Global behavior of solutions to generalized Gross-Pitaevskii equation

TL;DR

This paper investigates the long-term behavior of solutions to a generalized nonlinear Schrödinger equation with non-vanishing boundary conditions, showing conditions under which solutions exist globally and scatter to a non-vanishing state.

Contribution

It introduces a generalized Gross-Pitaevskii equation with modified nonlinearity near the non-vanishing state and analyzes its global solution behavior.

Findings

Solutions exist globally if nonlinearity decays rapidly near the non-vanishing state.

Solutions scatter to the non-vanishing element in both time directions.

Behavior is determined by the shape of the nonlinearity around the non-vanishing state.

Abstract

This paper is concerned with time global behavior of solutions to nonlinear Schr\"odinger equation with a non-vanishing condition at the spatial infinity. Under a non-vanishing condition, it would be expected that the behavior is determined by the shape of the nonlinear term around the non-vanishing state. To observe this phenomenon, we introduce a generalized version of the Gross-Pitaevskii equation, which is a typical equation involving a non-vanishing condition, by modifying the shape of nonlinearity around the non-vanishing state. It turns out that, if the nonlinearity decays fast as a solution approaches to the non-vanishing state, then the equation admits a global solution which scatters to the non-vanishing element for both time directions.