Zhang Yu-Feng

Lie Algebras for Constructing Nonlinear Integrable Couplings

Two new explicit Lie algebras are introduced for which the nonlinear integrable couplings of the Giachetti-Johnson (GJ) hierarchy and the Yang hierarchy are obtained, respectively. By employing the variational identity their Hamiltonian structures are also generated. The approach presented in the paper can also provide nonlinear integrable couplings of other soliton hierarchies of evolution equations.

Some Evolution Hierarchies Derived from Self-dual Yang—Mills Equations

We develop in this paper a new method to construct two explicit Lie algebras E and F. By using a loop algebra ? of the Lie algebra E and the reduced self-dual Yang?Mills equations, we obtain an expanding integrable model of the Giachetti?Johnson (GJ) hierarchy whose Hamiltonian structure can also be derived by using the trace identity. This provides a much simplier construction method in comparing with the tedious variational identity approach. Furthermore, the nonlinear integrable coupling of the GJ hierarchy is readily obtained by introducing the Lie algebra gN. As an application, we apply the loop algebra ? of the Lie algebra E to obtain a kind of expanding integrable model of the Kaup?Newell (KN) hierarchy which, consisting of two arbitrary parameters ? and ?, can be reduced to two nonlinear evolution equations. In addition, we use a loop algebra F? of the Lie algebra F to obtain an expanding integrable model of the BT hierarchy whose Hamiltonian structure is the same as using the trace identity. Finally, we deduce five integrable systems in R3 based on the self-dual Yang?Mills equations, which include Poisson structures, irregular lines, and the reduced equations.

Lie Algebras and Integrable Systems

A 3 × 3 matrix Lie algebra is first introduced, its subalgebras and the generated Lie algebras are obtained, respectively. Applications of a few Lie subalgebras give rise to two integrable nonlinear hierarchies of evolution equations from their reductions we obtain the nonlinear Schrodinger equations, the mKdV equations, the Broer-Kaup (BK) equation and its generalized equation, etc. The linear and nonlinear integrable couplings of one integrable hierarchy presented in the paper are worked out by casting a 3 × 3 Lie subalgebra into a 2 × 2 matrix Lie algebra. Finally, we discuss the elliptic variable solutions of a generalized BK equation.

A Corresponding Lie Algebra of a Reductive homogeneous Group and Its Applications

With the help of a Lie algebra of a reductive homogeneous space G/K, where G is a Lie group and K is a resulting isotropy group, we introduce a Lax pair for which an expanding(2+1)-dimensional integrable hierarchy is obtained by applying the binormial-residue representation(BRR) method, whose Hamiltonian structure is derived from the trace identity for deducing(2+1)-dimensional integrable hierarchies, which was proposed by Tu, et al. We further consider some reductions of the expanding integrable hierarchy obtained in the paper. The first reduction is just right the(2+1)-dimensional AKNS hierarchy, the second-type reduction reveals an integrable coupling of the(2+1)-dimensional AKNS equation(also called the Davey-Stewartson hierarchy), a kind of(2+1)-dimensional Schr¨odinger equation, which was once reobtained by Tu, Feng and Zhang. It is interesting that a new(2+1)-dimensional integrable nonlinear coupled equation is generated from the reduction of the part of the(2+1)-dimensional integrable coupling, which is further reduced to the standard(2+1)-dimensional diffusion equation along with a parameter. In addition, the well-known(1+1)-dimensional AKNS hierarchy, the(1+1)-dimensional nonlinear Schr¨odinger equation are all special cases of the(2+1)-dimensional expanding integrable hierarchy. Finally, we discuss a few discrete difference equations of the diffusion equation whose stabilities are analyzed by making use of the von Neumann condition and the Fourier method. Some numerical solutions of a special stationary initial value problem of the(2+1)-dimensional diffusion equation are obtained and the resulting convergence and estimation formula are investigated.

Solitary Wave Solutions for the Coupled Ito System and a Generalized Hirota–Satsuma Coupled KdV System*

We make use of an extended tanh-function method and symbolic computation to construct four kinds of travelling solitary wave solutions for the coupled Ito system and a generalized Hirota–Satsuma coupled KdV system.

On Integrable Properties for Two Variable-Coefficient Evolution Equations

With the help of the extended binary Bell polynomials, the new bilinear representations, Backlund transformations, Lax pair and infinite conservation laws for two types of variable-coefficient nonlinear integrable equations are obtained, respectively, which are more straightforward than previous corresponding results obtained. Finally, we obtain new multi-soliton wave solutions of a reduced soliton equations with variable coefficients.

Double Reduction and Exact Solutions of Zakharov—Kuznetsov Modified Equal width Equation with Power Law Nonlinearity via Conservation Laws

The conservation laws for the (1+2)-dimensional Zakharov—Kuznetsov modified equal width (ZK-MEW) equation with power law nonlinearity are constructed by using Noethers approach through an interesting method of increasing the order of this equation. With the aid of an obtained conservation law, the generalized double reduction theorem is applied to this equation. It can be shown that the reduced equation is a second order nonlinear ODE. Finally, some exact solutions for a particular case of this equation are obtained after solving the reduced equation.

Algebro-Geometric Solutions with Characteristics of a Nonlinear Partial Differential Equation with Three-Potential Functions*

With the help of a simple Lie algebra, an isospectral Lax pair, whose feature presents decomposition of element(1, 2) into a linear combination in the temporal Lax matrix, is introduced for which a new integrable hierarchy of evolution equations is obtained, whose Hamiltonian structure is also derived from the trace identity in which contains a constant γ to be determined. In the paper, we obtain a general formula for computing the constant γ. The reduced equations of the obtained hierarchy are the generalized nonlinear heat equation containing three-potential functions,the m Kd V equation and a generalized linear Kd V equation. The algebro-geometric solutions(also called finite band solutions) of the generalized nonlinear heat equation are obtained by the use of theory on algebraic curves. Finally, two kinds of gauge transformations of the spatial isospectral problem are produced.

A Few Expanding Integrable Models, Hamiltonian Structures and Constrained Flows

Two kinds of higher-dimensional Lie algebras and their loop algebras are introduced, for which a few expanding integrable models including the coupling integrable couplings of the Broer—Kaup (BK) hierarchy and the dispersive long wave (DLW) hierarchy as well as the TB hierarchy are obtained. From the reductions of the coupling integrable couplings, the corresponding coupled integrable couplings of the BK equation, the DLW equation, and the TB equation are obtained, respectively. Especially, the coupling integrable coupling of the TB equation reduces to a few integrable couplings of the well-known mKdV equation. The Hamiltonian structures of the coupling integrable couplings of the three kinds of soliton hierarchies are worked out, respectively, by employing the variational identity. Finally, we decompose the BK hierarchy of evolution equations into x-constrained flows and tn-constrained flows whose adjoint representations and the Lax pairs are given.

Double Integrable Couplings and Their Constructing Method

To extend the study scopes of integrable couplings, the notion of double integrable couplings is proposed in the paper. The zero curvature equation appearing in the constructing method built in the paper consists of the elements of a new loop algebra which is obtained by using perturbation method. Therefore, the approach given in the paper has extensive applicable values, that is, it applies to investigate a lot of double integrable couplings of the known integrable hierarchies of evolution equations. As for explicit applications of the method proposed in the paper, the double integrable couplings of the AKNS hierarchy and the KN hierarchy are worked out, respectively.