Interaction of solitons with saturable gain

We study the tail-overlap interaction of a well-separated pair of dissipative solitons in systems with slow saturable gain. Gain saturation introduces nonlocal terms in the soliton field equation that depend on the total energy and therefore invalidate the standard methods used to study overlap interactions. We solve this problem by making the overlap interactions local in a doubled phase space and use this approach to calculate the drift velocities of interacting Swift-Hohenberg solitons. We derive explicit equations of motion for the soliton position with both fixed and saturable gain for reflection-symmetric and asymmetric solitons. We show that gain saturation significantly affects the drift velocity of the pair as a whole, while relative drift is insensitive to this effect. Consequently, the interactions of reflection-symmetric solitons with fixed and saturable gain are identical, while gain saturation increases the overall drift velocity of a pair of asymmetric solitons by an exponentially large factor. The theoretical equations of motion are exact in the limit of large separations and are verified by comparison with numerical simulations of the field equation.