Transverse Evolution Operator for the Gross–Pitaevskii Equation in Semiclassical Approximation

The Gross–Pitaevskii equation with a local cubic nonlinearity that describes a many-dimensional system in an external field is considered in the framework of the complex WKB–Maslov method. Analytic asymptotic solutions are constructed in semiclassical approximation in a small parameter $\hbar$, $\hbar\to 0$, in the class of functions concentrated in the neighborhood of an unclosed surface associated with the phase curve that describes the evolution of surface vertex. The functions of this class are of the one-soliton form along the direction of the surface normal. The general constructions are illustrated by examples.