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An Isaac Newton Institute Workshop

The Theory of Highly Oscillatory Problems

Localization and propagation in high contrast highly oscillatory media via "non-classical" homogenization

29th March 2007

Authors: Valery P. Smyshlyaev (Bath University), Ilia V. Kamotski (Bath University)

Abstract

We discuss fundamental mathematical tools for analysis of localization and propagation effects in highly oscillatory media with high contrasts. With the underlying two small parameters of the oscillations and the contrast, there is a "critical" scaling when the phenomena at the micro and macro scales are coupled in a non-trivial way, with "unusual" effects observed in an asymptotically explicit way. The related mathematical tool is that of a "non-classical" (high-contrast) homogenization, accounting for "high-frequency" oscillations, as opposes to the "classical" homogenization whose scope is limited by dealing in effect with low frequencies only. Those tools include "non-classical" two-scale asymptotic expansions, two-scale operator and spectral convergence, and two-scale compactness (with the latter building on, among others, recent deep ideas of V.V. Zhikov [1]).

We illustrate this on the problem of wave localization in high contrast periodic media with a defect [2] (a problem relevant to photonic crystal fibres). We also discuss the use of these techniques to the problem of "slowing down" of wave packets in high contrast highly-oscillatory media (the so-called "slow light" effect, with relevance to coupled resonances and metastability), and other prospects. Joint work with Ilia V. Kamotski.

[1] V.V. Zhikov, On an extension of the method of two-scale convergence and its applications, (Russian) Mat. Sbornik 191 (2000), 31-72; English translation in Sbornik Math. 191 (2000), 973-1014; V.V. Zhikov, Gaps in the spectrum of some elliptic operators in divergent form with periodic coefficients. (Russian) Algebra i Analiz 16 (2004), 34-58;

[2] I.V. Kamotski and V.P. Smyshlyaev, Localised eigenstates due to defects in high contrast periodic media via homogenisation. BICS preprint 3/06. http://www.bath.ac.uk/math-sci/bics/preprints/BICS06_3.pdf (2006)

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