Yufeng Zhang

Bäcklund transformation and solutions of a (3+1)-dimensional nonlinear evolution equation

In this paper, Bell-polynomial method is used to study the integrability of a (3+1)-dimensional nonlinear evolution equation. We get the bilinear representation and Backlund transformation of the equation. The exact solutions are obtained by means of linear superposition principle and homoclinic test approach.

Two (2+1)-dimensional hierarchies of evolution equations and their Hamiltonian structures

Two kinds of appropriate isospectral problems are introduced by using a Lie algebra. With the help of the TAH scheme, we generate two new (2+1)-dimensional hierarchies of evolution equations, whose Hamiltonian structures are derived from the trace identity proposed by Tu Guizhang, Andrushkiw R.I. and Huang X.C. Finally, we propose some problems worth thinking about.

HAMILTONIAN STRUCTURES OF TWO INTEGRABLE COUPLINGS OF THE MODIFIED AKNS HIERARCHY

The extension of a three-dimensional Lie algebra into two higher-dimensional ones is used to deduce two new integrable couplings of the m-AKNS hierarchy. The Hamiltonian structures of the two integrable couplings are obtained, respectively. Specially, the complex Hamiltonian structure of the second integrable couplings is given.

An integrable hierarchy and Darboux transformations, bilinear Bäcklund transformations of a reduced equation

A new integrable hierarchy of evolution equations is obtained by making use of a Lie algebra and Tu-Ma scheme, from which a new generalized Broer-Kaup (gBK) equation is produced. Then two kinds of Darboux transformations, the bilinear presentation, the bilinear Backlund transformation and the new Lax pair of the gBK equation are generated, respectively, by employing the Bell polynomials.

A generalization of the simplest equation method and its application to (3+1)-dimensional KP equation and generalized Fisher equation

In this paper, the simplest equation method is used to construct exact traveling solutions of the -dimensional KP equation and generalized Fisher equation. We summarize the main steps of the simplest equation method. The Bernoulli and Riccati equation are used as simplest equations. This method is straightforward and concise, and it can be applied to other nonlinear partial differential equations.

A Few Integrable Couplings of Some Integrable Systems and (

Two high-dimensional Lie algebras are presented for which four ()-dimensional expanding integrable couplings of the D-AKNS hierarchy are obtained by using the Tu scheme; one of them is a united integrable coupling model of the D-AKNS hierarchy and the AKNS hierarchy. Then ()-dimensional DS hierarchy is derived by using the TAH scheme; in particular, the integrable couplings of the DS hierarchy are obtained.

2+1

In this paper, the Hirota’s bilinear form is employed to investigate the lump, periodic lump and interaction lump stripe solutions of the (2+1)-dimensional B-type Kadomtsev–Petviashvili (BKP) equat...

)-Dimensional Integrable Hierarchies

Abstract In this letter, the linear superposition principle is used to discuss the ( 3 + 1 ) -dimensional Boiti–Leon–Manna–Pempinelli equation with bilinear derivatives. As a result, we obtain new resonant soliton and complexiton solutions by discussing two different cases involved the parameters. These solutions are a class of N -wave solutions of linear combinations of exponential traveling waves.

Lump, periodic lump and interaction lump stripe solutions to the (2+1)-dimensional B-type Kadomtsev–Petviashvili equation

In the paper, we introduce the Lie algebras and the commutator equations to rewrite the Tu-d scheme for generating discrete integrable systems regularly. By the approach the various loop algebras of the Lie algebra A1 are defined so that the well-known Toda hierarchy and a novel discrete integrable system are obtained, respectively. A reduction of the later hierarchy is just right the famous Ablowitz–Ladik hierarchy. Finally, via two different enlarging Lie algebras of the Lie algebra A1, we derive two resulting differential-difference integrable couplings of the Toda hierarchy, of course, they are all various discrete expanding integrable models of the Toda hierarchy. When the introduced spectral matrices are higher degrees, the way presented in the paper is more convenient to generate discrete integrable equations than the Tu-d scheme by using the software Maple.

Resonant soliton and complexiton solutions for (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation

With the help of some reductions of the self-dual Yang Mills(briefly written as sdYM) equations, we introduce a Lax pair whose compatibility condition leads to a set of(2 + 1)-dimensional equations. Its first reduction gives rise to a generalized variable-coefficient Burgers equation with a forced term. Furthermore, the Burgers equation again reduces to a forced Burgers equation with constant coefficients, the standard Burgers equation, the heat equation,the Fisher equation, and the Huxley equation, respectively. The second reduction generates a few new(2 + 1)-dimensional nonlinear integrable systems, in particular, obtains a kind of(2 + 1)-dimensional integrable couplings of a new(2 + 1)-dimensional integrable nonlinear equation.