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SUMMARY:Whispering gallery waves diffraction by boundary inflection: an un
 solved canonical problem - Valery Smyshlyaev (University College London)
DTSTART:20190813T153000Z
DTEND:20190813T160000Z
UID:TALK128476@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:The problem of interest is that of a whispering gallery high-f
 requency asymptotic mode propagating along a concave part of a boundary an
 d approaching a boundary inflection point. Like Airy ODE and associated Ai
 ry function are fundamental for describing transition from oscillatory to 
 exponentially decaying asymptotic behaviors\, the boundary inflection prob
 lem leads to an arguably equally fundamental canonical boundary-value prob
 lem for a special PDE\, describing transition from a &ldquo\;modal&rdquo\;
  to a &ldquo\;scattered&rdquo\;  high-frequency asymptotic behaviour. The 
 latter problem was first formulated and analysed by M.M. Popov starting fr
 om 1970-s. The associated solutions have asymptotic behaviors of a modal t
 ype (hence with a discrete spectrum) at one end and of a scattering type (
 with a continuous spectrum) at the other end. Of central interest is to fi
 nd the map connecting the above two asymptotic regimes. The problem howeve
 r lacks separation of variables\, except in the asymptotical sense at both
  of the above ends.  &nbsp\;  <br>Nevertheless\, the problem asymptoticall
 y admits certain complex contour integral solutions\, see [1] and further 
 references therein. Further\, a non-standard perturbation analysis at the 
 continuous spectrum end can be performed\, ultimately describing the desir
 ed map connecting the two asymptotic representations. It also permits a re
 -formulation as a one-dimensional boundary integral equation\, whose regul
 arization allows its further asymptotic and numerical analysis. We briefly
  review all the above\, with an interesting open question being whether th
 e presence of an &lsquo\;incoming&rsquo\; and an &lsquo\;outgoing&rsquo\; 
 parts in the sought complex integral solution implies relevance of factori
 zation techniques of Wiener-Hopf type.  &nbsp\;  <br><span><br>[1] D. P. H
 ewett\, J. R. Ockendon\, V. P. Smyshlyaev\, Contour integral solutions of 
 the parabolic wave equation\, Wave Motion\, 84\, 90&ndash\;109 (2019) Pref
 ormatted version: <a target="_blank" rel="nofollow" href="http://www.newto
 n.ac.uk/files/webform/587.tex">http://www.newton.ac.uk/files/webform/587.t
 ex</a></span>
LOCATION:Seminar Room 1\, Newton Institute
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