Ward's conjecture

Template:Short description In mathematics, Ward's conjecture is the conjecture made by Template:Harvs that "many (and perhaps all?) of the ordinary and partial differential equations that are regarded as being integrable or solvable may be obtained from the self-dual gauge field equations (or its generalizations) by reduction".

Examples

Template:Harvs explain how a variety of completely integrable equations such as the Korteweg–De Vries equation (KdV) equation, the Kadomtsev–Petviashvili equation (KP) equation, the nonlinear Schrödinger equation, the sine-Gordon equation, the Ernst equation and the Painlevé equations all arise as reductions or other simplifications of the self-dual Yang–Mills equations:

$F=\star F$

where $F$ is the curvature of a connection on an oriented 4-dimensional pseudo-Riemannian manifold, and $\star$ is the Hodge star operator.

They also obtain the equations of an integrable system known as the Euler–Arnold–Manakov top, a generalization of the Euler top, and they state that the Kovalevsaya top is also a reduction of the self-dual Yang–Mills equations.

Penrose–Ward transform

Via the Penrose–Ward transform these solutions give the holomorphic vector bundles often seen in the context of algebraic integrable systems.

References

• Template:Citation http://www.ucl.ac.uk/~ucahrha/Publications/sdym-03.pdf
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