| |
 |
|
|
Science Forum Index » Physics - Research Forum » korteweg deVries equation and solitons
Page 1 of 1
|
| Author |
Message |
| verdigris |
 Posted: Mon Jan 29, 2007 2:23 pm |
|
|
|
Guest
|
The Korteweg -deVries equation accounts for the existence of solitons
in water. Al Osborne has done some interesting work on solitons in
water layers such as the giant solitons off the coast of sumatra
mentioned in the link below
http://www.physicscentral.com/people/2002/osborne.html
But do solitons exist on a smaller scale in the boundary layers of air
that flow over an aircraft wing? |
|
|
| Back to top |
|
| Guest |
| Posted: Sat Feb 24, 2007 8:36 pm |
|
|
|
|
On Tue, 30 Jan 2007, verdigris wrote:
Quote: The Korteweg -deVries equation accounts for the existence of solitons
in water. Al Osborne has done some interesting work on solitons in
water layers such as the giant solitons off the coast of sumatra
mentioned in the link below
http://www.physicscentral.com/people/2002/osborne.html
Yes, the classic soliton was observed by the observational equestrian (a
species of which he may have been the only member!) John Scott Russell, as
the article (from an APS sponsored website, physicscentral.com) mentions.
The key thing is that the solitons, which were later modeled by the famous
"sech" solutions of the KdV equation, are waves which occur in a channel
with shallow depth and narrow bore. The sech solutions are easily
discovered using a generalization of "dimensional analysis" which was
discovered by none other than Sophus Lie. The mathematical background for
his method turned out to be Lie theory; the geometrical intuition is
closely related to Klein's notion of geometry (indeed, Klein and Lie
worked very closely in the early stages of developing their landmark
ideas).
Quote: But do solitons exist on a smaller scale in the boundary layers of air
that flow over an aircraft wing?
The KdV itself models one-dimensional wave motion. It's actually a kind
of "linear factor" because it only supports one direction of propagation,
but the Boussinesq equation allows for both directions
eqworld.ipmnet.ru/en/solutions/npde/npde6101.pdf The
Kadomtsev-Petviashvili equation governs a two-dimensional generalization
of the KdV
http://mathworld.wolfram.com/Kadomtsev-PetviashviliEquation.html It is
worth mentioning some other famous equations admitting soliton solutions:
the cubic non-linear Schroedinger equation (NLS) and the Calabi equation,
which generalized the Calabi flow, which is related to the Ricci flow
currently famous due to its role in the Poincare conjecture.
Interestingly enough, the Calabi equation resembles some which turn up in
classical gravitation, and there is some speculation that an interesting
class of vacuum solutions to the EFE might exist which admit soliton
solutions. There is already a fine book on Gravitational Solitons, by
Belinksky and Verdaguer, so I hasten to add that if you denote by the word
"soliton" anything found by a generalization of the inverse scattering
transform, then such solutions already exist, and include some colliding
plane waves (CPW) and some cosmological models.
All of these equations may be found written in various "gauges", and they
also may appear in "potential form".
Returning to your question, if you take a shallow tray of water, with
sufficient care you can probably make some pretty convincing solitons.
Somewhere on the web I think I've even seen photographs of a successful
experiment along these lines in which one soliton overtakes another.
A good book you should enjoy: Brian J. Cantwell, Introduction to Symmetry
Analysis, University of Cambridge Press, 2002. I can give other citations
which offer more information about solitons, but this book covers boundary
layers very nicely. I don't know whether anyone suspects that solitons
might lurk in the aerodynamics of wings, but Cantwell might!
"T. Essel" |
|
|
| Back to top |
|
| |
|
Page 1 of 1
All times are GMT - 5 Hours
The time now is Mon Dec 01, 2008 11:55 pm
|
|