Doubly periodic solutions and breathers of the Hirota equation: recurrence, cascading mechanism and spectral analysis

Abstract

<p>The Hirota equation is an extension of the nonlinear Schrodinger equation by incorporating third-order dispersion. Doubly periodic solutions for the Hirota equation are established in terms of theta and elliptic functions. Cubic dispersion preserves numerical robustness, as slightly disturbed exact solutions as initial states still evolve to the analytical configurations. In contrast with the nonlinear Schrodinger case, the wavenumber of the envelope must satisfy an algebraic equation related to the magnitude of the quadratic / cubic terms and the period of the wave patterns. The long wave limits of these doubly periodic patterns will yield the widely studied Kuznetsov-Ma and Akhmediev breathers. A cascading mechanism for the Hirota equation is studied, which will elucidate the first formation of breather modes. Higher harmonics exponentially small initially will grow at a larger rate than the fundamental mode. Eventually the high frequency modes reach roughly the same magnitude at one moment in time (or spatial location), signifying the first occurrence of breather. Breathers then decay and modulation instability emerges again for sufficiently small amplitude. The cycle is repeated and constitutes a manifestation of the Fermi-Pasta-Ulam-Tsingou recurrence. These analytical doubly periodic solutions will permit the prediction of the period of recurrence. These results can be applied in hydrodynamic and optical contexts where third or higher-order dispersion is present.<br></p>