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 | |
| Publication status | Published - 1 Jan 2005 |
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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
Cite this
Exact solutions and symmetry operators for the nonlocal Gross-Pitaevskii equation with quadratic potential. / Shapovalov, Aleksandr Vasilievich; Trifonov, Andrey; Lisok, Alexander Leonidovich.
In: Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), Vol. 1, 007, 01.01.2005.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Exact solutions and symmetry operators for the nonlocal Gross-Pitaevskii equation with quadratic potential
AU - Shapovalov, Aleksandr Vasilievich
AU - Trifonov, Andrey
AU - Lisok, Alexander Leonidovich
PY - 2005/1/1
Y1 - 2005/1/1
N2 - 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.
AB - 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.
KW - Gross-pitaevskii equation
KW - Nonlinear evolution operator
KW - Semiclassical asymptotics
KW - Symmetry operators
KW - The cauchy problem
KW - Trajectory concentrated functions
KW - Wkb-maslov complex germ method
UR - http://www.scopus.com/inward/record.url?scp=84889234828&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84889234828&partnerID=8YFLogxK
U2 - 10.3842/SIGMA.2005.007
DO - 10.3842/SIGMA.2005.007
M3 - Article
VL - 1
JO - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
JF - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
SN - 1815-0659
M1 - 007
ER -