Equilibrium and nonequilibrium solitons in a lossy split-step system with lumped amplification

We propose a more realistic version of the recently introduced split-step model (SSM), which consists of periodically alternating dispersive and nonlinear segments, by adding uniformly distributed loss and lumped gain to it. In the case when the loss is exactly balanced by gain, a family of stable equilibrium solitons (ESs) is found. Unless the system's period L is very small, saturation is observed in the dependence of the amplitude of the established ES vs. that of the initial pulse. Stable nonequilibrium solitons (NESs) are found in the case when the gain slightly exceeds (by up to ≃ 3%) the value necessary to balance the loss. The existence of NESs is possible as the excessive energy pump is offset by permanent radiation losses, which is confirmed by computation of the corresponding Poynting vector. Unlike ESs that form a continuous family of solutions, NES is an isolated solution, which disappears in the limit L → 0, i.e., it cannot be found in the overpumped nonlinear Schrödinger equation. Interactions between ESs turn out to be essentially the same as in SSM without loss and gain, while interactions between NESs are different: two NESs perturb each other by the radiation jets emanating from them, even if they are separated by a large distance. Moving NESs survive collisions, changing their velocities.