Chirped chiral solitons in nonlinear Schrödinger equation with self-steepening and self-frequency shift
We find exact solutions to nonlinear Schrödinger equation in the presence of self-steepening and self-frequency shift. These include periodic solutions and localized solutions of dark-bright type which can be {\emph{chiral}}, and chirality being controlled by sign of self steepening term. A new form of self phase modulation, which can be tuned by higher order nonlinearities as also by the initial conditions, distinct from nonlinear Schrödinger equation, characterizes these solutions. In certain nontrivial parameter domain solutions are found to satisfy {\emph{linear}} Schrödinger equation, indicating possiblity of linear superposition in this nonlinear system. Dark and bright solitons exist in both anomalous and normal dispersion regimes and a duality between 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.