Weierstrass traveling wave solutions for dissipative BBM equation

TL;DR

This paper investigates how small dissipation affects wave solutions in the BBM equation, demonstrating the existence of various bounded traveling waves and expressing solutions using Weierstrass functions.

Contribution

It introduces a novel approach to find exact dissipative BBM solutions using elliptic functions and analyzes wave speed as a bifurcation parameter.

Findings

Existence of solitary, periodic, and elliptic wave solutions with dissipation.

Solutions expressed in terms of Weierstrass $ ext{wp}$ functions.

Wave speed acts as a bifurcation parameter.

Abstract

In this paper the effect of a small dissipation on waves is included to find exact solutions to the modified BBM equation. Using Lyapunov functions and dynamical systems theory, we prove that when viscosity is added to the BBM equation, in certain regions there still exist bounded traveling wave solutions in the form of solitary waves, periodic, and elliptic functions. By using the canonical form of Abel equation, the polynomial Appell invariant make the equation integrable in terms of Weierstrass $\wp$ functions. We will use a general formalism based on Ince's transformation to write the general solution of dissipative BBM in terms of $\wp$ functions, from which all the other known solutions can be obtained via simplifying assumptions. Using ODE analysis we show that the traveling wave speed is a bifurcation parameter that makes transition between different classes of waves.