Solitons, compactons and undular bores in Benjamin-Bona-Mahony-like systems

TL;DR

This paper investigates how dissipation affects traveling wave solutions in BBM-like systems, revealing transformations of solitons and compactons into undular bores, shocks, or their disappearance, depending on nonlinearity and viscosity.

Contribution

It provides a detailed analysis of the impact of dissipation on soliton, compacton, and anticompacton solutions in BBM-like equations, highlighting the conditions under which these waves transform or vanish.

Findings

Dissipation converts solitons into undular bores or shocks.

Anticompactons are transformed into undular bores by viscosity.

Compactons tend to vanish due to viscous effects.

Abstract

We examine the effect of dissipation on traveling waves in nonlinear dispersive systems modeled by Benjamin- Bona- Mahony (BBM)-like equations. In the absence of dissipation the BBM-like equations are found to support soliton and compacton/anticompacton solutions depending on whether the dispersive term is linear or nonlinear. We study the influence of increasing nonlinearity of the medium on the soliton- and compacton dynamics. The dissipative effect is found to convert the solitons either to undular bores or to shock-like waves depending on the degree of nonlinearity of the equations. The anticompacton solutions are also transformed to undular bores by the effect of dissipation. But the compactons tend to vanish due to viscous effects. The local oscillatory structures behind the bores and/or shock-like waves in the case of solitons and anticompactons are found to depend sensitively both on the coefficient of viscosity and solution of the unperturbed problem.