Darboux Transformations And Nonlinear Wave Solutions Of Generalized Nonlinear Schr(?)dinger-type Equations

The nonlinear Schr(?)dinger equation is an important integrable model in the field of mathematical physics.The equation can be applied to the fields of fluid dynamics,nonlinear optics,Bose-Einstein condensates,plasma physics,physical oceans and even financial mathematics.However,in order to better describe the theoretical mechanisms of the relevant physical phenomena,the effects of higher-order dispersion terms with nonlinear terms must be considered.In this thesis,we focus on the nonlinear wave solutions of generalized Schr(?)dinger-like equations containing higher-order dispersion terms and nonlinear terms and their dynamical properties,and extend the related theory to the nonlocal case.The main contents are organized as follows:In the first chapter,the research background of soliton is introduced,the theoretical basis and basic steps of the Darboux transformation are outlined,and the physical background and research significance of the main research model and the research object of this thesis are given.In the second chapter,the integrable reverse space-time nonlocal Lakshmanan-P orsezian-Daniel equation is proposed based on symmetry reduction,the first-order D arboux transformation of the equation and the N-order Darboux transformation are c onstructed,and nonlinear wave solutions such as solitons,complexiton and rogue wa ve solutions of the equation are constructed based on zero seed solutions and plane wave solution,and their dynamical properties are analyzed based on symbolic calcul ations.In the three chapter,the integrable reverse space-time nonlocal Fifth-order nonlinear Schr(?)dinger equation is constructed based on symmetry reduction,and its Nth-order Darboux transformation is constructed based on the Lax pair of the equation.The nonlinear wave solutions such as soliton,complexiton and rogue wave solutions of the equation are obtained using the zero seed solution and plane wave solution of the equation,and the dynamical properties of the solution are analyzed.In the fourth chapter,The main work of this thesis is summarized and an outlook on the upcoming work is provided.