Collision-induced amplitude dynamics in fast 2D solitons in the presence of the generic nonlinear loss

We study the dynamics of two-dimensional (2D) solitons and the fast collisions of two 2D solitons for the nonlinear Schr\"odinger equations in the presence of the generic weak nonlinear loss. We extend the perturbative technique for calculating the collision-induced dynamics of two one-dimensional (1D) solitons to derive the theoretical expression for the collision-induced amplitude shift in a fast collision of two 2D solitons. Our perturbative approach is based on the standard adiabatic perturbation theory for solitons and the calculations on the energy balance of perturbed solitons due to the generic nonlinear loss. The expression is set up for two types of potentials: the periodic and non-periodic potential. Furthermore, we also present the dependence of the collision-induced amplitude shift on the angle of the two 2D colliding-solitons. Our analytic calculations are confirmed by numerical simulations with the corresponding coupled nonlinear Schr\"odinger equations in the presence of the cubic loss and in the presence of the quintic loss.