Xia Tie-Cheng

Symbolic Computation and New Families of Exact Non-travelling Wave Solutions of (2+1)-dimensional Broer–Kaup Equations

The improved tanh function method [Commun. Theor. Phys. (Beijing, China) 43 (2005) 585] is further improved by generalizing the ansatz solution of the considered equation. As its application, the (2+1)-dimensional Broer–Kaup equations are considered and abundant new exact non-travelling wave solutions are obtained.

Lie Algebra and Lie Super Algebra for Integrable Couplings of C-KdV Hierarchy

Based on the constructed Lie algebra and Lie super algebra, the integrable couplings and super-integrable couplings of the C-KdV hierarchy are obtained respectively. Furthermore, its super-Hamiltonian structures are presented by using super-trace identity.

Two new integrable couplings of the soliton hierarchies with self-consistent sources

A kind of integrable coupling of soliton equations hierarchy with self-consistent sources associated with l(4) has been presented (Yu F J and Li L 2009 Appl. Math. Comput. 207 171; Yu F J 2008 Phys. Lett. A 372 6613). Based on this method, we construct two integrable couplings of the soliton hierarchy with self-consistent sources by using the loop algebra l(4). In this paper, we also point out that there are some errors in these references and we have corrected these errors and set up new formula. The method can be generalized to other soliton hierarchy with self-consistent sources.

Multi-component Dirac equation hierarchy and its multi-component integrable couplings system

A general scheme for generating a multi-component integrable equation hierarchy is proposed. A simple 3M-dimensional loop algebra is produced. By taking advantage of , a new isospectral problem is established and then by making use of the Tu scheme the multi-component Dirac equation hierarchy is obtained. Finally, an expanding loop algebra M of the loop algebra is presented. Based on the M, the multi-component integrable coupling system of the multi-component Dirac equation hierarchy is investigated. The method in this paper can be applied to other nonlinear evolution equation hierarchies.

Multi-component C-KdV Hierarchy of Soliton Equations and Its Multi-component Integrable Coupling System

A new simple loop algebra is constructed, which is devoted to establishing an isospectral problem. By making use of Tu scheme, the multi-component C-KdV hierarchy is obtained. Further,an expanding loop algebra of the loop algebra is presented. Based on , the multi-component integrable coupling system of the multi-component C-KdV hierarchy is worked out. The method can be used to other nonlinear evolution equations hierarchy.

The super-classical-Boussinesq hierarchy and its super-Hamiltonian structure

Based on the constructed Lie superalgebra, the super-classical-Boussinesq hierarchy is obtained. Then, its super-Hamiltonian structure is obtained by making use of super-trace identity. Furthermore, the super-classical-Boussinesq hierarchy is also integrable in the sense of Liouville.

Super-KN Hierarchy and Its Super-Hamiltonian Structure

Based on the basis of the constructed Lie super algebra, the super-isospectral problem of KN hierarchy is considered. Under the frame of the zero curvature equation, the super-KN hierarchy is obtained. Furthermore, its super-Hamiltonian structure is presented by using super-trace identity and it has super-bi-Hamiltonian structure.

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.

Multi-component Levi Hierarchy and Its Multi-component Integrable Coupling System

A simple 3M-dimensional loop algebra is produced, whose commutation operation defined by us is as simple and straightforward as that in the loop algebra A1. It follows that a general scheme for generating multi-component integrable hierarchy is proposed. By taking advantage of , a new isospectral problem is established, and then by making use of the Tu scheme the well-known multi-component Levi hierarchy is obtained. Finally, an expanding loop algebra M of the loop algebra is presented, based on the M, the multi-component integrable coupling system of the multi-component Levi hierarchy is worked out. The method in this paper can be applied to other nonlinear evolution equation hierarchies.

A new generalized fractional Dirac soliton hierarchy and its fractional Hamiltonian structure

Based on the differential forms and exterior derivatives of fractional orders, Wu first presented the generalized Tu formula to construct the generalized Hamiltonian structure of the fractional soliton equation. We apply the generalized Tu formula to calculate the fractional Dirac soliton equation hierarchy and its Hamiltonian structure. The method can be generalized to the other fractional soliton hierarchy.