  We study the amplitude dynamics of two-dimensional (2D) solitons in a fast
collision described by the coupled nonlinear Schr\"odinger equations with a
saturable nonlinearity and 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 dynamics in a fast collision of two 2D solitons. Our perturbative
approach is based on two major steps. The first step is the standard adiabatic
perturbation for the calculations on the energy balance of perturbed solitons
and the second step, which is the crucial one, is for the analysis of the
collision-induced change in the envelope of the perturbed 2D soliton.
Furthermore, we also present the dependence of the collision-induced amplitude
shift on the angle of the two 2D colliding-solitons. In addition, we show that
the current perturbative technique can be simply applied to study the
collision-induced amplitude shift in a fast collision of two perturbed 1D
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.
