Submitting Campus

Daytona Beach

Department

Department of Mathematics

Document Type

Article

Publication/Presentation Date

8-2013

Abstract/Description

In this paper the effect of a small dissipation on waves is included to find exact solutions to the modified Benjamin, Bona, and Mahony (BBM) equation by viscosity. 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 makes the equation integrable in terms of Weierstrass ℘ functions. We will use a general formalism based on Ince's transformation to write the general solution of dissipative BBM in terms of ℘ functions, from which all the other known solutions can be obtained via simplifying assumptions. Using ODE (ordinary differential equations) analysis we show that the traveling wave speed is a bifurcationparameter that makes transition between different classes of waves.

Publication Title

Journal of Mathematical Physics

DOI

https://doi.org/10.1063/1.4817342

Publisher

American Institute of Physics

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