Re: Solitons in One Post [Was: Solitons in one sentence]

tessel_at_tum.bot
Date: 07/09/04 

Date: 9 Jul 2004 04:49:05 -0400

On Wed, 7 Jul 2004, Gerard Westendorp wrote:

> there have been some interesting tv documentaries lately:
>
> http://www.bbc.co.uk/science/horizon/2002/freakwave.shtml
>
> Apparently, ocean waves of >30meter are much more common than thought
> possible. They have now been spotted using satellites.

Gosh. But these are -deep- water waves, so I don't see how they would
arise from the KdV right now. Did your source say anything about what
famous soliton equation is involved?

Be aware that some applied types have a much looser definition than more
careful folk. E.g AFAIK "solitons" in the Langmuir lattice are not true
solitons. Compare the Toda lattice and the sine-Gordon equation, where we
have true solitons.

> These waves are supposedly described by the non-linear Shrodinger
> equation...

Yes, yes, indeed! Gerard, since I last posted here on this topic, I've
found some more -fabulous- references which I urgently recommend. Alife
altering experiences and all that, heh :-/ I'd like to try to say
something about what's in them, but I don't have time. Suffice it to say
that they answer many of our questions from last time, and raise many new
and fascinating ones!!!

Arghghgh, right now my web browser is broken due to system work in
progress, so the complete journal refs and urls are missing below, but I
have the ArXiV references:

A beautifully written introduction, from the POV I have been advocating:

author = {Richard S. Palais},
title = {The Symmetries of Solitons},
journal = {Bull. of the A. M. S.}
volume = {?},
year = {1997}
note = {dg-ga/9708004}}

A fabulous book, out of print but available for download at Wood's website
(sorry, due to above problem I can't tell you what this is right now),
which complements the book I already cited:

editor = {A. P. Ford and J. C. Wood},
title = {Harmonic Maps and Integrable Systems},
series = {Aspects of Mathematics},
volume = {E23},
publisher = {Vieweg},
year = 1994,
note = {}}

This book contains excellent articles explaining supremely important
connections between solitons and many many things John Baez (and, more
obsurely alas, myself) have been talking about here in recent years, plus
harmonic maps (physicists may recognize another buzzword, "nonlinear-sigma
models"), and much more, as well as much more on aspects I've already
discussed such as

* Baecklund morphisms (and the origins of sine-Gordon in classical
differential geometry),

* Lie-Baecklund symmetries, infinite hierarchies of conservation laws and
how this relates to the remarkable persistence/stability of solitons,

* Toda lattice models and discrete --> continuous aspects (including
further hints of a connection with special Sturmian tilings, which
apparently Arnold knows all about but which -I- don't, arghghg).

The first article is a fine historical survey which gives an excellent
overview of IST and how the Schroedinger equation (see above!) enters into
soliton theory in a fundamental way.

To say just one thing about our previous conversation, I should have said
that in terms of the range of behavior of nonlinear dynamical systems,
completely integrable systems and "chaotic" systems are in a sense at
opposite extremes. Completely integrable systems are in a sense maximally
predictable; chaotic ones are in a sense maximally unpredictable
(although, unpredictable in highly predictable ways, statistically
speaking). Also, from Noether we would expect that the existence of an
infinite hierarcy of conservation laws is astonishing indeed, but should
be related to extreme stability (all those conserved quantities should
make it hard to change things too drastically), and this suspicion turns
out to be well justified.

Re more on how "chaos" appears in soliton theory:

author = {Yanguage (Charles) Li},
title = {Chaos in Partial Differential Equations},
journal = {Contemporary Mathematics},
volume = {?},
year = {?},
note = {math.AP/0205114}}

For more on connections with infinite-dimensional Kac-Moody algebras,
infinite dimensional Grassmannians, quantum algebra, etc., etc., (see
above), and complete integrability:

author = {Edward Frenkel},
title = {Five Lectures on Soliton Equations},
booktitle = {Surveys in Differential Geometry},
volume = 3,
publisher = {International Press},
year = 1997,
note = {q-alg./9712005}}

Sorry I'm out of time--- I hope that tomorrow I'll be able to try to
summarize some of what you can find in these references. I'd particularly
like to say more about

1. Baecklund morphisms and their role in classical differential geometry,

2. a famous mechanical model for the sine-Gordon equation, with links to
animated gifs which literally show what I'd be talking about, and how this
may give some intuition for your question regarding how an apparently fine
balance between dispersive effects (flattening wavecrests) and nonlinear
diffusive effects (steepening wavecrests) can actually be
"self-adjusting", and thus support -stable- solitary wave solutions.

"T. Essel" (hiding somewhere in cyberspace)

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