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   -> Volume 5, Issue 9


Answer: (WD 5.7 #18) Wavelets and the nonlinear Schroedinger eq.
 
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"Mariusz H. Jakubowski" (mj@cs.princeton.edu)
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PostPosted: Fri Oct 18, 1996 10:07 pm    
Subject: Answer: (WD 5.7 #18) Wavelets and the nonlinear Schroedinger eq.
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#17 Answer: (WD 5.7 #18) Wavelets and the nonlinear Schroedinger eq.

The following is a brief summary of the responses I received to my
question about wavelets and the nonlinear Schroedinger equation
(WD 5.7). Thanks to all who replied.

The papers listed below describe the application of wavelets to the
numerical solution of the NLSE. In particular, the papers present
"split-step wavelet" methods analogous to the split-step Fourier
method:

L. Gagnon, J. M. Lina, `Symmetric Daubechies Wavelets and Numerical
Solution of NLS Equations', J. Phys. A: Math. Gen. 27, 8207-8230
(1994). See http://fellini.crm.umontreal.ca .

I. Pierce and L.R. Watkins, `Modelling optical pulse propagation in
nonlinear media using wavelets', Proc. IEEE Intnl. Symp. on Time-
frequency and Time-scale analysis, Paris, France, June 1996, p. 361
ff. See http://www.sees.bangor.ac.uk/~iestyn/papers.htm
or ftp.sees.bangor.ac.uk/pub/iestyn/tfts96.ps .

L.R. Watkins and Y.R. Zhou, `Modelling propagation in optical fibres
using wavelets', IEEE J. Lightwave Technology, 12 (9), Sep. 1994, p.
1536 - 1542. See http://www.sees.bangor.ac.uk/~iestyn/papers.htm
or ftp.sees.bangor.ac.uk/pub/iestyn/jltpaper.ps.Z .

Matthias Holschneider ( hols@titan.univ-rennes1.fr ) informed me that
the traveling wavelets method is due to M. Holschneider, V. Perrier,
and C. Basdevant, and that the first paper about this appears "in the
CRAS." The paper from which I learned about the method is:

V. Perrier and C. Basdevant, `Travelling Wavelets Method',
Proc. of the US-French 'Wavelets and Turbulence' Workshop,
Princeton, June 1991.

The method is used in this paper to solve the linear advection-
diffusion equation and the Burgers equation. I adapted it to solve
the quadratic NLSE (I hope) for certain initial conditions, but I
have yet to try to make my implementation fast and general.

>From the responses I received, I gather that wavelet-based methods
for solving the NLSE offer some promise, but more work needs to be
done to see if they can outperform standard methods (such as the
split-step Fourier method) in general situations.

If anyone has done, or is aware of, further work on wavelets and the
NLSE, please send the information to the Wavelet Digest. The replies
I received indicate that there is some interest in this area.

Thanks,

-Mariusz H. Jakubowski
mj@cs.princeton.edu
http://www.cs.princeton.edu/~mj/
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