Dynamical Stability Of Gap Solitons In Quasi-1D Bose-einstein Condensate

Since the Bose Einstein condensate was implemented in experiment,it has become one of the most popular research topics in recent years.With the mean-field approximation theory,the Bose Einstein condensation is described by the Gross-Pitaevskii(GP)equation.As a nonlinear Schr?dinger equation,the research of GP equation is of great value.In the nonlinear periodic systems,the generation of gap solitons makes researchers interested in the study of gap solitons in Bose Einstein condensates.In this paper,based on the GP equation describing the Bose-Einstein condensate,we first linearize the GP equation and obtain the energy band structure.Then we discuss the stable solution of GP equation with Jacobi elliptic sine function potential in Bose-Einstein condensate.By using the Newton-conjugate gradient method,the single-peak and multi-peak gap solitons with periodic potential are obtained numerically.Changing its modulus and keeping the interaction term,observed that under the interaction of repulsion and attraction,the width of the gap soliton becomes smaller as the modulus increases,and the peak value does not change.Then,keeping the modulus unchanged and changing its interaction,it was found that as the repulsive interaction or the attraction interaction increases,the peak value becomes smaller and the width of the gap soliton does not change.Therefore,the change of modulus and interaction has a greater influence on the gap soliton.From the gap soliton,we numerically study the stability of its bandgap soliton.Then,its stability was studied by using linear stability analysis numerical method-Fourier configuration method.In the meantime,we using the dynamic evolution method of nonlinear equations to analyze the stability of the gap soliton.Through the study,we observed that the stability results of gap solitons are consistent with these methods,repulsive interactions and attractive interactions have different stability for gap solitons,moreover,the stability of the gap soliton gradually becomes stable as the modulus increases.In addition,we also found that the unimodal gap soliton has been in a stable state under the conditions we studied.