Local well-posedness for a quasilinear Schr\"odinger equation with degenerate dispersion

TL;DR

This paper establishes local well-posedness for a quasilinear Schrödinger equation with degenerate dispersion effects, demonstrating stability of certain solutions under specific initial conditions.

Contribution

It proves local well-posedness for degenerate initial data and shows stability of energy-minimizing breathers in the focusing case.

Findings

Local well-posedness for degenerate initial data

Stability of energy-minimizing breathers

Degenerate dispersion effects analyzed

Abstract

We consider a quasilinear Schr\"odinger equation on $\mathbb R$ for which the dispersive effects degenerate when the solution vanishes. We first prove local well-posedness for sufficiently smooth, spatially localized, degenerate initial data. As a corollary in the focusing case we obtain a short time stability result for the energy-minimizing compact breather.