Conservation laws and exact solutions for some nonlinear partial differential equations

Abstract

In this thesis we study some nonlinear partial diffrerential equations which appear in several physical phenomena of the real world. Exact solutions and conservation laws are obtained for such equations using various methods. The equations which are studied in this work are a generalized coupled (2+1)-dimensional hyperbolic system, a modi ed Kortweg-de Vries type equation, the higher-order modifi ed Boussinesq equation with damping term, coupled Korteweg-de Vries equations, coupled Boussinesq equations, a generalized Zakharov-Kuznetsov equation, a generalized Ablowitz-Kaup-Newell-Segur equation and a potential Kadomtsev- Petviashvili equation with p-power nonlinearity. We perform a complete Lie symmetry classi fication of a generalized coupled (2+1)- dimensional hyperbolic system, which models many physical phenomena in nonlinear sciences. The Lie group classi fication of the system provides us with elevendimensional equivalence Lie algebra and has several possible extensions. It is further shown that several cases arise in classifying the arbitrary parameters, the forms of which include amongst others the power and exponential functions. We obtain exact solutions of two nonlinear evolution equations, namely, modi ed Kortweg-de Vries equation and higher-order modifi ed Boussinesq equation with damping term. The (G'/G)–expansion method is employed to obtain the exact solutions. Travelling wave solutions of three types are obtained and these are the solitary waves, periodic and rational. In addition, the conservation laws for higher order modi fied Boussinesq equation with a damping term are constructed using the multiplier approach. The (G'/G)–expansion method is employed to derive the exact travelling wave solutions of coupled Korteweg-de Vries equations. The solutions obtained include the soliton solutions. Furthermore, the conservation laws for these equations are obtained. Travelling wave solutions of coupled Boussinesq equations are determined and conservation laws are obtained for the system using the new conservation theorem and multiplier approach. We study a generalized Zakharov–Kuznetsov equation in three variables, which has applications in the nonlinear development of ion-acoustic waves in a magnetized plasma. Conservation laws for this equation are constructed using the new conservation theorem. Furthermore, new exact solutions are obtained by employing the Lie symmetry method along with the simplest equation method. Conservation laws of a generalized Ablowitz-Kaup-Newell-Segur equation are constructed by using Noether theorem. The exact solutions are obtained using the Lie symmetry method together with the simplest equation method and direct integration. Finally, a potential Kadomtsev-Petviashvili equation with p–power nonlinearity, which arises in a number of signi cant nonlinear problems of physics and applied mathematics is studied. We carry out Noether symmetry classi cation on this equation. Four cases arise depending on the values of p and consequently we construct conservation laws for these cases with respect to the second-order Lagrangian. In addition, exact solutions for this equation are obtained using the Lie group analysis together with the Kudryashov method and direct integration.