Logarithmic Gross-Pitaevskii equation

Abstract

We consider the Schrödinger 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 traveling waves in the one dimensional case.

Keywords:

• Gross-Pitaevskii
• logarithmic nonlinearity
• Cauchy problem
• solitary wave
• traveling wave

Additional information

Funding

RC is supported by Centre Henri Lebesgue, program ANR-11-LABX-0020-0. A CC-BY public copyright license has been applied by the authors to the present document and will be applied to all subsequent versions up to the Author Accepted Manuscript arising from this submission.