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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:The Landau-Alber bifurcation: implications for the
analysis of metocean data\, and open questions in
Landau damping - Agis Athanassoulis (University o
f Dundee)
DTSTART;TZID=Europe/London:20220909T103000
DTEND;TZID=Europe/London:20220909T113000
UID:TALK177866AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/177866
DESCRIPTION:It is famously known that plane wave solutions of
the Nonlinear Schrö\;dinger equation (NLS) are
linearly unstable\; this fact is called the modul
ation instability for the NLS. While recent breakt
horughs have provided insights on the qualitative
aspects of nonlinear evolution of the modulation i
nstability\, many fundamental questions (such as g
lobal existence of solutions for general initial p
erturbations) are still open. The NLS is used as a
n approximate model in ocean waves\, where the mod
ulation instability is also known as the Benjamin-
Feir instability\, and is linked with concrete phy
sical phenomena including rogue waves. In this con
text of ocean engineering\, linear sea states are
often used as approximate stochastic solutions of
the NLS\, as only stochastic wavefileds -- and not
plane waves -- are relevant for marine safety app
lications. \;\nThe main points of this talk ar
e the following:\n\nThere exists a 2nd moment theo
ry of stochastic wavefileds under the NLS equation
that is compatible with the bulk of the metocean
data available\, namely the Alber equation. This a
llows concepts from the NLS to be generalised to a
stochastic setting yet remain tractable. The exis
ting well-posedness theory for the Alber equation
will be briefly reviewed [1].\nIn the stochastic s
etting\, the instability of the homogeneous soluti
on (i.e. 'generalised modulation instability') is
not the only possibility. In fact\, many homogeneo
us sea states will give rise to Landau damping\, u
nexpectedly stabilising themselves despite the pre
sence of infinite energy and a focusing nonlineari
ty. Still\, unstable sea states do exist. So a bif
urcation between Landau damping and generalised mo
dulation instability exists within measured sea st
ates [1-4]. Since I. E. Alber was the one who init
iated this line of work\, we call this the Landau-
Alber bifurcation. The rigorous result on the Land
au damping will be briefly outlined\, along with i
ts proof which builds crucially on an esoteric pro
perty of the Hilbert transform [1].\nThe linear st
ability analysis can surprisingly predict both pro
files of rogue waves in the unstable case\, and th
e likelihood of extreme events in the stable case.
These findings are in excellent agreement with em
pirical facts in the ocean engineering community [
2\,5].\n\nA number of open questions will be discu
ssed in the end.\nIncludes joint work with T. Saps
is (MIT)\, G. Athanassoulis (NTUA)\, M. Ptashnyk (
Heriot-Watt) and O. Gramstad (DNV).\nReferences:[1
]. Athanassoulis\, Agissilaos G.\, et al. "Strong
solutions for the Alber equation and stability of
unidirectional wave spectra." Kinetic & Related Mo
dels \;13.4 (2020).[2]. Athanassoulis\, Agissi
laos G.\, and Odin Gramstad. "Modelling of Ocean W
aves with the Alber Equation: Application to Non-P
arametric Spectra and Generalisation to Crossing S
eas." Fluids \;6.8 (2021): 291.[3]. Ribal\, A.
\, et al. "Recurrent solutions of the Alber equati
on initialized by Joint North Sea Wave Project spe
ctra." Journal of Fluid Mechanics \;719 (2013)
: 314-344.[4]. Alber\, I. E. "The effects of rando
mness on the stability of two-dimensional surface
wavetrains." Proceedings of the Royal Society of L
ondon. A. Mathematical and Physical Sciences \
;363.1715 (1978): 525-546.[5]. Dematteis\, Giovann
i\, Tobias Grafke\, and Eric Vanden-Eijnden. "Rogu
e waves and large deviations in deep sea." \;P
roceedings of the National Academy of Sciences 115
.5 (2018): 855-860.
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:
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