Deng-yuan Chen

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.

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

Exact solutions for the nonisospectral kadomtshev-petviashvili equation

The nonisospectral Kadomtshev–Petviashvili (KP) equation is solved by the Hirota method and Wronskian technique. Exact solutions that possess soliton characters with nonisospectral properties are obtained. In addition, rational and mixed solutions are derived. We also obtain a new molecular equation that admits a solution in the Wronskian form.

Lie algebraic structures of some (1+2)-dimensional Lax integrable systems

Abstract The paper proposes an approach to constructing the symmetries and their algebraic structures for isospectral and nonisospectral evolution equations of (1+2)-dimensional systems associated with the linear problem of Sato theory. To do that, we introduce the implicit representations of the isospectral flows {Km} and nonisospectral flows {σn} in the high dimensional cases. Three examples, the Kodomstev–Petviashvili system, BKP system and new CKP system, are considered to demonstrate our method.

Lie algebraic structures of (1+1)-dimensional Lax integrable systems

An approach of constructing isospectral flows Kl, nonisospectral flows σk and their implicit representations of a general Lax integrable system is proposed. By introducing product function matrices, it is shown that the two sets of flows and of related symmetries both constitute infinite‐dimensional Lie algebras with respect to the commutator ⟦⋅,⋅⟧ given in this paper. Algebraic properties for some well‐known integrable systems such as the AKNS system, the generalized Harry Dym system, and the n‐wave interaction system are obtained as particular examples.

SYMMETRIES AND LIE ALGEBRA OF THE DIFFERENTIAL DIFFERENCE KADOMSTEV PETVIASHVILI HIERARCHY

By introducing suitable non-isospectral flows, we construct two sets of symmetries for the isospectral differential–difference Kadomstev–Petviashvili hierarchy. The symmetries form an infinite dimensional Lie algebra.