Summetry solutions and conservation laws for certain partial dirrerential equations of mathematical physics

Abstract

In this thesis we study some nonlinear partial differential equations which appear in many physical phenomena of science and engineering. Exact solutions and conservation laws are obtained for such equations using various methods. In this work we study a generalized Benney-Luke equation with time dependent coefficients, the symmetric regularized long wave equation, the Klein-Gordon-Zakharov equations, a generalized (2+ 1 )-dimensional Burgers-Kadomtsev-Petviashvili equation, the Korteweg-de Vries-Burgers equation with power law nonlinearity, a four parameter Boussinesq system, an integrable coupling with Korteweg-de Vries equation, the coupled BBM equations and the convective Cahn-Hilliard equation. A complete Lie group classification is performed on a generalized Benney-Luke equation with time dependent coefficients. This equation models an approximation of the full water wave equation and is formally suitable for describing two-way water wave propagation in the presence of surface tension. The Lie group classification of this equation provides us with a two-dimensional principal Lie algebra and has several possible extensions. Six distinct cases arise in classifying the time dependent coefficients. They include amongst others the power and exponential functions. Group-invariant solutions are obtained for all cases. Exact solutions of two nonlinear evolution equations, namely, the symmetric regularized long wave equation and t he Klein-Gordon-Zakharov Equations are obtained using Lie symmetry analysis along with the simplest equation method and the exp-function method. Conservation laws are constructed using the multiplier approach. The (G'/G)-expansion method is used to obtain solutions of a generalized (2+ 1)dimensional Burgers-Kadomtsev-Petviashvili equation and a (2+ 1)-dimensional integrable coupling system with the Korteweg-de Vries equation. Exact travelling wave solutions of three types are obtained and these are the solitary waves, periodic and rational functions. Conservation laws in both cases are obtained using the new conservation theorem. Travelling wave solutions and conservation laws of the Korteweg-de Vries-Burgers equation with power law nonlinearity are derived via Lie symmetry analysis along with Kudryashov approach and the new conservation theorem, respectively. Conservation laws for a four parameter Boussineq system are derived using the Noether approach and its exact travelling wave solutions are obtained using the Kudryashov method. Exact solutions of the third-order coupled Benjamin-Bona-Mahony equations are derived by employing the Kudryashov method and conservation laws are obtained using the multiplier approach. Conservation laws and travelling wave solutions of the fourth-order Cahn-Hilliard equation are obtained. The conservation laws are constructed using the multiplier approach while Lie symmetry analysis along with the simplest equation method are used to obtain exact solutions.