Multiple-scale analysis for resonance reflection by a one-dimensional rectangular barrier in the Gross-Pitaevskii problem

TL;DR

This paper analyzes how nonlinearity affects quantum resonance reflection of Bose-Einstein condensates in a one-dimensional potential, providing analytical expressions for reflection coefficients near resonances.

Contribution

It introduces a multiple-scale analytical method to study nonlinear resonance reflection in the Gross-Pitaevskii equation, revealing how nonlinearity shifts and influences resonance conditions.

Findings

Derived an explicit formula for nonlinear reflection coefficient.

Identified conditions for zero reflection due to combined potential and nonlinearity.

Showed resonance shifts caused by nonlinearity in the system.

Abstract

We consider a quantum above-barrier reflection of a Bose-Einstein condensate by a one-dimensional rectangular potential barrier, or by a potential well, for nonlinear Schroedinger equation (Gross-Pitaevskii equation) with a small nonlinearity. The most interesting case is realized in resonances when the reflection coefficient is equal to zero for the linear Schroedinger equation. Then the reflection is determined only by small nonlinear term in the Gross-Pitaevskii equation. A simple analytic expression has been obtained for the reflection coefficient produced only by the nonlinearity. An analytical condition is found when common action of potential barrier and nonlinearity produces a zero reflection coefficient. The reflection coefficient is derived analytically in the vicinity of resonances which are shifted by nonlinearity.