The Dynamics of Waves in the Neighbourhood of the Benjamin-Feir Instability

Daniel Ratliff (Northumbria University, UK)

Abstract: The dynamics of dispersive nonlinear waves remains a problem that attracts significant interest, in part for their interesting stability properties. Arguably the most famous case is the Benjamin-Feir (BF) instability, where uniform wavetrains undergo a transition of stability due to a nonlinear frequency correction term . This transition occurs precisely when  where  is the linear dispersion relation and primes denotes differentiation. There are several emergent behaviours, such as an increase in the wave’s wavelength (frequency downshifting) or resonant wave bursting, which have been observed but elude mathematical insight as to why they occur.

In order to understand these phenomena, we explore the wave dynamics of from the standpoint of Whitham modulation theory and phase dynamics, ultimately uncovering that each of the possible transitions (either  or ) admits a different set of nonlinear dynamics governing the wave quantities whose solutions can be used to understand the wave behaviours near the BF threshold. Moreover, this work illustrates the role of mean flow is significant and is central to the emergence of permanent frequency downshifting and localised wave bursts one observes. We use this reasoning to explain, at least qualitatively, the experimental observations from wave-tank experiments in water waves and fluid conduits.