Travelling wave solutions for nonlinear dispersive equations with dissipation

Abstract

It is shown that in order to obtain travelling wave solution many nonlinear dispersive equations with dissipative terms can be reduced by means of elementary transformations to the 1st order Abel o.d.e., and consequently, to the Emden-Fowler equation, if both nonlinear and dissipative terms are polynomials. These reductions can be integrated in closed form in terms of the Weierstrass elliptic function p, containing kink solutions as appropriate limits. Additional conditions for the equation coefficients and wave parameters are established for the wave existence, and their physical meaning is analysed. Some of nonlinear o.d.e., particularly, of higher order, do not provide a reduction to the Abel equation, therefore a different algorithm is proposed to obtain some exact solutions in terms of elliptic p-function.

Keywords:

• nonlinear wave equations
• travelling waves
• solitons
• exact solutions
• the Abel differential equation
• the Weierstrass elliptic function