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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Hamiltonian dynamics of degenerate quartets of dee
p-water waves - Raphael Stuhlmeier (University of
Plymouth)
DTSTART;TZID=Europe/London:20221208T163000
DTEND;TZID=Europe/London:20221208T170000
UID:TALK183818AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/183818
DESCRIPTION:In the weakly nonlinear theory of waves on the sur
face of deep water\, the simplest interaction take
s place between quartets of waves. This interactio
n was first observed using perturbation methods (e
.g. by Stokes (1847)\, and later by Phillips and o
thers in the 1960s)\, which assume the water wave
problem can be expanded in terms of a small parame
ter. Today many model equations exist which captur
e the salient features of nonlinear interaction &n
dash\; one of these\, the Zakharov equation\, will
be the starting point for this talk.\nThe Zakharo
v equation has been used to derive the nonlinear S
chrö\;dinger equation (NLS)\, and many of its
modifications\, in a limit of narrow bandwidth. It
has also been used to study the modulational (Ben
jamin-Feir) instability of water waves (e.g. Yuen
& Lake (1982))\, where it provides a refinement of
the thresholds derived from the NLS. Such instabi
lity criteria have classically been derived from l
inearisation\, and subsequent behaviour obtained t
hrough numerical solution of the underlying equati
ons.\nI 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. Inspire
d by related work in optics (Capellini & Trillo (1
991)) this problem can be recast as a planar Hamil
tonian system in terms of the dynamic phase and a
single modal amplitude. In this simple form\, the
full\, nonlinear dynamics are readily apparent wit
hout recourse to numerical solutions.\nThe dynamic
al system is characterised by two free parameters:
the wave action and the separation between the ca
rrier and the side-bands\; the latter serves as a
bifurcation parameter. Fixed points of our system
correspond to non-trivial\, steady-state nearly-re
sonant degenerate quartets\, of the type recently
found by Liao et al (2016). I will explain the con
nection between saddle-points and the instability
of uniform and bichromatic wave trains\, and show
that heteroclinic orbits correspond to breather-li
ke solutions of this simplified system.\n \;\n
This work is joint with David Andrade.
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:
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