Dym equation

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

$u_t = u^3u_{xxx}.\,$

It is often written in the equivalent form

$v_t=(v^{-1/2})_{xxx}.\,$

The Dym equation first appeared in Kruskal ^[1] and is attributed to an unpublished paper by Harry Dym.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

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.

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. 

• Vassiliou, P.J. (2001), "Harry Dym equation", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/H/h130050.htm