×

zbMATH — the first resource for mathematics

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.

MSC:
35Q51 Soliton equations
35Q53 KdV equations (Korteweg-de Vries equations)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] N. J. Zabusky, Phys. Rev. Lett. 15 pp 240– (1965) · Zbl 1201.35174 · doi:10.1103/PhysRevLett.15.240
[2] M. J. Ablowitz, in: Solitons and the Inverse Transform Method (1981) · Zbl 0472.35002 · doi:10.1137/1.9781611970883
[3] A. Oron, Phys. Rev. A 39 pp 2063– (1989) · doi:10.1103/PhysRevA.39.2063
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.