Dym equation
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 for some function v of one space variable and time
The Dym equation first appeared in Kruskal [1] and is attributed to an unpublished paper by Harry Dym.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
Painlevé property
.
The Dym equation has strong links to the
Sturm–Liouville operator
.
The Liouville transformation transforms this operator Schrödinger operator.[2]
Thus by the inverse Liouville transformation solutions of the Korteweg–de Vries equation are transformed
into solutions of the Dym equation. An explicit solution of the Dym equation, valid in a finite interval, is found by an auto-Bäcklund transform[2]
![{\displaystyle u(t,x)=\left[-3\alpha \left(x+4\alpha ^{2}t\right)\right]^{2/3}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68ea162cafaef6e96b93d9b45c9bf710093cc7f4)
Notes
- 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.
- ^ a b 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.
References
- ISBN 0-8176-5985-4.
- Kichenassamy, Satyanad (1996). Nonlinear wave equations. Marcel Dekker. ISBN 0-8247-9328-5.
- Gesztesy, Fritz; ISBN 0-521-75307-4.
- ISBN 0-387-94007-3.
- Vassiliou, P.J. (2001) [1994], "Harry Dym equation", Encyclopedia of Mathematics, EMS Press