The nonlinear Benjamin-Feir instability - a discrete Hamiltonian approach #

Raphael Stuhlmeier, David Andrade

15:10 Monday in 2Q42.

Part of the Advances in water waves and free-surface flows session.

Abstract #

In the weakly nonlinear theory of waves on the surface of deep water, the simplest interaction takes place between quartets of waves. This interaction was first observed using perturbation methods (e.g. by Stokes (1847), and later by Phillips and others in the 1960s), which assume the water wave problem can be expanded in terms of a small parameter. Today many model equations exist which capture the salient features of nonlinear interaction – one of these, the Zakharov equation, will be the starting point for this talk. The Zakharov equation has been used to derive the nonlinear Schrödinger equation (NLS), and many of its modifications, in a limit of narrow bandwidth. It has also been used to study the modulational (Benjamin-Feir) instability of water waves (e.g. Yuen & Lake (1982)), where it provides a refinement of the thresholds derived from the NLS. Such instability criteria have classically been derived from linearisation, and subsequent behaviour obtained through numerical solution of the underlying equations. I will describe an approach to the Benjamin-Feir instability based on the degenerate quartets of the discretised Zakharov equation which is free of any restriction on spectral bandwidth. Inspired by related work in optics this problem can be recast as a planar Hamiltonian system in terms of the dynamic phase and a single modal amplitude. In this simple form, the full, nonlinear dynamics are readily apparent without recourse to numerical solutions. The dynamical system is characterised by two free parameters: the wave action and the separation between the carrier and the side-bands; the latter serves as a bifurcation parameter. Fixed points of our system correspond to non-trivial, steady-state nearly-resonant degenerate quartets, of the type recently found by Liao et al (2016). I will explain the connection between saddle-points and the instability of uniform and bichromatic wave trains, and show that heteroclinic orbits correspond to discrete breather solutions of this simplified system.