Collective excitations of a 1D Bose-Einstein condensate in an anharmonic trap

We investigate the collective excitations of a one-dimension Bose-Einstein condensate trapped in an anharmonic potential by solving the time-dependent Tonks-Girardeau equation. The governing equations of motions for the low-energy excitations are obtained using variational approaches. The motion of a 1D BEC in a harmonic trap is just like the motion of one particle in a harmonic trap. And quartic distortion of the potential causes the blue-shift and red-shift on the excitation frequency while cube distortion only causes the red-shift