Friday, February 22, 2013

1302.5408 (Stefan C. Mancas et al.)

Weierstrass solutions for dissipative BBM equation    [PDF]

Stefan C. Mancas, Greg Spradlin, Harihar Khanal
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. We will use a general formalism based on Ince's transformation to write the general solution of dissipative BBM in terms of Weierstrass $\wp$ 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.
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