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SUMMARY:Two examples of beyond all-orders asymptotics in diffraction and h
 omogenisation - Valery Smyshlyaev (University College London)
DTSTART:20221031T145000Z
DTEND:20221031T154000Z
UID:TALK185180@talks.cam.ac.uk
DESCRIPTION:I will briefly review two different problems to which I was pe
 rsonally exposed\, both dealing with exponentially small effects in the as
 ymptotic expansions.\nOne is that of diffraction of a whispering gallery (
 WG) high-frequency asymptotic mode propagating along a concave part of a b
 oundary and approaching a boundary inflection point. The problem leads to 
 an arguably as fundamental canonical boundary-value problem for a special 
 PDE describing transition from a "modal" to a "scattered" asymptotic behav
 iour\, as Airy functions are for transition from oscillatory to exponentia
 lly decaying asymptotic patterns. The problem was formulated by M.M. Popov
  starting from 1970-s and remains largely open despite considerable progre
 ss since. In [1]\, we constructed an exponentially small shadow asymptotic
 s matching with the incoming WG wave. On its other end it appears to surge
  and break-up on approaching the tangent at the inflection point\, indicat
 ing at a physically expected "searchlight" beam. In a recent paper [2] we 
 reviewed the problem and uncovered some asymptotic properties of the searc
 hlight.\nIn homogenisation\, we have shown in [3] that a two-scale asympto
 tic expansion can sometimes be constructed not only in "all-orders" but ev
 en with a property that upon an optimal truncation (i.e. when the number o
 f terms is chosen to depend in a particular way on the underlying small pa
 rameter $\\epsilon$) the error of the approximation becomes exponentially 
 small.\nReferences:\n[1] V. M. Babich\, V.P. Smyshlyaev\, Scattering probl
 em for the Schroedinger equation in the case of a potential linear in time
  and coordinate. I. Asymptotics in the shadow zone\, Journal of Soviet Mat
 hematics\, 32\, 103-112 (1986).\n[2] V.P. Smyshlyaev\, I. V. Kamotski\, Se
 archlight asymptotics for high-frequency scattering by boundary inflection
 \, St. Petersburg Mathematical Journal\, 33\, 387-403 (2022).\n[3] V. Kamo
 tski\, K. Matthies\, V.P. Smyshlyaev\, Exponential homogenization of linea
 r second order elliptic PDEs with periodic coefficients\, SIAM J. Math. An
 al.\, 38\, 1565-1587 (2007).
LOCATION:Seminar Room 1\, Newton Institute
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