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Two examples of beyond all-orders asymptotics in diffraction and homogenisation

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AR2W02 - Mathematics of beyond all-orders phenomena

I will briefly review two different problems to which I was personally exposed, both dealing with exponentially small effects in the asymptotic expansions. One is that of diffraction of a whispering gallery (WG) high-frequency asymptotic mode propagating along a concave part of a boundary 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 behaviour, as Airy functions are for transition from oscillatory to exponentially decaying asymptotic patterns. The problem was formulated by M.M. Popov starting from 1970-s and remains largely open despite considerable progress since. In [1], we constructed an exponentially small shadow asymptotics 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, indicating at a physically expected “searchlight” beam. In a recent paper [2] we reviewed the problem and uncovered some asymptotic properties of the searchlight. In homogenisation, we have shown in [3] that a two-scale asymptotic expansion can sometimes be constructed not only in “all-orders” but even with 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 parameter $\epsilon$) the error of the approximation becomes exponentially small. References: [1] V. M. Babich, V.P. Smyshlyaev, Scattering problem for the Schroedinger equation in the case of a potential linear in time and coordinate. I. Asymptotics in the shadow zone, Journal of Soviet Mathematics, 32, 103-112 (1986). [2] V.P. Smyshlyaev, I. V. Kamotski, Searchlight asymptotics for high-frequency scattering by boundary inflection, St. Petersburg Mathematical Journal, 33, 387-403 (2022). [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).

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