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 ttamiltonian 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 ）-dimensionai 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）-dimensionai Sehr6dinger 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）-dimensionaJ diffusion equation along with a parameter. In addition, the well-known （1＋1）-dimensional AKNS hierarchy, the （1＋1）-dimensional nonlinear Schr6dinger 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 yon 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.