On the second-order approximate symmetry classification and optimal systems of subalgebras for a forced Korteweg–de Vries equation

Abstract

We investigate a forced Korteweg–de Vries (fKdV) equation, u,t+cu,x+αuu,x+βu,xxx=βF(t), which arises in the modelling of tsunami generation by submarine landslides. Approximate symmetries are found for the fKdV equation using the method as proposed by Fushchich and Shtelen [6]. Symmetries are found to second order in the perturbation parameter using the MAPLE symmetry package ASP[11], an add-on to the symmetry package DESOLVII[18]. ASP also allows particular forms of the arbitrary function F(t) to be found which extend the symmetry algebra and hence a full approximate symmetry classification of the fKdV equation is obtained. Optimal systems of one-dimensional subalgebras are also determined. Corresponding approximate invariant solutions to the fKdV equation are then constructed for particular F(t) using DESOLVII routines.