Symmetry analysis, conservation laws and exact solutions of certain nonlinear partial differential equations

Abstract

In this research work we study some nonlinear partial differential equations which model many physical phenomena in science, engineering and finance. Closedform solutions and conservation laws are obtained for such equations using various methods. The nonlinear partial differential equations that are investigated in this thesis are; a variable coeffcients Gardner equation, a generalized (2+1)- dimensional Kortweg-de Vries equation, a coupled Korteweg-de Vries-Burgers system, a Kortweg-de Vries{modified Kortweg-de Vries equation, a generalized improved Boussinesq equation, a Kaup-Boussinesq system, a classical model of Prandtl's boundary layer theory for radial viscous ow, a generalized coupled (2+1)-dimensional Burgers system, an optimal investment-consumption problem under the constant elasticity of variance model and the Zoomeron equation. We perform Lie group classi cation of a variable coe cients Gardner equation, which describes various interesting physics phenomena, such as the internal waves in a strati ed ocean, the long wave propagation in an inhomogeneous two-layer shallow liquid and ion acoustic waves in plasma with a negative ion. The Lie group classi cation of the equation provides us with four-dimensional equivalence Lie algebra and has several possible extensions. It is further shown that several cases arise in classifying the arbitrary parameters. Conservation laws are obtained for certain cases. A generalized (2+1)-dimensional Korteweg-de Vries equation is investigated. This equation was recently constructed using Lax pair generating technique. The extended Jacobi elliptic method is employed to construct new exact solutions for this equation and obtain cnoidal and snoidal wave solutions. Moreover, conservation laws are derived using the multiplier method.