Coherent patterns and self-induced diffraction of electrons on a thin   nonlinear layer
  Electron scattering on a thin layer where the potential depends self-consistently on the wave function has been studied. When the amplitude of the incident wave exceeds a certain threshold, a soliton-shaped brightening (darkening) appears on the layer causing diffraction of the wave. Thus the spontaneously formed transverse pattern can be viewed as a self-induced nonlinear quantum screen. Attractive or repulsive nonlinearities result in different phase shifts of the wave function on the screen, which give rise to quite different diffraction patterns. Among others, the nonlinearity can cause self-focusing of the incident wave into a ``beam'', splitting in two ``beams'', single or double traces with suppressed reflection or transmission, etc. 
O.M. Bulashenko, V.A. Kochelap, and L.L. Bonilla