Transverse evolution operator for the Gross-Pitaevskii equation in semiclassical approximation

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{stroke}, h{stroke} → 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.

Original language	English
Article number	019
Journal	Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
Volume	1
DOIs	https://doi.org/10.3842/SIGMA.2005.019
Publication status	Published - 2005

Keywords

Gross-pitaev-skii equation
Semiclassical asymptotics
Solitons
Symmetry operators
Wkb-maslov complex germ method

ASJC Scopus subject areas

Analysis
Geometry and Topology
Mathematical Physics

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10.3842/SIGMA.2005.019

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title = "Transverse evolution operator for the Gross-Pitaevskii equation in semiclassical approximation",

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{stroke}, h{stroke} → 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.",

keywords = "Gross-pitaev-skii equation, Semiclassical asymptotics, Solitons, Symmetry operators, Wkb-maslov complex germ method",

author = "Alexey Borisov and Shapovalov, {Aleksandr Vasilievich} and Andrey Trifonov",

year = "2005",

doi = "10.3842/SIGMA.2005.019",

language = "English",

volume = "1",

journal = "Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)",

issn = "1815-0659",

publisher = "Department of Applied Research, Institute of Mathematics of National Academy of Science of Ukraine",

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T1 - Transverse evolution operator for the Gross-Pitaevskii equation in semiclassical approximation

AU - Borisov, Alexey

AU - Shapovalov, Aleksandr Vasilievich

AU - Trifonov, Andrey

PY - 2005

Y1 - 2005

N2 - 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{stroke}, h{stroke} → 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.

AB - 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{stroke}, h{stroke} → 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.

KW - Gross-pitaev-skii equation

KW - Semiclassical asymptotics

KW - Solitons

KW - Symmetry operators

KW - Wkb-maslov complex germ method

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