Damped solitons in an extended nonlinear Schrödinger equation with a spatial stimulated Raman scattering and decreasing dispersion

Dynamics of solitons is considered in the framework of an extended nonlinear Schrödinger equation (NLSE), which is derived from a system of the Zakharov׳s type for the interaction between high- and low-frequency (HF and LF) waves. The resulting NLSE includes a pseudo-stimulated-Raman-scattering (pseudo-SRS) term, i.e., a spatial-domain counterpart of the SRS term, which is a known ingredient of the temporal-domain NLSE in optics. Inhomogeneity of the spatial second-order dispersion (SOD) and linear losses of HF waves was also included. It is shown that wavenumber downshift by the pseudo-SRS may be compensated by the upshift provided by SOD whose local strength is an exponentially decaying function of the coordinate. An analytical soliton solution with a permanent shape is found in an approximate form, and is verified by the comparison with numerical results.