Berry phases for the nonlocal Gross-Pitaevskii equation with a quadratic potential

TL;DR

This paper constructs localized solutions to a nonlocal Gross-Pitaevskii equation with a quadratic potential using the complex germ method, and generalizes the Berry phase concept to this nonlinear context, providing explicit formulas.

Contribution

It introduces a method to find localized solutions and extends the Berry phase concept to the nonlinear Gross-Pitaevskii equation with explicit results.

Findings

Constructed asymptotic localized solutions in the adiabatic regime.

Generalized Berry phase for nonlinear Gross-Pitaevskii equation.

Explicit formulas for the Berry phases of the solutions.

Abstract

A countable set of asymptotic space -- localized solutions is constructed by the complex germ method in the adiabatic approximation for the nonstationary Gross -- Pitaevskii equation with nonlocal nonlinearity and a quadratic potential. The asymptotic parameter is 1/T, where $T\gg1$ is the adiabatic evolution time. A generalization of the Berry phase of the linear Schr\"odinger equation is formulated for the Gross-Pitaevskii equation. For the solutions constructed, the Berry phases are found in explicit form.