Is there an extension of WKB Theory to systems of equations? I have taken a course on Asymptotic Methods and learned an introductory WKB Theory for specific second order differential equations but I was wondering if the theory has been extended to second order systems of equations. In particular, I am looking at a simple system:
eps*x'' + a*y - x = 0
eps*y'' + b*x - y = 0
Furthermore, are there extensions of WKB Theory to nonlinear problems in a single variable, such as
esp*x'' + f(x) = 0.
Thanks in advance.
For system of equations of the form (below X,M,S,Y depends on x)
eps^2 X''m = Mmn Xn
one may try the ansatz Xn = exp( S/eps) * Yn, to find that S'2 must
be a local (x-dependent) eigenvalue of M, with Y a corresponding eigenvector. I did not check how/if this works to higher orders in eps, but it does not seems impossible (although messy).
The nonlinear case does not work well, athough some progress may be made for special solutions of f.i. the non-linear Schrödinger equation (where higher order phase-factors cancel).
Both are easily integrated. The first one is a linear system. An the second: kinetic energy plus work is constant
One of the highest experts in WKB was Michael Fedoryuk, see eg.
Asymptotic Methods in Analysis - Springer
link.springer.com/.../10.1007%2F978-3-642-6...
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автор: MV Fedoryuk - 1989
Recast the approach to modelling nonlinear oscillators as a coordinate transform as explained in Chapter 3 of my recent book "Model emergent dynamics in complex systems" SIAM, Philadelphia, Dec 2014. Then for arbitrary number of interacting nonlinear systems use the generalisation of a 'normal form' coordinate transform to decouple as far as possible the dynamics of the natural normal modes of the complex systems as described in Chapter 17 (see the exercises). Alternatively, appropriately submit your complex system to my web service at http://www.maths.adelaide.edu.au/anthony.roberts/gencm.php and it too computes the modulation equations for multiple normal modes of an oscillating system.
WKB methods can work for much richer class of equations that you suggest. Even wiki provides examples of the equation of higher order and corresponding ansatz: https://en.wikipedia.org/wiki/WKB_approximation
There is a lot of material about application of WKB method even for nonlinear waves. See, for example, this book:
Babich V., Buldyrev V., Molotkov I. (2002) The Space-Time Ray Method. Linear and Nonlinear Waves. - Cambridge University Press, ISBN 978-0-521-62735-1
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