  We demonstrate that pulses of linear physical systems, weakly perturbed by
nonlinear dissipation, exhibit soliton-like behavior in fast collisions. The
behavior is demonstrated for linear waveguides with weak cubic loss and for
systems described by linear diffusion-advection models with weak quadratic
loss. We show that in both systems, the expressions for the collision-induced
amplitude shifts due to the nonlinear loss have the same form as the expression
for the amplitude shift in a fast collision between two optical solitons in the
presence of weak cubic loss. Our analytic predictions are confirmed by
numerical simulations with the corresponding coupled linear evolution models
with weak nonlinear loss. These results open the way for studying dynamics of
fast collisions between pulses of weakly perturbed linear physical systems in
an arbitrary spatial dimension.
