Reservoir engineering for classical nonlinear fields

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 paper, 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.