Full nonlinear evolution of modulation instability in the model of long wave-short wave resonance: spectral recurrence, two types of breathers, and their excitation.

Abstract

The nonlinear stage of modulation instability (MI) starting with either purely periodic or localized periodic modulation is investigated analytically and numerically in the model of long wave-short wave (LS) resonance. Exact multiparametric solutions of Akhmediev breathers (ABs) and super-regular breathers (SRBs) are constructed based on an eigenvalue analysis. Physical spectra of ABs in analytic form are derived. The spectra of the short-wave component show expansion of energy from a single component to the sidebands and recurrence back to the same mode. However, the long-wave component shows complex oscillations in the sidebands, but the central mode remains constant. The evolution of LS-resonance-induced SRB is studied through its exact link with the MI. Our predictions based on exact analytical equations are confirmed by direct numerical simulations. We show that even nonideal single-wave excitations may lead asymptotically to the exact solutions. The connection of our exact solutions with results for the nonintegrable LS resonance model is also considered.