Dissipative perturbations for the K(n,n) Rosenau-Hyman equation

TL;DR

This paper studies how dissipative perturbations affect compacton solutions in the K(n,n) Rosenau-Hyman equation, deriving evolution equations for their velocity and amplitude, and validating the approach numerically.

Contribution

It applies the adiabatic perturbation method to compactons with dissipative terms, deriving evolution equations for key parameters and validating the results numerically.

Findings

Derived evolution equations for compacton velocity and amplitude.

Validated the perturbation method numerically for a fourth-order dissipative term.

Showed that dissipative terms influence compacton dynamics while preserving mass.

Abstract

Compactons are compactly supported solitary waves for nondissipative evolution equations with nonlinear dispersion. In applications, these model equations are accompanied by dissipative terms which can be treated as small perturbations. We apply the method of adiabatic perturbations to compactons governed by the K(n,n) Rosenau-Hyman equation in the presence of dissipative terms preserving the "mass" of the compactons. The evolution equations for both the velocity and the amplitude of the compactons are determined for some linear and nonlinear dissipative terms: second-, fourth-, and sixth-order in the former case, and second- and fourth-order in the latter one. The numerical validation of the method is presented for a fourth-order, linear, dissipative perturbation which corresponds to a singular perturbation term.