Some conservation laws for a forced KdV equation

Abstract

The Korteweg–de Vries equation with a forcing term is established by recent studies as a simple mathematical model by which the physics of a shallow layer of fluid subject to external forcing is described. It serves as an analytical model of tsunami generation by sub-marine landslides. In this paper, we derive the classical Lie symmetries admitted by the forced KdV equation. Looking for travelling wave solutions we find that the forced KdV equation has abundant exact solutions that can be expressed in terms of the Jacobi elliptic functions. Hence the Korteweg–de Vries equation with a forcing term has plenty of periodic waves and solitary waves. These solutions are derived from the solutions of a simple nonlinear ordinary differential equation. The first author has introduced the concept of weak self-adjoint equations. This definition generalizes the concept of self-adjoint and quasi self-adjoint equations that were introduced by Ibragimov in 2006. In a previous paper we found a class of weak self-adjoint forced Korteweg–de Vries equations which are neither self-adjoint nor quasi self-adjoint. By using a general theorem on conservation laws proved by Ibragimov, the Lie symmetries and the new concept of weak self-adjointness we find some conservation laws for some of these partial differential equations.