Solution to the initial value problem for the quantum nonlinear Schrüdinger equation

The quantum nonlinear Schrödinger equation, which models the propagation of light through dispersive nonlinear waveguides, provides an integrable quantum field theory that has been solved by a number of methods. Most recently, Gutkin [J. Func. Anal. 77, 327 (1988)] has developed an interwining operator technique for obtaining the time evolution of the field operators. Using Gutkin’s formalism, we show how to obtain the exact time evolution of an initial Schrödinger state vector. As an example, the two-particle case is carried out explicitly. Numerical results for the time evolution of the position probability distribution are presented for the case when the two particles initially occupy the same Gaussian wave packet.