The Gross–Pitaevskii Equation with a Nonlocal Interaction in a Semiclassical Approximation on a Curve

Abstract

We propose an approach to constructing semiclassical solutions for the generalized multidimensional Gross–Pitaevskii equation with a nonlocal interaction term. The key property of the solutions is that they are concentrated on a one-dimensional manifold (curve) that evolves over time. The approach reduces the Cauchy problem for the nonlocal Gross–Pitaevskii equation to a similar problem for the associated linear equation. The geometric properties of the resulting solutions are related to Maslov’s complex germ, and the symmetry operators of the associated linear equation lead to the approximation of the symmetry operators for the nonlocal Gross–Pitaevskii equation.