Lie group analysis and conserved vectors of multidimensional nonlinear Partial differential equations

Abstract

This thesis studies the applications of Lie group analysis and conserved vectors to multi-dimensional nonlinear partial differential equations. Exact solutions and conservation laws are obtained for such equations. The equations which are considered in this thesis are the generalized Chaffee-Infante equation in (3+1) dimensions, the (2+1)-dimensional combined potential Kadomtsev-Petviashvili-b-type Kadomtsev- Petviashvili equation and a generalized (2+1)-dimensional generalized Korteweg-de Vries equation. The generalized Chaffee-Infante equation with power-law nonlinearity in (3+1) dimensions is analyzed. Ansatz methods are utilized to provide topological and nontopological soliton solutions of this equation. Soliton solutions to nonlinear evolution equations have several practical applications in many areas of mathematical physics such as plasma physics and diffusion process. It will be shown that for certain values of the parameters, the power-law nonlinearity Chaffee-Infante equation permits soliton solutions. The requirements and restrictions for existence of soliton solutions are mentioned. Conservation laws are derived for the aforementioned equation. The Lie symmetry method investigates a (2+1)-dimensional combined potential Kadomtsev-Petviashvili-b-type Kadomtsev-Petviashvili equation. Symmetry reduction is performed and group invariant solutions are obtained. Furthermore, the multiplier method is employed to derive conservation laws for a (2+1)-dimensional combined potential Kadomtsev-Petviashvili-b-type Kadomtsev-Petviashvili equation. Finally a generalized nonlinear (2+1)-dimensional equation is investigated from the view point of symmetry analysis in conjunction with ansatz methods and multipleexp function method. Moreover, conservation laws based on the multiplier approach will be investigated.