Logarithmic Gross-Pitaevskii equation

TL;DR

This paper studies the logarithmic Gross-Pitaevskii equation, proving global well-posedness in the energy space and characterizing solitary and traveling waves in one dimension.

Contribution

It establishes the global well-posedness of the logarithmic Gross-Pitaevskii equation and characterizes its solitary and traveling wave solutions in one dimension.

Findings

Global well-posedness in energy space for small dimensions

Characterization of solitary waves in 1D

Analysis of traveling waves in 1D

Abstract

We consider the Schr{\"o}dinger equation with a logarithmic nonlinearty and non-trivial boundary conditions at infinity. We prove that the Cauchy problem is globally well posed in the energy space, which turns out to correspond to the energy space for the standard Gross-Pitaevskii equation with a cubic nonlinearity, in small dimensions. We then characterize the solitary and travelling waves in the one dimensional case.