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SUMMARY:Hamiltonian dynamics of degenerate quartets of deep-water waves - 
 Raphael Stuhlmeier (University of Plymouth)
DTSTART:20221208T163000Z
DTEND:20221208T170000Z
UID:TALK183818@talks.cam.ac.uk
DESCRIPTION:In the weakly nonlinear theory of waves on the surface of deep
  water\, the simplest interaction takes place between quartets of waves. T
 his interaction was first observed using perturbation methods (e.g. by Sto
 kes (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. Tod
 ay many model equations exist which capture the salient features of nonlin
 ear interaction &ndash\; one of these\, the Zakharov equation\, will be th
 e starting point for this talk.\nThe Zakharov equation has been used to de
 rive the nonlinear Schr&ouml\;dinger equation (NLS)\, and many of its modi
 fications\, 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 fr
 om the NLS. Such instability criteria have classically been derived from l
 inearisation\, and subsequent behaviour obtained through numerical solutio
 n of the underlying equations.\nI will describe an approach to the Benjami
 n-Feir instability based on the degenerate quartets of the discretised Zak
 harov equation which is free of any restriction on spectral bandwidth. Ins
 pired by related work in optics (Capellini & Trillo (1991)) this problem c
 an 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 d
 ynamics are readily apparent without recourse to numerical solutions.\nThe
  dynamical system is characterised by two free parameters: the wave action
  and the separation between the carrier and the side-bands\; the latter se
 rves as a bifurcation parameter. Fixed points of our system correspond to 
 non-trivial\, steady-state nearly-resonant degenerate quartets\, of the ty
 pe recently found by Liao et al (2016). I will explain the connection betw
 een saddle-points and the instability of uniform and bichromatic wave trai
 ns\, and show that heteroclinic orbits correspond to breather-like solutio
 ns of this simplified system.\n&nbsp\;\nThis work is joint with David Andr
 ade.
LOCATION:Seminar Room 1\, Newton Institute
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