Study On Novel Steady State Solitons In Higher Order Nonlinear Systems

Nonlinear phenomena are ubiquitous in the physical world.Nonlinear equations are also an indispensable theoretical model in many related fields.It is of great significance to explore the soliton solutions of nonlinear equations.Up to now,the soliton theory has become more and more perfect after continuous development.A large number of researchers have devoted themselves to solving the soliton solutions of nonlinear equations.However,the analytical models are mostly based on one-dimensional low-order cases,and there are relatively few studies on the Soliton solutions of high-order nonlinear equations.To solve this problem,novel Solitons in higher-order nonlinear systems are studied in this thesis.Firstly,the F-expansion method is used to solve the(1 + 1)dimensional Schrodinger equation with high-order dispersion and nonlinear interaction.Through appropriate parameter settings,the analytical solution with typical kink soliton characteristics is obtained,and its stability is verified and analyzed by adding disturbance and G’ G method.The theoretical results can be used to guide the observation of kink solitons in higher-order dispersion and higher-order nonlinear interaction systems.At the same time,this thesis studies the three-dimensional quantum system with high-order dispersion and nonlinear interaction.Based on the self-similarity method and the bright soliton solution of the(1 + 1)dimensional nonlinear Schrodinger equation,the bright soliton solution of the(3+ 1)dimensional nonlinear Schrodinger equation under high-order dispersion and high-order nonlinear interaction is derived,and the typical characteristics of the bright soliton of the three-dimensional system are shown,Finally,the velocity analysis of the system shows that the waveform of bright solitons will not change with time.At the same time,the kink soliton solution and bright soliton solution are obtained by solving the system with high-order dispersion and high-order nonlinear interaction.Then the soliton behavior in the acoustic horizon is studied.Firstly,based on the Fermi system of ultracold atom,the evolution of the system is deduced and calculated by using the one-dimensional Gross-Pitaevskii equation in the elongated simple harmonic trap and the Feshbach resonance and variational method,The equation describing the condition of the acoustic horizon and the corresponding Hawking radiation temperature is obtained.Then,the results of the previous numerical analysis are compared with the theoretical results obtained in this thesis,and the consistent results are obtained,which also shows the applicability of the theoretical results obtained in this thesis.Aiming at the theoretical expansion of the research problem,this thesis analyzes the introduction of high-order nonlinear interaction effect into the quantitative calculation of sonic horizon dynamics under simple harmonic confinement potential.Based on one-dimensional Gross-Pitaevskii equation model,the judgment equation of sonic horizon is derived by using Variational method,and the influence of high-order nonlinear interaction on the distribution width of the system is analyzed.The results of numerical simulation are compared with the results of numerical simulation with and without high-order nonlinear interaction term.Our results show that the introduction of high-order nonlinear interaction is more consistent with the results of numerical simulation than without high-order nonlinear interaction.The research results of this thesis can guide the experimental observation of related phenomena of sonic black holes in high-order nonlinear systems.