Dym equation

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

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

It is often written in the equivalent form

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

The Dym equation first appeared in Kruskal and is attributed to an unpublished paper by Harry Dym.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

References

• Cercignani, Carlo (1998). Scaling limits and models in physical processes. Basel: Birkhäuser Verlag. ISBN 0817659854. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

• Kichenassamy, Satyanad (1996). Nonlinear wave equations. Marcel Dekker. ISBN 0824793285.

• Gesztesy, Fritz (2003). Soliton equations and their algebro-geometric solutions. Cambridge University Press. ISBN 0521753074. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

• Olver, Peter J. (1993). Applications of Lie groups to differential equations, 2nd ed. Springer-Verlag. ISBN 0387940073.