Lie Group Analysis of a Nonlinear Coupled System of Korteweg-de Vries Equations

In this paper, we consider coupled Korteweg-de Vries equations that model the propagation of shallow water waves, ion-acoustic waves in plasmas, solitons, and nonlinear perturbations along internal surfaces between layers of different densities in stratified fluids, for example propagation of solitons of long internal waves in oceans. The method of Lie group analysis is used to on the system to obtain symmetry reductions. Soliton solutions are constructed by use of a linear combination of time and space translation symmetries. Furthermore, we compute conservation laws in two ways that is by multiplier method and by an application of new conservation theorem developed by Nail Ibragimov.

ReLie: A Reduce Program for Lie Group Analysis of Differential Equations

Lie symmetry analysis provides a general theoretical framework for investigating ordinary and partial differential equations. The theory is completely algorithmic even if it usually involves lengthy computations. For this reason, along the years many computer algebra packages have been developed to automate the computation. In this paper, we describe the program ReLie, written in the Computer Algebra System Reduce, since 2008 an open source program for all platforms. ReLie is able to perform almost automatically the needed computations for Lie symmetry analysis of differential equations. Its source code is freely available too. The use of the program is illustrated by means of some examples; nevertheless, it is to be underlined that it proves effective also for more complex computations where one has to deal with very large expressions.