Long-time relaxation processes in the nonlinear Schrödinger equation

The nonlinear Schrödinger equation, known in low-temperature physics as the Gross-Pitaevskii equation, has a large family of excitations of different kinds. They include sound excitations, vortices, and solitons. The dynamics of vortices strictly depends on the separation between them. For large sep...

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Bibliographic Details
Published in:Journal of experimental and theoretical physics 2011-03, Vol.112 (3), p.469-478
Main Authors: Ovchinnikov, Yu. N., Sigal, I. M.
Format: Article
Language:English
Subjects:
Classical and Quantum Gravitation
Electronic Properties of Solid
Elementary Particles
Particle and Nuclear Physics
Physics
Physics and Astronomy
Quantum Field Theory
Relativity Theory
Solid State Physics
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Summary:The nonlinear Schrödinger equation, known in low-temperature physics as the Gross-Pitaevskii equation, has a large family of excitations of different kinds. They include sound excitations, vortices, and solitons. The dynamics of vortices strictly depends on the separation between them. For large separations, some kind of adiabatic approximation can be used. We consider the case where an adiabatic approximation can be used (large separation between vortices) and the opposite case of a decay of the initial state, which is close to the double vortex solution. In the last problem, no adiabatic parameter exists (the interaction is strong). Nevertheless, a small numerical parameter arises in the problem of the decay rate, connected with an existence of a large centrifugal potential, which leads to a small value of the increment. The properties of the nonlinear wave equation are briefly considered in the Appendix A.
ISSN:1063-7761
1090-6509
DOI:10.1134/S1063776111020087