Compacton solutions in a class of generalized Fifth-order Korteweg-de Vries equations

Cooper, Fred ; Hyman, James M. ; Khare, Avinash (2001) Compacton solutions in a class of generalized Fifth-order Korteweg-de Vries equations Physical Review E - Statistical, Nonlinear and Soft Matter Physics, 64 (2). 026608_1-026608_13. ISSN 1539-3755

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Official URL: http://pre.aps.org/abstract/PRE/v64/i2/e026608

Related URL: http://dx.doi.org/10.1103/PhysRevE.64.026608

Abstract

Solitons play a fundamental role in the evolution of general initial data for quasilinear dispersive partial differential equations, such as the Korteweg-de Vries (KdV), nonlinear Schrödinger, and the Kadomtsev-Petviashvili equations. These integrable equations have linear dispersion and the solitons have infinite support. We have derived and investigate a new KdV-like Hamiltonian partial differential equation from a four-parameter Lagrangian where the nonlinear dispersion gives rise to solitons with compact support (compactons). The new equation does not seem to be integrable and only mass, momentum, and energy seem to be conserved; yet, the solitons display almost the same modal decompositions and structural stability observed in integrable partial differential equations. The compactons formed from arbitrary initial data, are nonlinearly self-stabilizing, and maintain their coherence after multiple collisions. The robustness of these compactons and the inapplicability of the inverse scattering tools, that worked so well for the KdV equation, make it clear that there is a fundamental mechanism underlying the processes beyond integrability. We have found explicit formulas for multiple classes of compact traveling wave solutions. When there are more than one compacton solution for a particular set of parameters, the wider compacton is the minimum of a reduced Hamiltonian and is the only one that is stable.

Item Type:Article
Source:Copyright of this article belongs to The American Physical Society.
ID Code:83266
Deposited On:20 Feb 2012 06:28
Last Modified:19 May 2016 00:10

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