Rogue waves for a long wave–short wave resonance model with multiple short waves

Abstract

A resonance between long and short waves will occur if the phase velocity of the long wave matches the group velocity of the short wave. In this paper, a system with two distinct packets of short waves in resonance with a common long wave is studied. Breather solutions are calculated by the Hirota bilinear method, and rogue wave modes (unexpectedly large displacements from an otherwise calm background state) are obtained from the breathers through a long wave limit. The location and magnitude of the maximum displacement are determined quantitatively. Remarkably this coupling enables a rogue wave to attain a larger magnitude than that in a configuration with just one single short wave component. Furthermore, as the wavenumber varies, a transition from an elevation rogue wave to a depression rogue wave is possible. This transformation of the wave profile is elucidated in terms of the properties of the carrier envelope. The connection with the modulation instability of the background plane wave is investigated. Some numerical simulations are performed to demonstrate both the robust nature and unstable behavior for these rogue waves, depending on the parameters of the system. Dynamics and properties of rogue waves with three or more short wave components are also considered.