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Nonlinear Schrödinger equations and simple Lie algebras. (English) Zbl 0563.35062

This paper deals with a generalisation and classification of the nonlinear Schrödinger (NLS) equation. A system of integrable, generalised NLS equations is associated with each Hermitian symmetric space. Each of these NLS equations is a reduction of a generalised second order ”N-wave” equation associated with a reductive homogeneous space which is no longer symmetric. The nonlinear terms are related to the curvature and torsion tensors of the appropriate geometrical space. The Hamiltonian structure is shown to be canonical for all these equations. This is done using the r-matrix techniques. This Hamiltonian structure does not degenerate throughout the reduction. Also it is shown that each of the NLS equations is gauge equivalent to a generalised ferromagnet. Further possible developments are indicated.
Reviewer: V.K.Kumar

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
35J10 Schrödinger operator, Schrödinger equation
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
53D50 Geometric quantization
17B99 Lie algebras and Lie superalgebras
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