Soliton-defect collisions in the nonlinear Schrödinger equation

We report results of systematic numerical simulations of collisions between a soliton and an attractive defect described by the perturbing term in the nonlinear Schrödinger equation proportional to the δ-function. This model has applications in nonlinear optics and other physical systems. In the parametric range where the defect's strength is of the same order of magnitude as the soliton's amplitude, i.e., the interaction is essentially nonlinear, only two outcomes of the collision are possible: transmission and capture. We have found a border between the corresponding parametric regions. The border is represented in a universal form which does not depend on any free parameter. In this essentially nonlinear regime of the interaction, the soliton keeps its integrity, being transmitted or captured as a whole, practically without emission of radiation. At larger values of the defect's strength, the interaction becomes nearly linear. In this case, the soliton is split into reflected, transmitted, and trapped wave packets. Using the corresponding linear Schrödinger equation, we calculate analytically the shares of the reflected and transmitted energy in the limiting case when the defect's strength and the soliton's velocity are essentially larger than its amplitude.