Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 1 (2005), 019, 17 pages      math-ph/0511081      http://dx.doi.org/10.3842/SIGMA.2005.019

Transverse Evolution Operator for the Gross-Pitaevskii Equation in Semiclassical Approximation

Alexey Borisov a, Alexander Shapovalov a, b, c and Andrey Trifonov b, c
a) Tomsk State University, 36 Lenin Ave., 634050 Tomsk, Russia
b) Tomsk Polytechnic University, 30 Lenin Ave., 634050 Tomsk, Russia
c) Math. Phys. Laboratory, Tomsk Polytechnic University, 30 Lenin Ave., 634050 Tomsk, Russia

Received July 27, 2005, in final form November 13, 2005; Published online November 22, 2005

Abstract
The Gross-Pitaevskii equation with a local cubic nonlinearity that describes a many-dimensional system in an external field is considered in the framework of the complex WKB-Maslov method. Analytic asymptotic solutions are constructed in semiclassical approximation in a small parameter h, h -> 0, in the class of functions concentrated in the neighborhood of an unclosed surface associated with the phase curve that describes the evolution of surface vertex. The functions of this class are of the one-soliton form along the direction of the surface normal. The general constructions are illustrated by examples.

Key words: WKB-Maslov complex germ method; semiclassical asymptotics; Gross-Pitaevskii equation; solitons; symmetry operators.

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