Lie group analysis of nonlinear partial differential equations in multiple space dimensions

Abstract

In this thesis we study the applications of Lie group analysis to certain nonlinear partial differential equations in multiple space dimensions of mathematical physics. Exact solutions and conservation laws are obtained for such equations. The equations which are considered in this thesis are the Krichever-Novikov equation, the (2+ 1 )-dimensional nonlinear Kadomtsov-Petviashivilli-Benjamin-BonaMahony equation, the generalized (2,1)-dimensional nonlinear Zakharov-KuznetsovBenjamin-Bona-Mahony equation , the generalized two-dimensional nonlinear Kadomtsev-Petviashivilli-rnodified equal width equation. the coupled Drinfekl-SokolovSatsuma- Hirota system, the (2+ 1 )-dimensional nonlinear Kadomtsev-PetviashviliJoseph-Egri equation with power law nonlinearity and the Camassa-Holm KadomtscvPetviashvili equation with power law nonlinearity. Lie group classification is performed on the Krichever-Novikov equation. We show that the equation admits a six-dimensional equivalence Lie algebra. It is also shown that the principal Lie algebra is two-dimensional. Several possible extensions of the principal Lie algebra are computed and heir associated symmetry reductions and exact solutions arc obtained using the simplest equation method. Also. conservation laws are derived using the new conservation theorem due to Ibragimov. The (2+ 1 )-dimensional nonlinear Kadomtsov-Petviashivilli-Benjamin-Bona-Mahony equation is studied using Lie group analysis and simplest equation method. Also conservation laws for this equation are constructed by using the multiplier method. The generalized (2+ 1 )-dimensional nonlinear Zakharov-K uznetsov-Benj amin-Bona: Joseph-Egri equation is analysed using Lie group analysis in conjunction with the simplest equation method and the (G' / G)-expansion method to obtain exact solutions. Conserved quantities of the underlying equation are constructed by using two distinct approches, viz., the new conservation theorem due to Ibragimov and the multiplier method. A generalized two-dimensional nonlinear Kadomtsev-Petviashivilli-modified equal width equation is investigated. The symmetries and adjoint representations for this equation are given and an optimal system of one-dimensional subalgebras is derived. The similarity reductions and exact solutions with the aid of ( G' / G)expansion method are obtained based on the optimal systems of one-dimensional subalgebras and conservation laws are constructed by using the multiplier method. The coupled Drinfeld-Sokolov-Satsuma-Hirota system is analyzed using Lie symmetry analysis. The similarity reductions and exact solutions with the aid of simplest equation method and Jacobi elliptic function method are obtained for the coupled Drinfeld-Sokolov-Satsuma-Hirota system. In addition to this, the conservation laws are derived using the new conservation theorem due to Ibragimov and the multiplier method. Lie group analysis and the Exp-function method arc used to carry out the integration of the two-dimensional nonlinear Kadomtsov-Petviashivilli-Joseph-Egri equation with power law nonlinearity. Conservation laws are constructed by using the multiplier method. Finally, we investigate the Camassa-Holm Kadomtsev-Petviashvili equation with power law nonlinearity. Lie group analysis, simplest equation method and the Exp-function method are used to carry out the integration of the underlying equation. Also conservation laws are constructed by using the multiplier method.