Abstract
We find exact solutions to the nonlinear Schr\"odinger equation (NLSE) in the presence of self-steepening and a self-frequency shift. These include periodic solutions and localized solutions of dark-bright type which can be chiral, the chirality being controlled by the sign of the self-steepening term. A form of self-phase-modulation that can be tuned by higher-order nonlinearities as well as by the initial conditions, distinct from the nonlinear Schr\"odinger equation, characterizes these solutions. In certain nontrivial parameter domains, solutions are found to satisfy the linear Schr\"odinger equation, indicating the possibility of linear superposition in this nonlinear system. Dark and bright solitons exist in both the anomalous and normal dispersion regimes, and a duality between the dark-bright type of solution and kinematic higher-order chirping is also seen. Localized kink solutions similar to NLSE solitons, but with very different self-phase-modulation, are identified.
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