Lie symmetry analysis of certain nonlinear evolution equations of mathematical physics

Abstract

In this work we study the applications of Lie symmetry analysis to certain nonlinear evolution equations of mathematical physics. Exact solutions and conservation laws are obtained for such equations. The equations which are considered in this thesis are a generalized Korteweg-de Vries-Burgers equation, a two-dimensional integrable generalization of the Kaup-Kupershmidt equation, a coupled Korteweg-de Vries system, a generalized coupled variable-coefficient modified Korteweg-de Vries system, a new coupled Korteweg-de Vries system and a new coupled Kadomtsev-Petviashvili system. The generalized Korteweg-de Vries-Burgers equation is investigated from the point of view of Lie group classification. We show that this equation admits a four-dimensional equivalence Lie algebra. It is also shown that the principal Lie algebra consists of a single translation symmetry. Several possible extensions of the principal Lie algebra are computed and their associated symmetry reductions and exact solutions are obtained. The Lie symmetry method is performed on a two-dimensional integrable generalization of the Kaup-Kupershmidt equation. Exact solutions are obtained using the Lie symmetry method in conjunction with the extended tanh method and the extended Jacobi elliptic function method. In addition to exact solutions we also present conservation laws which are derived using the multiplier approach. A coupled Korteweg-de Vries system and a generalized coupled variable-coefficient modified Korteweg-de Vries system are investigated using Lie symmetry analysis. The similarity reductions and exact solutions with the aid of simplest equations and Jacobi elliptic function methods are obtained for the coupled Korteweg-de Vries system and the generalized coupled variable-coefficient modified Korteweg-de Vries system. In addition to this, the conservation laws for the two systems are derived using the multiplier approach and the conservation theorem due to Ibragimov. Finally, a new coupled Korteweg-de Vries system and a new coupled Kadomtsev Petviashvili system are analyzed using Lie symmetry method. Exact solutions are obtained using the Lie symmetry method in conjunction with the simplest equation, Jacobi elliptic function and (G'/G)-expansion methods. Conservation laws are also obtained for both the systems by employing the multiplier approach.