nonlinear superposition formula

The so-called Blaszak–Marciniak lattice is considered. By the dependent variable transformation, the Blaszak–Marciniak lattice is transformed into a quart-linear form. By introducing an auxiliary variable, we further transform it into the bilinear form. A corresponding Backlund transformation is obtained. Furthermore, a nonlinear superposition formula is proved rigorously. As an application of the obtained results, soliton solutions are derived.

A nonisospectral extension of the Volterra hierarchy to 2+1 dimensions

We give a new nonisospectral generalization of the Volterra lattice equation to 2+1 dimensions. We use this to construct a new nonisospectral lattice hierarchy in 2+1 dimensions, along with its underlying linear problem. Reductions yield a variety of new integrable hierarchies, including generalizations of known discrete Painleve hierarchies, all along with their corresponding linear problems. This represents an extension of previously developed techniques to the discrete case.

New 2+1 dimensional nonisospectral Toda lattice hierarchy

We give a new 2+1 dimensional nonisospectral generalization of the Toda lattice hierarchy. Reductions yield a variety of new integrable hierarchies along with their underlying linear problems, including new 1+1 dimensional differential-delay hierarchies (nonisospectral and isospectral), new ordinary differential-delay hierarchies, and new discrete Painleve hierarchies. We also show that a reduction in components yields our previously obtained 2+1 dimensional nonisospectral Volterra lattice hierarchy.

Sech-polynomial travelling solitary-wave solutions of odd-order generalized KdV equations

Abstract A (2 m + 1)th-order generalized KdV equation is considered, where m is a positive integer. Four new explicit travelling solitary-wave solutions are obtained for the case m = 4. The case of arbitrary m ≥ 3 is considered; it is shown that, subject to certain restrictions on the coefficients in the equation, there are always at least two sech-polynomial type solutions.

New Matrix Lax Representation for a Blaszak–Marciniak Four-Field Lattice Hierarchy and Its Infinitely Many Conservation Laws

In this article, by means of considering a 4×4 discrete isospectral problem, and constructing a proper continuous time evolution equation, and using discrete zero curvature equation, a Blaszak–Marciniak four-field lattice hierarchy is re-derived. Thus a new matrix Lax representation for the hierarchy is obtained. From the new matrix Lax representation, we demonstrate the existence of infinitely many conservation laws for the lattice hierarchy and give the corresponding conserved densities and the associated fluxes formulaically. Thus its integrability is further confirmed.

A BACKLUND TRANSFORMATION AND NONLINEAR SUPERPOSITION FORMULA FOR THE BELOV-CHALTIKIAN LATTICE

The so-called Belov-Chaltikian lattice is considered. By the dependent variable transformation, the Belov-Chaltikian lattice is transformed into a trilinear form. By introducing an auxiliary variable, we further transform it into the bilinear form. A corresponding Backlund transformation for it is obtained. Furthermore, a nonlinear superposition formula is proved rigorously. As an application of the obtained results, soliton solutions are derived.

Infinitely many conservation laws for the Blaszak–Marciniak four-field integrable lattice hierarchy

Abstract In this Letter, by means of new matrix Lax representation for the Blaszak–Marciniak four-field integrable lattice hierarchy, we demonstrate the existence of infinitely many conservation laws for the lattice hierarchy and give the corresponding conserved density and the associated flux formulaically. So, its integrability is further confirmed.

Non-isospectral lattice hierarchies in 2 + 1 dimensions and generalized discrete Painleve hierarchies

Abstract In a recent paper we introduced a new 2 + 1-dimensional non-isospectral extension of the Volterra lattice hierarchy, along with its corresponding hierarchy of underlying linear problems. Here we consider reductions of this lattice hierarchy to hierarchies of discrete equations, which we obtain once again along with their hierarchy of underlying linear problems. We obtain a generalized discrete first Painlevé hierarchy which includes as special cases, after further summation, both the standard discrete first Painlevé hierarchy and a new extended version of the discrete thirty-fourth Painlevé hierarchy.

Nonisospectral negative Volterra flows and mixed Volterra flows: Lax pairs, infinitely many conservation laws and integrable time discretization

In this paper, by means of the discrete zero curvature representation, nonisospectral negative Volterra flows and mixed Volterra flows are proposed. By means of solving corresponding discrete spectral equations, we demonstrate the existence of infinitely many conservation laws for the two nonisospectral flows and obtain the formulae of the corresponding conserved densities and associated fluxes. Integrable time discretizations for several isospectral equations of the two flows are also presented.

On the gauge equivalent structure of the modified nonlinear Schrödinger equation

Abstract By using new Lax pairs, we prove that the continuous limits of the gauge equivalent structures of the discrete nonlinear Schrodinger equation for κ=1 and −1 are exactly the classical ones of the nonlinear Schrodinger equation for κ=1 and −1 respectively.