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Compactons: Solitons with finite wavelength. (English) Zbl 0952.35502
Summary: To understand the role of nonlinear dispersion in pattern formation, we introduce and study Korteweg-de Vries-like equations with nonlinear dispersion: $u_t+(u^m)_x+(u^n)_{xxx}=0,$ $m,n>1$. 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.

35Q51Soliton-like equations
35Q53KdV-like (Korteweg-de Vries) equations
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