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\title{Whispering gallery waves diffraction by boundary inflection: an unsolved canonical problem}
\author{%Smyshlyaev V.P.,
\textbf{Smyshlyaev V.P.}} % presenting author is marked out by \textbf
{University College London, Department of Mathematics, Gower Street, London WC1E 6BT
\\ e-mail: {\tt v.smyshlyaev@ucl.ac.uk} }
% for name index
\index{Smyshlyaev, V.P.}
The problem of interest is that of a whispering gallery high-frequency asymptotic mode propagating along a concave part of a boundary and approaching a boundary inflection point. Like Airy ODE and associated Airy function are fundamental for describing transition from oscillatory to exponentially decaying asymptotic behaviors, the boundary inflection problem leads to an arguably equally fundamental canonical boundary-value problem for a special PDE, describing transition from a ``modal'' to a ``scattered'' high-frequency asymptotic behaviour. The latter problem was first formulated and analysed by M.M. Popov starting from 1970-s. The associated solutions have asymptotic behaviors of a modal type (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 find the map connecting the above two asymptotic regimes. The problem however lacks separation of variables, except in the asymptotical sense at both of the above ends.
Nevertheless, the problem asymptotically admits certain complex contour integral solutions, see \cite{1} and further references therein. Further, a non-standard perturbation analysis at the continuous spectrum end can be performed,
ultimately describing the desired map connecting the two asymptotic representations.
It also permits a re-formulation as a one-dimensional boundary integral equation, whose regularization allows its further asymptotic and numerical analysis. We briefly review all the above, with an interesting open question being whether the presence of an `incoming' and an `outgoing' parts in the sought complex integral solution implies relevance of factorization techniques of Wiener-Hopf type.
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\begin{thebibliography}{9}
\bibitem{1} D.\,P.\;Hewett, J.\,R.\;Ockendon, V.\,P.\;Smyshlyaev,
Contour integral solutions of the parabolic wave equation,
\textit{Wave Motion}, \textbf{84}, 90--109 (2019).
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