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some function v of one space variable and time The Dym equation first appeared in Kruskal [1] and is attributed to an unpublished paper by Harry Dym.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 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 obeys an infinite number of conservation laws; it does not possess the Painlevé property.
An explicit solution of the Dym equation, valid in a finite interval, is found by an auto-Bäcklund transform[2]