On the stability of the compacton waves for the degenerate KdV and NLS models

Abstract

In this paper, we consider the degenerate semi-linear Schr\"odinger and Korteweg-deVries equations in one spatial dimension. We construct special solutions of the two models, namely standing wave solutions of NLS and traveling waves, which turn out to have compact support, compactons. We show that the compactons are unique bell-shaped solutions of the corresponding PDE's and for appropriate variational problems as well. We provide a complete spectral characterization of such waves, for all values of $p$. Namely, we show that all waves are spectrally stable for $2<p\leq 8$, while a single mode instability occurs for $p>8$. This extends previous work of Germain, Harrop-Griffits and Marzuola, who have established orbital stability for some specific waves, in the range $p<8$.