  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.
