We study the higher-order generalized nonlinear Schrödinger (NLS) equation describing the propagation of ultrashort optical pulse in optical fibres. By using Darboux transformation, we derive the superregular breather solution that develops from a small localized perturbation. This type of solution can be used to characterize the nonlinear stage of the modulation instability (MI) of the condensate. In particular, we show some novel characteristics of the nonlinear stage of MI arising from higher-order effects: (i) coexistence of a quasi-Akhmediev breather and a multipeak soliton; (ii) two multipeak solitons propagation in opposite directions; (iii) a beating pattern followed by two multipeak solitons in the same direction. It is found that these patterns generated from a small localized perturbation do not have the analogues in the standard NLS equation. Our results enrich Zakharov’s theory of superregular breathers and could provide helpful insight on the nonlinear stage of MI in presence of the higher-order effects.
Statistical Properties of the Nonlinear Stage of Modulation Instability in Fiber Optics
