Critical soliton speed for quantum reflection by a reflectionless potential well

We consider the scattering of the nonlinear Schr\"odinger equation bright soliton by a reflectionless P\"oschl-Teller potential well. We show that at the sharp transition between quantum reflection and full transmission, a single-node bound state localized at the center of the potential forms and is fully occupied. The profile and energy of the trapped mode are calculated both numerically and analytically using a variational calculation. The critical speed for quantum reflection is then determined from a delicate balance between the energy of the incident soliton and the energy of the trapped mode. We investigate the stability of the trapped mode against perturbations in its profile and position which explains the sharpness of the transition and sheds some light on the physics of quantum reflection. We show that quantum reflection by exciting the multinode trapped modes is also possible.