Symmetry and Intertwining Operators for the Nonlocal Gross-Pitaevskii Equation

TL;DR

This paper explores symmetry properties of a nonlocal multidimensional Gross-Pitaevskii equation, deriving symmetry operators and exact solutions through semiclassical asymptotics and intertwining operators.

Contribution

It introduces a novel approach to construct symmetry operators for the nonlocal Gross-Pitaevskii equation using semiclassical analysis and algebraic conditions.

Findings

Explicit symmetry operators for 1D reduced Gross-Pitaevskii equation

Families of exact solutions constructed from symmetry operators

Method applicable to nearly linear nonlocal equations

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.