Local well-posedness for a quasilinear Schrödinger equation with degenerate dispersion

Abstract

We consider a quasilinear Schrödinger 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.

31 pages, 1 figure, comments welcome!!