Nonlinear spatial evolution of degenerate quartets of water waves

• Published in: Wave motion 2024-10, Vol.130, p.103381, Article 103381
• Main Authors: Heffernan, Conor, Chabchoub, Amin, Stuhlmeier, Raphael
• Format: Article
• Language: English
• Subjects: Benjamin–Feir instability; Breathers; Modulational instability; Resonant wave interactions; Spatial wave evolution; Wave-wave interaction
• ISSN: 0165-2125
• DOI: 10.1016/j.wavemoti.2024.103381

In this manuscript we investigate the Benjamin–Feir (or modulation) instability for the spatial evolution of water waves from the perspective of the discrete, spatial Zakharov equation, which captures cubically nonlinear and resonant wave interactions in deep water without restrictions on spectral bandwidth. Spatial evolution, with measurements at discrete locations, is pertinent for laboratory hydrodynamic experiments, such as in wave flumes, which rely on time-series measurements at fixed gauges installed along the facility. This setting is likewise appropriate for experiments in electromagnetic and plasma waves. Through a reformulation of the problem for a degenerate quartet, we bring to bear techniques of phase-plane analysis which elucidate the full dynamics without recourse to linear stability analysis. In particular we find hitherto unexplored breather solutions and discuss the optimal transfer of energy from carrier to sidebands. We show that the maximal energy transfer consistently occurs for smaller side-band separation than the fastest linear growth rate. Finally, we discuss the observability of such discrete solutions in light of numerical simulations. •A full description of spatial modulation instability without bandwidth restriction.•Describe degenerate quartet interaction via a planar Hamiltonian dynamical system.•Discover new, discrete breather solutions to the reduced spatial Zakharov equation.•Find solutions for maximal energy conversion from carrier to side-bands.•Show that fastest growing modes do not exhibit the largest total energy exchange.