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.


35Q51 Soliton equations
35Q53 KdV equations (Korteweg-de Vries equations)
Full Text: DOI


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