
SIGMA 9 (2013), 066, 21 pages arXiv:1302.3326
http://dx.doi.org/10.3842/SIGMA.2013.066
Symmetry and Intertwining Operators for the Nonlocal GrossPitaevskii 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 integrodifferential multidimensional
GrossPitaevskii 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 GrossPitaevskii 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
GrossPitaevskii 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 GrossPitaevskii equation.
The symmetry operators are found explicitly, and the corresponding families of exact solutions
are obtained.
Key words:
symmetry operators; intertwining operators; nonlocal GrossPitaevskii equation; semiclassical asymptotics; exact solutions.
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