Fan En-Gui

Characteristic Numbers of Matrix Lie Algebras

A notion of characteristic number of matrix Lie algebras is defined, which is devoted to distinguishing various Lie algebras that are used to generate integrable couplings of soliton equations. That is, the exact classification of the matrix Lie algebras by using computational formulas is given. Here the characteristic numbers also describe the relations between soliton solutions of the stationary zero curvature equations expressed by various Lie algebras.

Explicit N-Fold Darboux Transformations and Soliton Solutions for Nonlinear Derivative Schrodinger Equations

An explicit N-fold Darboux transformation for a coupled of derivative nonlinear Schrodinger equations is constructed with the help of a gauge transformation of spectral problems. As a reduction, the Darboux transformation for well-known Gerdjikov–Ivanov equation is further obtained, from which a general form of N-soliton solutions for Gerdjikov–Ivanov equation is given.

New Integrable Couplings of Generalized Kaup—Newell Hierarchy and Its Hamiltonian Structures

A new isospectral problem is firstly presented, then we derive integrable system of soliton hierarchy. Also we obtain new integrable couplings of the generalized Kaup?Newell soliton equations hierarchy and its Hamiltonian structures by using Tu scheme and the quadratic-form identity. The method can be generalized to other soliton hierarchy.

Complexiton solutions of the (2+1)-dimensional dispersive long wave equation

In this pager a pure algebraic method implemented in a computer algebraic system, named multiple Riccati equations rational expansion method, is presented to construct a novel class of complexiton solutions to integrable equations and nonintegrable equations. By solving the (2+1)-dimensional dispersive long wave equation, it obtains many new types of complexiton solutions such as various combination of trigonometric periodic and hyperbolic function solutions, various combination of trigonometric periodic and rational function solutions, various combination of hyperbolic and rational function solutions, etc.

A hierarchy of nonlinear evolution equations, its bi-Hamiltonian structure, and finite-dimensional integrable systems

An isospectral problem and the associated hierarchy of nonlinear evolution equations is presented. As a reduction, a new generalized nonlinear Schrodinger equation is obtained. It is shown that the hierarchy possesses bi-Hamiltonian structure and is integrable in Liouville sense. Moreover, the eigenvalue problem can be nonlinearized as a finite-dimensional completely integrable system under the Bargmann constraint between the potentials and the eigenvalues.

Isospectral and Nonisospectral Lattice Hierarchies Associated with a Discrete Spectral Problem and Their Infinitely Many Conservation Laws

In this paper, based on the discrete zero curvature representation, isospectral and nonisospectral lattice hierarchies are proposed. By means of solving corresponding discrete spectral equations, we demonstrate the existence of infinitely many conservation laws for this two hierarchies and obtain the formulae of the corresponding conserved densities and associated fluxes.

A new completely integrable Liouville's system, its Lax representation and Bi-Hamiltonian structure

A new isospectral problem and the corresponding hierarchy of nonlinear evolution equations is presented. As a reduction, the well-known MKdV equation is obtained. It is shown that the hierarchy of equations is integrable in Liouvilles sense and possesses Bi-Hamiltonian structure. Under the constraint between the potentials and eigenfunctions, the eigenvalue problem can be nonlinearized as a finite dimensional completely integrable system.

Quasi-Hamiltonian Structure Associated with an Integrable Coupling System

Starting from a spectral problem, a corresponding soliton hierarchy is proposed, and we construct an integrable coupling system with five dependent variables for the hierarchy by using a class of semi-direct sums of Lie algebras. Moreover, it is shown that the coupling system possesses quasi-Hamiltionian structures, and that infinitely many conserved quantities are obtained.

Hamiltonian System and Infinite Conservation Laws Associated with a New Discrete Spectral Problem

Starting from a new discrete spectral problem, the corresponding hierarchy of nonlinear lattice equations is proposed. It is shown that the lattice soliton hierarchy possesses the bi-Hamiltonian structures and infinitely many common commuting conserved functions. Further, infinite conservation laws of the hierarchy are presented.

Pfaffianization of the variable-coefficient Kadomtsev?Petviashvili equation

This paper constructs more general exact solutions than N-soliton solution and Wronskian solution for variablecoefficient Kadomtsev–Petviashvili (KP) equation. By usingthe Hirota method and Pfaffian technique, it finds the Grammian determinant-type solution for the variable-coefficient KP equation (VCKP), the Wronski-type Pfaffian solution and the Gram-type Pfaffian solutions for the Pfaffianized V CKP equation.