Solitary waves, periodic and elliptic solutions to the Benjamin, Bona & Mahony (BBM) equation modified by viscosity

TL;DR

This paper investigates solitary and periodic wave solutions of the viscous-modified BBM equation, revealing bifurcation phenomena and providing explicit solutions using elliptic functions and dynamical systems analysis.

Contribution

It introduces a comprehensive analysis of viscous effects on BBM wave solutions, including explicit elliptic function solutions and bifurcation analysis based on dynamical systems theory.

Findings

Existence of periodic and solitary wave solutions in terms of Jacobi elliptic functions.

Identification of a transcritical bifurcation depending on wave velocity.

Numerical results supporting the analytical findings.

Abstract

In this paper, we use a traveling wave reduction or a so-called spatial approximation to comprehensively investigate periodic and solitary wave solutions of the modified Benjamin, Bona & Mahony equation (BBM) to include both dissipative and dispersive effects of viscous boundary layers. Under certain circumstances that depend on the traveling wave velocity, classes of periodic and solitary wave like solutions are obtained in terms of Jacobi elliptic functions. An ad-hoc theory based on the dissipative term is presented, in which we have found a set of solutions in terms of an implicit function. Using dynamical systems theory we prove that the solutions of \eqref{BBMv} experience a transcritical bifurcation for a certain velocity of the traveling wave. Finally, we present qualitative numerical results.