The coupled Hirota system as an example displaying discrete breathers: Rogue waves, modulation instability and varying cross-phase modulations

Discrete dynamical systems constitute an elegant branch of nonlinear science, where ingenious techniques provide penetrating insight for vibrations and wave motion on lattices. In terms of applications, such systems can model oscillators with hard quartic nonlinearities and switching of optical pulses on discrete arrays. A two-component Hirota system is investigated as an extension of the widely studied Ablowitz-Ladik equation by including discrete third order dispersion. Breathers (periodic pulsating modes) are derived analytically, and are used to establish conservation laws. Rogue waves (unexpectedly large displacements from equilibrium configurations) exhibit unusual features in undergoing oscillations above and below the mean level, and may even reverse polarity. Coupling produces new regimes of modulation instabilities for discrete evolution equations. The robustness of these novel rogue waves, in terms of sensitivity to initial conditions, is elucidated by numerical simulations. Self-phase modulations and cross-phase modulations of the same or opposite signs will generate nonlinear corrections of the frequency, due to the intensity of the wave train itself and the one in the accompanying waveguide respectively. Such effects have a crucial influence on the evolution of discrete and continuous multi-component dynamical systems.