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SIGMA 9 (2013), 066, 21 pages arXiv:1302.3326
https://doi.org/10.3842/SIGMA.2013.066
Symmetry and Intertwining Operators for the Nonlocal Gross-Pitaevskii Equation
Aleksandr L. Lisok a, Aleksandr V. Shapovalov a, b and Andrey Yu. Trifonov a, b
a) Mathematical Physics Department, Tomsk Polytechnic University, 30 Lenin Ave., Tomsk, 634034 Russia
b) Theoretical Physics Department, Tomsk State University, 36 Lenin Ave., Tomsk, 634050 Russia
Received February 15, 2013, in final form October 26, 2013; Published online November 06, 2013
Abstract
We consider the symmetry properties of an integro-differential multidimensional
Gross-Pitaevskii equation with a nonlocal nonlinear (cubic) term in the context of symmetry analysis
using the formalism of semiclassical asymptotics.
This yields a semiclassically reduced nonlocal Gross-Pitaevskii equation, which can be treated as a nearly
linear equation, to determine the principal term of the semiclassical asymptotic solution.
Our main result is an approach which allows one to construct a class of symmetry operators for the reduced
Gross-Pitaevskii equation.
These symmetry operators are determined by linear relations including intertwining operators and additional
algebraic conditions.
The basic ideas are illustrated with a 1D reduced Gross-Pitaevskii equation.
The symmetry operators are found explicitly, and the corresponding families of exact solutions
are obtained.
Key words:
symmetry operators; intertwining operators; nonlocal Gross-Pitaevskii equation; semiclassical asymptotics; exact solutions.
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