| In this thesis,the nonlocal residual symmetries of two nonlinear evolution equations with variable coefficients are constructed with the help of the truncated Painlevé analysis and their group invariant solutions are obtained by applying symmetry reduction theory.In addition,the exact solutions of the coupled Lakshmanan-Porsezian-Daniel equations with variable coefficients are constructed by applying the consistent tanh function expansion method.The first chapter focuses on the introduction of symmetry theory and several mathematical-physical methods for studying nonlinear evolution equations covered in this thesis,and describes the basic framework of this thesis.In Chapter 2,we construct the nonlocal symmetries of the variable-coefficient WhithamBroer-Kaup(WBK)equation and the variable-coefficient coupled Lakshmanan-PorsezianDaniel(LPD)equation using the truncated Painlevé analysis.In order to obtain the symmetry reduction solutions with the help of the nonlocal symmetries,the nonlocal residual symmetries of the original equations are localized to the Lie point symmetries of the extended equations by introducing auxiliary field quantities,and then the symmetry reduction method is applied to obtain the similarity reduction solutions.In Chapter 3,the soliton-elliptic periodic wave interaction solutions are derived for the system of coupled Lakshmanan-Porsezian-Daniel equations with variable coefficients using the consistent tanh function expansion method.By choosing different variable coefficients,exact solutions of this equation of different forms are constructed and the dynamical properties of these solutions are shown by evolution diagrams.The last chapter is a summary of this thesis. |
