Study On Generalized Darboux Transformation And Localized Wave Solutions For Several Nonlinear Evolution Equations

The nonlinear evolution equation plays an important role in nonlinear science fields such as fiber optic communication,molecular biology,and electromagnetism.It’s frequently used to describe complicated nonlinear phenomena.The essential law of the equation can be explored by studying the analytical solution of the nonlinear evolution equation,and the natural phenomena can be deeply understood in life.With the deepening of the research,the multi-component coupled nonlinear development equation can better demonstrate the motion law of microscopic particles,and it can obtain more diversified analytical solutions.In this paper,localized wave solutions of several nonlinear evolution equations are derived via generalized Darboux transformation.Three-dimensional evolution plots are drawn by numerical simulation,and their dynamical characteristics are explored.The following are the specific research contents:In Chapter 1,the research background and significance of nonlinear evolution equations,the research progress of nonlinear localized waves and generalized Darboux transformation are introduced,and the structure and main work of this paper are described.In Chapter 2,the coupled fourth-order nonlinear Schr?dinger equation in birefringent fiber is studied.Select the plane wave solutions are the seed solutions,and the Nth-order localized wave solutions of the equation are obtained through iteration via generalized Darboux transformation.Based on the numerical simulation,the third-order localized wave evolution plots are obtained,and the dynamical characteristics of the interaction between the third-order rogue wave and the three bright and dark solitons and the three rows of breathers are analyzed.The results show that the localized wave solutions of the free parameters,the separation function and the higher-order linear and nonlinear effect intensity coefficients play a decisive role in the propagation of the localized wave in birefringent optical fiber.In Chapter 3,based on the three-coupled fourth-order nonlinear Schr?dinger equation in the α-helix protein,the zero solutions are selected as the seed solutions.The recursive expression of the Nth-order soliton solution of the equation is derived via generalized Darboux transformation.Different spectral parameters and free parameters are selected for numerical simulation,and the dynamical evolution plots of the four solitons interaction are obtained.The dynamical characteristics of elastic collision,inelastic collision and the bound state among the four solitons are further analyzed.The results show that the biological energy is exchanged during the collision of the four solitons to achieve energy transfer.In Chapter 4,based on the coupled variable-coefficient Hirota equation in inhomogeneous optical fiber and its Lax pair,the plane wave solutions are selected as the seed solutions.The Nth-order localized wave solutions of the equation are constructed via generalized Darboux transformation.A series of novel dynamical evolution plots are obtained by numerical simulation.The results show that the coefficients of group velocity dispersion,the nonlinear terms referring to self-phase modulation and cross-phase modulation,and the third-order dispersion have effects on the propagation shape,propagation period,and propagation velocity of localized waves in inhomogeneous optical fiber.The final chapter summarizes the content of this study and looks forward to the future research work and direction.