This talk is based on the degenerate semi-linear Schrödinger and Korteweg-de Vries equations in one spatial dimension. We construct variationally special solutions of the two models, that is,  standing wave solutions of NLS and traveling waves for KDV, which turn out to have compact support, hence compactons. We show that the compactons are unique bell-shaped solutions of the corresponding PDE's and for appropriate variational problems as well.

 We also 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 the previous work of Germain, Harrop-Griffiths and Marzuola, who have previously established orbital stability for some specific waves, in the range \(p<8\). This is a joint work with Atanas stefanov and Sevdhan Hakkaev.