Study On The Formation Of Sonic Black Hole In Ultracold Atomic Gas With High-Order Nonlinear Interactions

There are many incredible physical phenomena in physics,in which Bose-Einstein condensation is among the most fascinating ones.In this thesis,we theoretically study the evolution of vortex solitons which is the typical nonlinear phenomena in Bose-Einstein condensates and the formation of sonic black hole in ultracold Fermi gas.Using a two-dimensional coupled Gross-Pitaevskii equation model and variational method,we firstly derived the analytical vortex soliton solution for the dilute component and determined that,under a certain parametric setting,the system's distribution width,which is a time-dependent parametric function,oscillated periodically such that the vortex ring showing up in the system expands and contracts alternately and stays in a quasi-stable dynamic state.The results of this theoretical research will play a key role in the observation of vortex solitons in coupled ultracold atomic systems.With the successful application of Feshbach resonance technology on the experiment,the strength of interaction in condensation system can be modulated continuously from minus infinity to positive infinity.We found that higher-order corrections are indispensable elements of the GPE even at the mean-field level.As a result,we analytically derived the solitary vortex solution of two-dimensional Bose-Einstein condensate via variational approach and a two-dimensional GPE incorporating the higher-order nonlinear effect and the effects of higher-order nonlinear interactions on the dynamics of solitary vortex were analyzed.We found that the circular symmetric vortex is not qualitatively modified by higher-order nonlinear effects if the nonlinear interactiom strength is not high.For asymmetric vortex evolution,we found that,the evolutionary pattern obtained using higher-order nonlinear corrections coincides better with that generated by a purely numerical simulation than the pattern obtained using only leading-order nonlinear interactions,demonstrating the necessity of incorporating the higher-order nonlinear term.We have made a lot of analytical derivations and numerical simulations in the ultracold atomic system with higher-order nonlinear interactions,and finally observed acoustic analogues of event horizon in quantized superflow,which opens a new perspective for the investigation of Hawking radiation and event horizon in astrophysics.In this paper,we study the sonic horizon formation dynamics in Bose-Einstein condensate systems with nonlinear interaction modulation based on the Gross-Pitaevskii equation incorporating higher-order nonlinear term.Through the variational method,we derive the evolution equation for the BEC mass distribution parameters from which the criteria formula of sonic horizon occurrence is derived through the difference formula of the system flow speed and sound speed.The dynamical “potential” curve on behalf of sonic horizon typical occurrence is compared with that where higher-order nonlinear interaction is not considered.We identify the principal stabilization effect of the nonlinear interaction for the system oscillation mode at leading and quintic order nonlinearity,which is crucial for the occurrence of the sonic horizon.In addition,we identify that the repulsiveitive higher-order nonlinear interaction tends to stabilize the oscillation mode of the system,while the attractive interaction tends to make the evolution mode of the sonic black hole metastable.