Compactons: Solitons with finite wavelength

The understand the role of nonlinear dispersion in pattern formation, we introduce and study Korteweg--de Vries--like equations wtih nonlinear dispersion: ${\mathit{u}}_{\mathit{t}}$+(${\mathit{u}}^{\mathit{m}}$${)}_{\mathit{x}}$+(${\mathit{u}}^{\mathit{n}}$${)}_{\mathit{x}\mathit{x}\mathit{x}}$=0, m,n>1. The solitary wave solutions of these equations have remarkable properties: They collide elastically, but unlike the Korteweg--de Vries (m=2, n=1) solitons, they have compact support. When two ``compactons'' collide, the interaction site is marked by the birth of low-amplitude compacton-anticompacton pairs. These equations seem to have only a finite number of local conservation laws. Nevertheless, the behavior and the stability of these compactons is very similar to that observed in completely integrable systems.