Symmetries, solutions and conservation laws of a class of nonlinear dispersive wave equations

• We study a damped externally excited Korteweg-de Vries equation with a forcing term.
• We derive the classical Lie symmetries admitted by the equation.
• Conservation laws are established from the property of nonlinear self-adjointness.
• Some exact solutions are also obtained.

In this paper we consider a damped externally excited Korteweg-de Vries (KdV) equation with a forcing term. We derive the classical Lie symmetries admitted by the equation. We then find that the damped externally excited KdV equation has some exact solutions which are periodic waves and solitary waves. These solutions are derived from the solutions of a simple nonlinear ordinary differential equation. By using a general theorem on conservation laws and the multiplier method, we construct some conservation laws for some of these partial differential equations.