Nonlinear Schr�dinger equations and simple Lie algebras
Copyright policy )Nonlinear Schr�dinger equations and simple Lie algebras
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.
- University of Manchester United Kingdom
Hermitian symmetric space, 58G40, Partial differential equations of mathematical physics and other areas of application, 17B65, Lie algebras and Lie superalgebras, Hamiltonian structure, isotropic ferromagnet, 58F07, 35Q20, Geometric quantization, Schrödinger operator, Schrödinger equation, simple Lie algebras, 82A05, Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics, gauge equivalence, nonlinear Schrödinger equation
Hermitian symmetric space, 58G40, Partial differential equations of mathematical physics and other areas of application, 17B65, Lie algebras and Lie superalgebras, Hamiltonian structure, isotropic ferromagnet, 58F07, 35Q20, Geometric quantization, Schrödinger operator, Schrödinger equation, simple Lie algebras, 82A05, Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics, gauge equivalence, nonlinear Schrödinger equation
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