Exact Solutions and Symmetry Operators for the Nonlocal Gross-Pitaevskii Equation with Quadratic Potential

TL;DR

This paper develops an exact solution method for the multidimensional nonlocal Gross-Pitaevskii equation with quadratic potential, utilizing the complex WKB-Maslov approach to find symmetry operators and solve the Cauchy problem.

Contribution

It introduces an exact solution framework and symmetry operators for the nonlocal Gross-Pitaevskii equation with quadratic potential, extending previous approximate methods.

Findings

Exact solutions for the nonlocal Gross-Pitaevskii equation are constructed.

Explicit form of the nonlinear evolution operator is derived.

Symmetry operators mapping solutions to solutions are obtained.

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.