BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//talks.cam.ac.uk//v3//EN
BEGIN:VTIMEZONE
TZID:Europe/London
BEGIN:DAYLIGHT
TZOFFSETFROM:+0000
TZOFFSETTO:+0100
TZNAME:BST
DTSTART:19700329T010000
RRULE:FREQ=YEARLY;BYMONTH=3;BYDAY=-1SU
END:DAYLIGHT
BEGIN:STANDARD
TZOFFSETFROM:+0100
TZOFFSETTO:+0000
TZNAME:GMT
DTSTART:19701025T020000
RRULE:FREQ=YEARLY;BYMONTH=10;BYDAY=-1SU
END:STANDARD
END:VTIMEZONE
BEGIN:VEVENT
CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Two examples of beyond all-orders asymptotics in d
iffraction and homogenisation - Valery Smyshlyaev
(University College London)
DTSTART;TZID=Europe/London:20221031T145000
DTEND;TZID=Europe/London:20221031T154000
UID:TALK185180AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/185180
DESCRIPTION:I will briefly review two different problems to wh
ich I was personally exposed\, both dealing with e
xponentially small effects in the asymptotic expan
sions.\nOne is that of diffraction of a whispering
gallery (WG) high-frequency asymptotic mode propa
gating along a concave part of a boundary and appr
oaching a boundary inflection point. The problem l
eads to an arguably as fundamental canonical bound
ary-value problem for a special PDE describing tra
nsition from a "modal" to a "scattered" asymptotic
behaviour\, as Airy functions are for transition
from oscillatory to exponentially decaying asympto
tic patterns. The problem was formulated by M.M. P
opov starting from 1970-s and remains largely open
despite considerable progress since. In [1]\, we
constructed an exponentially small shadow asymptot
ics matching with the incoming WG wave. On its oth
er end it appears to surge and break-up on approac
hing the tangent at the inflection point\, indicat
ing at a physically expected "searchlight" beam. I
n a recent paper [2] we reviewed the problem and u
ncovered some asymptotic properties of the searchl
ight.\nIn homogenisation\, we have shown in [3] th
at a two-scale asymptotic expansion can sometimes
be constructed not only in "all-orders" but even w
ith a property that upon an optimal truncation (i.
e. when the number of terms is chosen to depend in
a particular way on the underlying small paramete
r $\\epsilon$) the error of the approximation beco
mes exponentially small.\nReferences:\n[1] V. M. B
abich\, V.P. Smyshlyaev\, Scattering problem for t
he Schroedinger equation in the case of a potentia
l linear in time and coordinate. I. Asymptotics in
the shadow zone\, Journal of Soviet Mathematics\,
32\, 103-112 (1986).\n[2] V.P. Smyshlyaev\, I. V.
Kamotski\, Searchlight asymptotics for high-frequ
ency scattering by boundary inflection\, St. Peter
sburg Mathematical Journal\, 33\, 387-403 (2022).\
n[3] V. Kamotski\, K. Matthies\, V.P. Smyshlyaev\,
Exponential homogenization of linear second order
elliptic PDEs with periodic coefficients\, SIAM J
. Math. Anal.\, 38\, 1565-1587 (2007).
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:
END:VEVENT
END:VCALENDAR