Weimin Xue

Negaton and positon solutions of the soliton equation with self-consistent sources

The Korteweg–de Vries (KdV) equation with self-consistent sources (KdVES) is used as a model to illustrate this method. We present a generalized binary Darboux transformation (GBDT) with an arbitrary time-dependent function for the KdVES as well as the formula for N-times repeated GBDT. This GBDT provides non-auto-Backlund transformation between two KdV equations with different degrees of sources and enables us to construct more general solutions with N arbitrary t-dependent functions. By taking the special t-function, we obtain multisoliton, multipositon, multinegaton, multisoliton–positon, multinegaton–positon and multisoliton–negaton solutions of the KdVES.

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.

Soliton solutions to the Jimbo–Miwa equations and the Fordy–Gibbons–Jimbo–Miwa equation

Abstract This Letter considers two Jimbo–Miwa equations and the Fordy–Gibbons–Jimbo–Miwa equation in their bilinear form. Three-soliton solutions to these equations are explicitly derived by the Hirota method with the assistance of Mathematica.

A multilevel successive iteration method for nonlinear elliptic problems

In this paper, a multilevel successive iteration method for solving nonlinear elliptic problems is proposed by combining a multilevel linearization technique and the cascadic multigrid approach. The error analysis and the complexity analysis for the proposed method are carried out based on the two-grid theory and its multilevel extension. A superconvergence result for the multilevel linearization algorithm is established, which, besides being interesting for its own sake, enables us to obtain the error estimates for the multilevel successive iteration method. The optimal complexity is established for nonlinear elliptic problems in 2-D provided that the number of grid levels is fixed.

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.

Infinitely many conservation laws of two Blaszak–Marciniak three-field lattice hierarchies

Abstract In this Letter, by means of the matrix Lax representations for two Blaszak–Marciniak three-field lattice hierarchies, we demonstrate the existence of infinitely many conservation laws for the two lattice hierarchies and give the corresponding conserved density and the associated flux formulaically.

Infinitely many conservation laws and integrable discretizations for some lattice soliton equations

In this paper, by means of the Lax representations, we demonstrate the existence of infinitely many conservation laws for the general Toda-type lattice equation, the relativistic Volterra lattice equation, the Suris lattice equation and some other lattice equations. The conserved density and the associated flux are given formulaically. We also give an integrable discretization for a lattice equation with n dependent coefficients.

New Lax Representation and Integrable Discretization of the Relativistic Volterra Lattice

From a proper 2×2 discrete isospectral problem, the relativistic Volterra lattice introduced by Suris and Ragnisco is rederived. So, a new Lax matrix for the relativistic Volterra lattice is given....

The bi-Hamiltonian structures of some new Lax integrable hierarchies associated with 3 × 3 matrix spectral problems

Abstract Some new Lax integrable hierarchies associated with 3 × 3 matrix spectral problems and their bi-Hamiltonian structures are obtained.

Convergence of Finite Element Approximations and Multilevel Linearization for Ginzburg–Landau Model of d-Wave Superconductors

In this paper, we consider the finite element approximations of a recently proposed Ginzburg–Landau-type model for d-wave superconductors. In contrast to the conventional Ginzburg–Landau model the scalar complex valued order-parameter is replaced by a multicomponent complex order-parameter and the free energy is modified according to the d-wave paring symmetry. Convergence and optimal error estimates and some superconvergent estimates for the derivatives are derived. Furthermore, we propose a multilevel linearization procedure to solve the nonlinear systems. It is proved that the optimal error estimates and superconvergence for the derivatives are preserved by the multilevel linearization algorithm.