Study On Soliton Dynamics Models With High-order Nonlinear Interactions

After quantum mechanics and relativity theory,nonlinear science is another important discovery in the field of natural science.Among them,the soliton theory is an interesting branch of the field,the research of the soliton theory has been the focus of scientists in recent years.It is well known that there are stable soliton solutions in the traditional nonlinear Schr(?)dinger equation(NLSE)or the cubic nonlinear Gross-Pitaevskii equation(GPE).And the propagation of optical solitons in optical fibers satisfies NLSE.In this thesis,we study the soliton dynamics from a theoretical perspective.Since the one-dimensional case of NLSE is the most classical,the theoretical solvable models are mostly concentrated in the one-dimensional case,while the high-dimensional integrability is rarely discussed.In this thesis,based on the self-similarity method and the obtained soliton solutions of one-dimensional power-law nonlinear systems,the typical bright soliton solutions of three-dimensional power-law nonlinear quantum systems are derived and their stability is analyzed,which explained the typical bright soliton characteristics.The results of this theory can be used to guide the observation of soliton behavior in high-dimensional power-law nonlinear systems.Recently,it has been shown that stable soliton behavior also exists in high-dimensional systems with high-order nonlinear interactions,such as stable optical soliton operating in a hollow optical cable filled with cold atom gas.Since ultracold atomic system modulation is flexible and relatively easy,it has became the hot spot of the research of the cold atomic physics theory.In this thesis,we investigate the soliton dynamics of two-dimensional Bose-Einstein Condensates(BEC)with fifth-order nonlinear particle interactions.Based on the Gross-Pitaevskii equation model,we first derive the bright soliton solution of the system by using the self-similarity method and the modified variational method.We find that the kinematic quantities obtained by the two methods are very consistent.Finally,according to the bright soliton solution obtained,the dynamics of the formation of the two-dimensional sonic horizon is calculated,and the periodic formation and evolution of the two-dimensional sonic horizon are quantitatively analyzed and illustrated.The results obtained can be used to guide the observation of the sonic black hole related phenomena of the quintic-order nonlinear equation in two-dimensional Bose-Einstein condensates.Our theoretical results will lay a foundation for the further study of two-dimensional simulated sonic black holes in the two-dimensional quantum system described by GPE model,such as Hawking radiation.The results can also be used to guide the experimental observation of soliton behavior and related layer formation in a two-dimensional BEC with quintic-order nonlinearity.