Exact solutions and symmetry operators for the nonlocal Gross-Pitaevskii equation with quadratic potential

Abstract

The complex WKB-Maslov method is used to consider an approach to the semiclassical integrability of the multidimensional Gross-Pitaevskii equation with an external field and nonlocal nonlinearity previously developed by the authors. Although the WKB-Maslov method is approximate in essence, it leads to exact solution of the Gross- Pitaevskii equation with an external and a nonlocal quadratic potential. For this equation, an exact solution of the Cauchy problem is constructed in the class of trajectory concentrated functions. A nonlinear evolution operator is found in explicit form and symmetry operators (mapping a solution of the equation into another solution) are obtained for the equation under consideration. General constructions are illustrated by examples.

Original language	English
Article number	007
Journal	Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
Volume	1
DOIs	https://doi.org/10.3842/SIGMA.2005.007
Publication status	Published - 1 Jan 2005

Keywords

Gross-pitaevskii equation
Nonlinear evolution operator
Semiclassical asymptotics
Symmetry operators
The cauchy problem
Trajectory concentrated functions
Wkb-maslov complex germ method

ASJC Scopus subject areas

Analysis
Mathematical Physics
Geometry and Topology

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10.3842/SIGMA.2005.007

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title = "Exact solutions and symmetry operators for the nonlocal Gross-Pitaevskii equation with quadratic potential",

abstract = "The complex WKB-Maslov method is used to consider an approach to the semiclassical integrability of the multidimensional Gross-Pitaevskii equation with an external field and nonlocal nonlinearity previously developed by the authors. Although the WKB-Maslov method is approximate in essence, it leads to exact solution of the Gross- Pitaevskii equation with an external and a nonlocal quadratic potential. For this equation, an exact solution of the Cauchy problem is constructed in the class of trajectory concentrated functions. A nonlinear evolution operator is found in explicit form and symmetry operators (mapping a solution of the equation into another solution) are obtained for the equation under consideration. General constructions are illustrated by examples.",

keywords = "Gross-pitaevskii equation, Nonlinear evolution operator, Semiclassical asymptotics, Symmetry operators, The cauchy problem, Trajectory concentrated functions, Wkb-maslov complex germ method",

author = "Shapovalov, {Aleksandr Vasilievich} and Andrey Trifonov and Lisok, {Alexander Leonidovich}",

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