Gap solitons and their stabilities in a quasi one-dimensional Bose-Einstein condensate under three-body interaction

We study the gap solitons and their stability properties in a Bose-Einstein condensation (BEC) under three-body interaction loaded in a Jacobian elliptic sine potential, which can be described by a cubic-quintic Gross-Pitaevskii equation (GPE) in the mean-field approximation. Firstly, the GPE is transformed into a stationary cubic-quintic nonlinear Schrodinger equation (NLSE) by the multi-scale method. A class of analytical solution of the NLSE is presented to describe the gap solitons. It is shown analytically that the amplitude of the gap soliton decreases as the two-body or three-body interaction strength increases. Secondly, many kinds of gap solitons, including the fundamental soliton and the sub-fundamental soliton, are obtained numerically by the Newton-Conjugate-Gradient (NCG) method. There are two families of fundamental solitons: one is the on-site soliton and the other is the off-site soliton. All of them are bifurcated from the Bloch band. Both in-phase and out-phase dipole solitons for off-site solitons do exist in such a nonlinear system. The numerical results also indicate that the amplitude of the gap soliton decreases as the nonlinear interaction strength increases, which accords well with the analytical prediction. Finally, long-time dynamical evolution for the GPE is performed by the time-splitting Fourier spectrum method to investigate the dynamical stability of gap solitons. It is shown that the on-site solitons are always dynamically stable, while the off-site solitons are always unstable. However, both stable and unstable in-phase or out-phase dipole solitons, which are not bifurcated from the Bloch band, indeed exist. For a type of out-phase soliton, there is a critical value \begin{document}$ q_c$\end{document} when the chemical potential μ is fixed. The solitons are linearly stable as \begin{document}$ q>q_c$\end{document} , while they are linearly unstable for \begin{document}$ q . Therefore, the modulus q plays an important role in the stability of gap solitons. One can change the dynamical behavior of gap solitons by adjusting the modulus of external potential in experiment. We also find that there exists a kind of gap soliton, in which the soliton is dynamically unstable if only the two-body interaction is considered, but it becomes stable when the three-body interaction is taken into account. This indicates that the three-body interaction has influence on the stability of gap solitons.