Da-jun Zhang

The N-Soliton Solutions for the Modified KdV Equation with Self-Consistent Sources

The N -soliton solutions for the modified KdV equation with self-consistent sources are obtained through Hirotas method and Wronskian technique respectively. Some novel determinantal identities are presented to treat the nonlinear term in the time evolution and finish the Wronskian verifications. The uniformity of these two kinds of N -soliton solutions is proved.

Integrable properties of the general coupled nonlinear Schrödinger equations

In this paper, a general integrable coupled nonlinear Schrodinger system is investigated. In this system, the coefficients of the self-phase modulation, cross-phase modulation, and four-wave mixing terms are more general while still maintaining integrability. The N-soliton solutions in this system are obtained by the Riemann–Hilbert method. The collision dynamics between two solitons is also analyzed. It is shown that this collision exhibits some new phenomena (such as soliton reflection) which have not been seen before in integrable systems. In addition, the recursion operator and conservation laws for this system are also derived.

The conservation laws of some discrete soliton systems

Abstract A systematic approach to constructing an infinite number of conservation laws for discrete soliton systems is proposed, and three examples are given.

Hamiltonian structure of discrete soliton systems

We describe an approach for investigating the Hamiltonian structures of the lattice isospectral evolution equations associated with a general discrete spectral problem. By using the so-called implicit representations of the isospectral flows, we demonstrate the existence of the recursion operator L, which is a strong and hereditary symmetry of the flows. It is then proven that every equation in the isospectral hierarchy possesses the Hamiltonian structure if L has a skew-symmetric factorization and the first equation (ut = K(0)) in the hierarchy satisfies some simple condition. We obtain related properties, such as the implectic-symplectic factorization of L, the Liouville complete integrability and the multi-Hamiltonian structures of the isospectral hierarchy. Four examples are given.

The Multisoliton Solutions of the KP Equation with Self-consistent Sources

The KP equation with self-consistent sources is derived through the linear problem of the KP system. The multisoliton solutions for the KP equation with self-consistent sources are obtained by using Hirota method and Wronskian technique. The coincidence of these solutions is shown by direct computation.

The N-soliton solutions of the sine-Gordon equation with self-consistent sources

The hierarchy of the sine-Gordon equation with self-consistent sources is derived by using the eigenfunctions of recursion operator. The bilinear form of the sine-Gordon equation with self-consistent sources is given and the N-soliton solutions are obtained through Hirota method and Wronskian technique, respectively. Some novel determinantal identities are presented to treat the nonlinear term in the time evolution and finish the Wronskian verifications.

The N-soliton solutions of some soliton equations with self-consistent sources

The hierarchy of the mKdV–sine-Gordon equation with self-consistent sources is derived. The N -soliton solutions of the mKdV–sine-Gordon equation with N self-consistent sources are obtained through Hirota method and Wronskian technique, respectively, from which we also reduce solutions for some soliton equations with self-consistent sources, such as one-dimensional atomic grid equation with self-consistent sources, the sine-Gordon equation with self-consistent sources, the mKdVequation with self-consistent sources and the KdVequation with self-consistent sources (KdVESCS). Finally, the mixed rational-soliton solutions in Wronskian form for the KdVESCS are discussed. 2003 Elsevier Science Ltd. All rights reserved.

Negatons, positons, rational-like solutions and conservation laws of the Korteweg–de Vries equation with loss and non-uniformity terms

Solitons, negatons, positons, rational-like solutions and mixed solutions of a non-isospectral equation, the Korteweg–de Vries equation with loss and non-uniformity terms, are obtained through the Wronskian technique. The non-isospectral characteristics of the motion behaviours of some solutions are described with some figures made by using Mathematica. We also derive an infinite number of conservation laws of the equation.

The Novel Multi-Soliton Solutions of the MKdV-Sine Gordon Equations.

!j 1⁄4 B kj Ak j ; e Ajl 1⁄4 ðkj klÞ ðkj þ klÞ ; ð9bÞ and the sum over 1⁄4 0; 1 refers to each of the j; j 1⁄4 1; 2; ;N. Obviously, the solution of SG equation and MKdV equation can be found from (2) and (9). In addition, the solution of this equation possesses an another representation, the Wronskian form, which is essentially the same as (9). In this paper, an alternative choice of f ð1Þ of our interest is f ð1Þ 1⁄4 XN

Soliton solutions for ABS lattice equations: II. Casoratians and bilinearization

In Part I soliton solutions to the ABS list of multi-dimensionally consistent difference equations (except Q4) were derived using connection between the Q3 equation and the NQC equations, and then by reductions. In that work, the central role was played by a Cauchy matrix. In this work we use a different approach, and we derive the N-soliton solutions following Hirotas direct and constructive method. This leads to Casoratians and bilinear difference equations. We give here details for the H-series of equations and for Q1; the results for Q3 have been given earlier.