Solitons in the nonlinear Schrödinger equation with the pseudo-Raman term, anti-diffusion gain and nonlinear loss
We introduce a nonconservative nonlinear Schrödinger equation (NLSE) which includes additional nonlinear terms accounting for the pseudo-stimulated-Raman-scattering (pseudo-SRS) effect and cubic loss. An essentially novel feature of the model is the linear anti-diffusion gain, which leads to the obvious blow-up in the infinite or spatially periodic system. We consider the absorptive boundary conditions, which suppress the blow-up driven by effective convective instability in the setup. The NSLE model is derived, in the extended adiabatic approximation, from the underlying Zakharov system, which includes the viscosity term acting on its low-frequency component and the anti-diffusion gain in the high-frequency (HF) one. The latter term appears if the expansion, with respect to the deviation of the wavenumber, is performed not around the minimum-loss point (as in the usual situation), but around a maximum-loss point. Dynamics of moving solitons in the model is addressed by means of analytical and numerical methods. The sech approximation for the soliton predicts it as a stable fixed point (FP) of the system of balance equations for the HF norm and wave momentum, derived in the moving reference frame. The FP attraction basin is identified too. The predictions are corroborated by direct simulations of the full NSLE with the absorptive boundary conditions.