Solitons In Higher-order Nonlinear Bose-einstein Condensates

In recent years,nonlinear phenomena have always been one of the most attractive research topics in the field of physical science,and the study of soliton has always been an important research direction in the field of nonlinear science,involving many fields.Up to now,there are more and more theories and researches on solitons,and the content is more and more rich.A large number of scientists devote themselves to solving solitons in nonlinear systems.However,through investigation and research,it is found that most of the current research is limited to low-dimensional and low-order nonlinear systems,and there is less research content for higher-order systems,which are closer to the reality itself,and the results obtained will be more accurate.Therefore,aiming at the current research status,this thesis mainly studies the higher-order system and analyzes the behavior dynamics of solitons in the higher-order nonlinear system.This thesis first introduces the theoretical content and development status of the research object,the soliton,the Bose-Einstein condensate and the ultra-cold atomic vortex.Then,for the high-order nonlinear system in this thesis,we focus on the main equation models and research methods.After the preparation work is completed,we first model the one-dimensional system with high order nonlinear interaction,derive its kinking soliton solution by F-expansion method,and prove the stability and practicability of the solution,at the same time set different parameters,and show its characteristics through graphs.Then,the author studied the seventh-order nonlinear optical system with high-order dispersion effect,established a model based on the typical optical system,updated the basis function,also used the F-expansion method to solve the high-order equation,deduced the bright soliton solution,and demonstrated the bright soliton characteristics through numerical analysis and corresponding diagrams,deduced the analytic form near the boundary of the bright soliton waveform.In the last chapter,we studied the evolution of vortices in a quintic nonlinear system.Based on the coupled fifth-order Gross-Pitaevskii equation model and the variational method,we obtained the vortex solution in this higher-order system.Later,we found that the evolution process of vortices is actually a metastable structure,and the radius of vortices decreases with the increase of time.Periodic respiratory oscillations are generated,and the angular frequency of respiratory oscillations is twice the harmonic capture frequency under infinitesimal nonlinear intensity.At the same time,it is found that the effect of higher order nonlinear term on vibration period is quantitative rather than qualitative.Combined with practical experiments,we analyze the quasi-stable oscillations of the deduced vortex evolution model and illustrate its metastable characteristics graphically.The results obtained in this thesis can be used to guide the further study of the analytical solutions of higher-order nonlinear systems.