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

We investigate a forced Korteweg-de Vries (fKdV) equation, u(,t) + cu(,x) + alpha uu(,x) + beta u(,xxx) = beta F(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. (C) 2013 Elsevier B.V. All rights reserved.