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 ttam...

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Published inCommunications in theoretical physics Vol. 63; no. 5; pp. 535 - 548
Main Author 张玉峰 吴立新 芮文娟
Format Journal Article
LanguageEnglish
Published 01.05.2015
Subjects
AKNS方程
AKNS族
Diffusion
Hierarchies
Joining
Lie groups
Mathematical analysis
Mathematical models
Reduction
Schroedinger equation
同构群
层次结构
李代数
部分还原
非线性Schrodinger方程
非线性耦合方程
Online AccessGet full text
ISSN0253-6102
1572-9494
DOI10.1088/0253-6102/63/5/535

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Summary: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.
Bibliography:ZHANG Yu-Feng , WU Li-Xin , RUI Wen-Juan （ 1College of Sciences, China University of Mining and Technology, Xuzhou 221116, China 2School of Environment Science and Spatial Informatics, China University of Mining and Technology Xuzhou 221116, China）
（2＋1）-dimensional hierarchy, Lie algebra, Hamiltonian structure
11-2592/O3
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.
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ISSN:0253-6102
1572-9494
DOI:10.1088/0253-6102/63/5/535