  Reservoir engineering has become a prominent tool to control quantum systems.
Recently, there have been first experiments applying it to many-body systems,
especially with a view to engineer particle-conserving dissipation for quantum
simulations using bosons. In this work, we explore the dissipative dynamics of
these systems in the classical limit. We derive a general equation of motion
capturing the effective nonlinear dissipation introduced by the bath and apply
it to the special case of a Bose-Hubbard model, where it leads to an
unconventional type of dissipative nonlinear Schr\"odinger equation. Building
on that, we study the dynamics of one and two solitons in such a dissipative
classical field theory.
