The study of the stability of 1D confined Bose–Einstein condensates based on one novel orthogonal basis

One novel orthogonal basis of the 1D Gross–Pitaevskii equation with square well, describing the properties of the nonuniform Bose–Einstein condensates, was introduced. Based on this orthogonal basis, we investigated the dynamical stability of the ground state and the first excited state for the nonuniform condensates. We found that the ground state is stable even for large negative nonlinear interaction. But the first excited state performs dynamical instability when the nonlinear interaction is over the critical value. Our numerical results support our analytical analysis. Some underlying physics are discussed.