Scattering of Solitons and Dark Solitons by Potential Walls in the   Nonlinear Schroedinger Equation
  Scattering of solitons and dark solitons by potential walls is studied in the nonlinear Schroedinger equation under various conditions. We investigate the conditions under which solitons are split into two solitons at the potential wall. We find that a soliton can be trapped in an interspace between two potential walls. A dark soliton can also be scattered at the potential wall. Similarly to a bright soliton, a dark soliton can pass through more easily the potential wall, as the width of the dark soliton is larger. A dark soliton can run away spontaneously from an interspace between the two potential walls. We also study the motion of a two-dimensional soliton in a two-dimensional quintic nonlinear Schroedinger equation. We find the coherent tunneling through a potential wall, and the refraction corresponding to Newton's refraction theory. 
Hidetsugu Sakaguchi and Mitsuaki Tamura