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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Whispering gallery waves diffraction by boundary i
nflection: an unsolved canonical problem - Valery
Smyshlyaev (University College London)
DTSTART;TZID=Europe/London:20190813T163000
DTEND;TZID=Europe/London:20190813T170000
UID:TALK128476AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/128476
DESCRIPTION:The problem of interest is that of a whispering ga
llery high-frequency asymptotic mode propagating a
long a concave part of a boundary and approaching
a boundary inflection point. Like Airy ODE and ass
ociated Airy function are fundamental for describi
ng transition from oscillatory to exponentially de
caying asymptotic behaviors\, the boundary inflect
ion problem leads to an arguably equally fundament
al canonical boundary-value problem for a special
PDE\, describing transition from a &ldquo\;modal&r
dquo\; to a &ldquo\;scattered&rdquo\; high-freque
ncy asymptotic behaviour. The latter problem was f
irst formulated and analysed by M.M. Popov startin
g from 1970-s. The associated solutions have asymp
totic behaviors of a modal type (hence with a disc
rete spectrum) at one end and of a scattering type
(with a continuous spectrum) at the other end. Of
central interest is to find the map connecting th
e above two asymptotic regimes. The problem howeve
r lacks separation of variables\, except in the as
ymptotical sense at both of the above ends.
\;

Nevertheless\, the problem asymptotically
admits certain complex contour integral solutions\
, see [1] and further references therein. Further\
, a non-standard perturbation analysis at the cont
inuous spectrum end can be performed\, ultimately
describing the desired map connecting the two asym
ptotic representations. It also permits a re-formu
lation as a one-dimensional boundary integral equa
tion\, whose regularization allows its further asy
mptotic and numerical analysis. We briefly review
all the above\, with an interesting open question
being whether the presence of an &lsquo\;incoming&
rsquo\; and an &lsquo\;outgoing&rsquo\; parts in t
he sought complex integral solution implies releva
nce of factorization techniques of Wiener-Hopf typ
e. \;

[1] D. P. Hewett\, J. R
. Ockendon\, V. P. Smyshlyaev\, Contour integral s
olutions of the parabolic wave equation\, Wave Mot
ion\, 84\, 90&ndash\;109 (2019) Preformatted versi
on: http://ww
w.newton.ac.uk/files/webform/587.tex
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:info@newton.ac.uk
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