Dong Huan-He

A Multi-component Matrix Loop Algebra and Its Application

A set of multi-component matrix Lie algebra is constructed. It follows that a type of new loop algebra AM-1 is presented. An isospectral problem is established. Integrable multi-component hierarchy is obtained by Tu pattern, which possesses tri-Hamiltonian structures. Furthermore, it can be reduced to the well-known AKNS hierarchy and BPT hierarchy. Therefore, the major result of this paper can be regarded as a unified expression integrable model of the AKNS hierarchy and the BPT hierarchy.

A Hierarchy of Integrable Nonlinear Lattice Equations and New Integrable Symplectic Map

A discrete spectral problem is discussed, and a hierarchy of integrable nonlinear lattice equations related to this spectral problem is devised. The new integrable symplectic map and finite-dimensional integrable systems are given by nonlinearization method. The binary Bargmann constraint gives rise to a Backlund transformation for the resulting integrable lattice equations.

Generation of Solitary Rossby Waves by Unstable Topography

The effect of topography on generation of the solitary Rossby waves is researched. Here, the topography, as a forcing for waves generation, is taken as a function of longitude variable x and time variable t, which is called unstable topography. With the help of a perturbation expansion method, a forced mKdv equation governing the evolution of amplitude of the solitary Rossby waves is derived from quasi-geostrophic vorticity equation and is solved by the pseudospectral method. Basing on the waterfall plots, the generational features of the solitary Rossby waves under the influence of unstable topography and stable topography are compared and some conclusions are obtained.

A subalgebra of Lie algebra A2 and its associated two types of loop algebras, as well as Hamiltonian structures of integrable hierarchy

In this paper, a subalgebra A2 of the Lie algebra A2 is constructed, which gives a corresponding loop algebra A¯2 by properly choosing the gradation of the basis elements. It follows that an isospectral problem is established and a new Liouville integrable Hamiltonian hierarchy is obtained. By making use of a matrix transformation, a subalgebra A2 of the Lie algebra A1 is presented, which possesses the same communicative operations of basis elements as those in A2. Again we expand the Lie algebra A1 into a high-dimensional loop algebra G, and a type of expanding integrable system of the hierarchy obtained above is worked out. Furthermore, Hamiltonian structures of hierarchy are presented by use of the quadratic form identity.

Exact Periodic Wave Solution of Extended (2+1)-Dimensional Shallow Water Wave Equation with Generalized Dp̄-operators

With the aid of binary Bell polynomial and a general Riemann theta function, we introduce how to obtain the exact periodic wave solutions by applying the generalized D-operators in term of the Hirota direct method when the appropriate value of is determined. Furthermore, the resulting approach is applied to solve the extended (2+1)-dimensional Shallow Water Wave equation, and the periodic wave solution is obtained and reduced to soliton solution via asymptotic analysis.

Nonlinear integrable couplings of a nonlinear Schrodinger-modified Korteweg de Vries hierarchy with self-consistent sources

By means of the Lie algebra B-2, a new extended Lie algebra F is constructed. Based on the Lie algebras B-2 and F, the nonlinear Schrodinger-modified Korteweg de Vries (NLS-mKdV) hierarchy with self-consistent sources as well as its nonlinear integrable couplings are derived. With the help of the variational identity, their Hamiltonian structures are generated.

Two New Expanding Lie Algebras and Their Integrable Models

A new Lax integrable hierarchy is obtained by constructing an isospectral problem with constrained conditions. Two kinds of integrable couplings are obtained by constructing two new expanding Lie algebras of the Lie algebra B2, respectively.

The quadratic-form identity for constructing Hamiltonian structures of the Guo hierarchy

The trace identity is extended to the quadratic-form identity. The Hamiltonian structures of the multi-component Guo hierarchy, integrable coupling of Guo hierarchy and (2+1)-dimensional Guo hierarchy are obtained by the quadratic-form identity. The method can be used to produce the Hamiltonian structures of the other integrable couplings or multi-component hierarchies.

A New Loop Algebra and Corresponding Computing Formula of Constant γ in Quadratic-Form Identity

A new loop algebra containing four arbitrary constants is presented, whose commutation operation is concise, and the corresponding computing formula of constant ? in the quadratic-form identity is obtained in this paper, which can be reduced to computing formula of constant ? in the trace identity. As application, a new Liouville integrable hierarchy, which can be reduced to AKNS hierarchy is derived.

Multi-component Harry--Dym hierarchy and its integrable couplings as well as their Hamiltonian structures

This paper obtains the multi-component Harry–Dym (H–D) hierarchy and its integrable couplings by using two kinds of vector loop algebras 3 and 6. The Hamiltonian structures of the above system are given by the quadratic-form identity. The method can be used to produce the Hamiltonian structures of the other integrable couplings or multi-component hierarchies.