Localized nonlinear matter waves in a Bose-Einstein condensate with spatially inhomogeneous two- and three-body interactions

We investigate the profiles and stability of the localized nonlinear matter waves in a Bose-Einstein condensate with spatially inhomogeneous two-and three-body interactions in various external potentials both analytically and numerically. Several families of localized stationary solutions of the one-dimensional mean-field Gross-Pitaevskii equation with cubic-quintic nonlinearities are obtained by similarity transformation. It is shown that the whole Bose-Einstein condensate can have an infinite number of localized nonlinear matter waves determined by certain non-negative integers. The linear and dynamical stabilities of the localized stationary solutions are studied numerically, and some stable localized nonlinear matter waves are found. We demonstrate that the attractive inhomogeneous two- and three-body interactions support linearly and dynamically stable localized nonlinear matter waves, but attractive two-body interaction along with repulsive three-body interaction destroys the linear stability. Moreover, an anharmonic potential is proposed to stabilize the localized nonlinear matter waves in the Bose-Einstein condensate with spatially inhomogeneous interactions.