  We investigate the dynamics of fast two-pulse collisions in linear
diffusion-advection systems with weak quadratic loss in spatial dimension 2. We
introduce a two-dimensional perturbation method, which generalizes the
perturbation method used for studying two-pulse collisions in spatial dimension
1. We then use the generalized perturbation method to show that a fast
collision in spatial dimension 2 leads to a change in the pulse shape in the
direction transverse to the advection velocity vector. Moreover, we show that
in the important case of a separable initial condition, the longitudinal part
in the expression for the amplitude shift has a simple universal form, while
the transverse part does not. Additionally, we show that anisotropy in the
initial condition leads to a complex dependence of the amplitude shift on the
orientation angle between the pulses. Our perturbation theory predictions are
in very good agreement with results of extensive numerical simulations with the
weakly perturbed diffusion-advection model. Thus, our study significantly
enhances and generalizes the results of previous works on fast collisions in
diffusion-advection systems, which were limited to spatial dimension 1.
