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Matched asymptotics for a generalisation of a model equation for optical tunnelling

Published online by Cambridge University Press:  16 July 2009

Jinsong Liu  and
Alastair Wood
Jinsong Liu
Affiliation:
School of Mathematical Sciences, Dublin city University, Dublin 9, Ireland
Alastair Wood
Affiliation:
School of Mathematical Sciences, Dublin city University, Dublin 9, Ireland
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Abstract

We apply the method of matched asymptotic expansions to link the outgoing wave solution at infinity of the differential equation y″(x)+(lgr;+єxn)y(x) = 0, x∈(0, ∞),λ∈Cє>0, n∈N across the turning point x = (—λ/є)1/n nearest to the positive real axis to a linear homogeneous boundary condition at the origin. The equation with n = 1 models the leakage of energy from the core of a bent fibre optic waveguide, the rate of leakage corresponding to Im λ, which was shown to be exponentially small like O(exp[— 1/є]) by Paris & Wood (1989). The extension n = 2 by Brazel et al. (1990) obtained Im λ = O(exp[— 1/є1/2]). Both these papers involve delicate analysis of the asymptotics of special functions near to Stokes' lines. When n > 2 no special functions are available, and completely different methods must be employed to obtain the result Im λ = O(exp[— 1/є1/n]).

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 2 , Issue 3 , September 1991 , pp. 223 - 231
Copyright
Copyright © Cambridge University Press 1991

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