Global well-posedness for intermediate NLS with nonvanishing conditions at infinity

TL;DR

This paper proves local and global well-posedness for the intermediate nonlinear Schrödinger equation with nonvanishing boundary conditions, accommodating dark solitons in a suitable functional space.

Contribution

It introduces the first well-posedness results for the equation in a space tailored to dark solitons with nonvanishing conditions at infinity.

Findings

Established local and global well-posedness in a Zhidkov-type space

Extended results to a generalized defocusing equation

First functional setting adapted to dark soliton solutions

Abstract

The intermediate nonlinear Schr\"odinger equation models quasi-harmonic internal waves in two-fluid layer system, and admits dark solitons, that is, solutions with nonvanishing boundary conditions at spatial infinity. These solutions fall outside existing well-posedness theories. We establish local and global well-posedness in a Zhidkov-type space naturally suited to such non-trivial boundary conditions, and extend these results to a generalized defocusing equation. This appears to be the first well-posedness result for the equation in a functional setting adapted to its dark soliton structure.