Volterra integral equation: Difference between revisions

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[[File:Dym eq Backlund solution animation.gif|thumb|right|300px|Dym eq Backlund transform solution animation]]
[[File:Harry Dym nlpde 3d animation.gif|thumb|right|300px|Dym equation 3d animation]]
 
In [[mathematics]], and in particular in the theory of [[soliton]]s, the '''Dym equation''' ('''HD''') is the third-order [[partial differential equation]]
 
:<math>u_t = u^3u_{xxx}.\,</math>
 
It is often written in the equivalent form
 
: <math>v_t=(v^{-1/2})_{xxx}.\,</math>
 
The Dym equation first appeared in Kruskal <ref>[[Martin Kruskal]] ''Nonlinear Wave Equations''. In [[Jürgen Moser]], editor, Dynamical Systems, Theory and Applications, volume 38 of Lecture Notes in Physics, pages 310–354. Heidelberg. Springer. 1975.</ref> and is attributed to an unpublished paper by [[Harry Dym]].
 
The Dym equation represents a system in which [[Dispersion relation|dispersion]] and [[nonlinearity]] are coupled together.  HD is a [[completely integrable]] [[nonlinear]] [[evolution equation]] that may be solved by means of the [[inverse scattering transform]].  It is interesting because it obeys an [[Infinity|infinite]] number of [[conservation law]]s; it does not possess the [[Painlevé property]].
 
The Dym equation has strong links to the [[Korteweg–de Vries equation]]. The [[Lax pair]] of the Harry Dym equation is associated with the [[Sturm–Liouville operator]].
The Liouville transformation transforms this operator [[isospectral|isospectrally]] into the [[Schrödinger]] operator.<ref>[[Fritz Gesztesy]] and [[Karl Unterkofler]], Isospectral deformations for Sturm–Liouville and Dirac-type operators and associated nonlinear evolution equations,   Rep. Math. Phys. 31 (1992), 113–137.</ref>
Thus by the inverse Liouville transformation solutions of the Korteweg–de Vries equation are transformed
into solutions of the Dym equation. In that paper an expicit solution of the Dym equation is found by an auto-[[Bäcklund transform]]
: <math> u(t,x) = (- 3 \alpha (x + 4 \alpha^2 t )^{2/3} . </math>
 
==Notes==
<references/>
 
==References==
*{{Cite book
| last      = Cercignani
| first      = Carlo
| author-link = Carlo Cercignani
| coauthors  = David H. Sattinger
| title      = Scaling limits and models in physical processes
| publisher  = Basel: Birkhäuser Verlag
| year      = 1998
| pages      =
| isbn      = 0-8176-5985-4
}}
 
*{{Cite book
| last      = Kichenassamy
| first      = Satyanad
| title      = Nonlinear wave equations
| publisher  = Marcel Dekker
| year      = 1996
| pages      =
| isbn      = 0-8247-9328-5
}}
 
*{{Cite book
| last      = Gesztesy
| first      = Fritz
| coauthors  = [[Helge Holden|Holden, Helge]]
| title      = Soliton equations and their algebro-geometric solutions
| publisher  = Cambridge University Press
| year      = 2003
| pages      =
| isbn      = 0-521-75307-4
}}
 
*{{Cite book
| last      = Olver
| first      = Peter J.
| title      = Applications of Lie groups to differential equations, 2nd ed
| publisher  = Springer-Verlag
| year      = 1993
  | pages      =
  | isbn      = 0-387-94007-3
}}
 
* {{springer|id=H/h130050|title=Harry Dym equation|first=P.J.|last=Vassiliou}}
 
 
 
{{DEFAULTSORT:Dym Equation}}
[[Category:Solitons]]
[[Category:Exactly solvable models]]

Revision as of 01:33, 18 November 2013

Dym eq Backlund transform solution animation
Dym equation 3d animation

In mathematics, and in particular in the theory of solitons, the Dym equation (HD) is the third-order partial differential equation

It is often written in the equivalent form

The Dym equation first appeared in Kruskal [1] and is attributed to an unpublished paper by Harry Dym.

The Dym equation represents a system in which dispersion and nonlinearity are coupled together. HD is a completely integrable nonlinear evolution equation that may be solved by means of the inverse scattering transform. It is interesting because it obeys an infinite number of conservation laws; it does not possess the Painlevé property.

The Dym equation has strong links to the Korteweg–de Vries equation. The Lax pair of the Harry Dym equation is associated with the Sturm–Liouville operator. The Liouville transformation transforms this operator isospectrally into the Schrödinger operator.[2] Thus by the inverse Liouville transformation solutions of the Korteweg–de Vries equation are transformed into solutions of the Dym equation. In that paper an expicit solution of the Dym equation is found by an auto-Bäcklund transform

Notes

  1. ↑ Martin Kruskal Nonlinear Wave Equations. In Jürgen Moser, editor, Dynamical Systems, Theory and Applications, volume 38 of Lecture Notes in Physics, pages 310–354. Heidelberg. Springer. 1975.
  2. ↑ Fritz Gesztesy and Karl Unterkofler, Isospectral deformations for Sturm–Liouville and Dirac-type operators and associated nonlinear evolution equations, Rep. Math. Phys. 31 (1992), 113–137.

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