Waves in Quantum Systems with Nonlinearity and Walls

The nonlinear cubic Shrödinger equation is integrated numerically for quantum systems confined by potential walls of a well. If nonlinear potential describing nonlinearity is distributed on the whole width of the well, soliton generation is possible. For the nonlinear potential distributed on the right half of the well with free motion on the left one the propagation of waves occurs. Solutions for the probability density, expectation positions and velocities of the wave were analyzed under different initial conditions including the Gaussian and specified wave packets.