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

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.