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A corresponding Lie algebra of a reductive homogeneous group and its applications. (English) Zbl 1314.37053

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 Hamiltonian structure is derived from the trace identity for deducing \((2+1)\)-dimensional integrable hierarchies, which was proposed by G. Tu et al. [J. Math. Phys. 32, No. 7, 1900–1907 (1991; Zbl 0737.58027)]. 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 Schrödinger equation, which was once reobtained by G. Tu et al. [“Binormial and residue representation of \((2+1)\)-dimensional integrable systems”, J. Weifang University 6, 1–7 (2014)]. 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 Schrödinger 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.

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
17B80 Applications of Lie algebras and superalgebras to integrable systems

Citations:

Zbl 0737.58027
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