Evolution of a damped soliton in a higher-order nonlinear Schrödinger equation

Evolution of a soliton under the action of a small dissipative term, or the term accounting for the so-called intra-pulse Raman scattering, is analyzed within the framework of the recently discovered exactly integrable nonlinear Schrödinger equation including higher-order dispersive terms. In this equation, the one-soliton solution can be one-humped or two-humped. It is demonstrated analytically that, depending on the initial state, the Raman term can transform the one hump into two, and vice versa. In the case of the usual dissipative term, the final state is always with one hump.