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| [[File:Dym eq Backlund solution animation.gif|thumb|right|300px|Dym eq Backlund transform solution animation]]
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| [[File:Harry Dym nlpde 3d animation.gif|thumb|right|300px|Dym equation 3d animation]]
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| In [[mathematics]], and in particular in the theory of [[soliton]]s, the '''Dym equation''' ('''HD''') is the third-order [[partial differential equation]]
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| :<math>u_t = u^3u_{xxx}.\,</math>
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| It is often written in the equivalent form
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| : <math>v_t=(v^{-1/2})_{xxx}.\,</math>
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| 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 and is attributed to an unpublished paper by [[Harry Dym]].
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| 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]].
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| 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]].
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| 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>
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| Thus by the inverse Liouville transformation solutions of the Korteweg–de Vries equation are transformed
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| into solutions of the Dym equation. In that paper an expicit solution of the Dym equation is found by an auto-[[Bäcklund transform]]
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| : <math> u(t,x) = (- 3 \alpha (x + 4 \alpha^2 t )^{2/3} . </math> | |
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| ==Notes== | |
| <references/>
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| ==References==
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| *{{Cite book
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| | last = Cercignani
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| | first = Carlo
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| | author-link = Carlo Cercignani
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| | coauthors = David H. Sattinger
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| | title = Scaling limits and models in physical processes
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| | publisher = Basel: Birkhäuser Verlag
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| | year = 1998
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| | pages =
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| | isbn = 0-8176-5985-4
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| }}
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| *{{Cite book
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| | last = Kichenassamy
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| | first = Satyanad
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| | title = Nonlinear wave equations
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| | publisher = Marcel Dekker
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| | year = 1996
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| | pages =
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| | isbn = 0-8247-9328-5
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| }}
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| *{{Cite book
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| | last = Gesztesy
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| | first = Fritz
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| | coauthors = [[Helge Holden|Holden, Helge]]
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| | title = Soliton equations and their algebro-geometric solutions
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| | publisher = Cambridge University Press
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| | year = 2003
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| | pages =
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| | isbn = 0-521-75307-4
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| }}
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| *{{Cite book
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| | last = Olver
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| | first = Peter J.
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| | title = Applications of Lie groups to differential equations, 2nd ed
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| | publisher = Springer-Verlag
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| | year = 1993
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| | pages =
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| | isbn = 0-387-94007-3
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| }}
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| * {{springer|id=H/h130050|title=Harry Dym equation|first=P.J.|last=Vassiliou}}
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| {{DEFAULTSORT:Dym Equation}}
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| [[Category:Solitons]]
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| [[Category:Exactly solvable models]]
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Hello from Germany. I'm glad to be here. My first name is Lavonne.
I live in a city called Boxberg in east Germany.
I was also born in Boxberg 36 years ago. Married in August year 2003. I'm working at the college.
my website - Fifa 15 coin generator