Dym equation
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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The Dym equation first appeared in Kruskal [1] and is attributed to an unpublished paper by Harry Dym.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]
[edit] Notes
- ^ Kruskal, M. Nonlinear Wave Equations. In J. Moser, editor, Dynamical Systems, Theory and Applications, volume 38 of Lecture Notes in Physics, pages 310-354. Heidelberg. Springer. 1975.
- ^ F. Gesztesy and K. Unterkofler, Isospectral deformations for Sturm-Liouville and Dirac-type operators and associated nonlinear evolution equations, Rep. Math. Phys. 31 (1992), 113-137.
[edit] References
- Cercignani, Carlo; David H. Sattinger (1998). Scaling limits and models in physical processes. Basel: Birkhäuser Verlag. ISBN 0817659854.
- Kichenassamy, Satyanad (1996). Nonlinear wave equations. Marcel Dekker. ISBN 0824793285.
- Gesztesy, Fritz; Holden, Helge (2003). Soliton equations and their algebro-geometric solutions. Cambridge University Press. ISBN 0521753074.
- Olver, Peter J. (1993). Applications of Lie groups to differential equations, 2nd ed. Springer-Verlag. ISBN 0387940073.
