In mathematics, Ward's conjecture is the conjecture made by 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".
explain how a variety of completely integrable equations such as the KortewegâÂÂDe Vries equation (KdV) equation, the KadomtsevâÂÂPetviashvili 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 YangâÂÂMills equations:
where is the curvature of a connection on an oriented 4-dimensional pseudo-Riemannian manifold, and is the Hodge star operator.
They also obtain the equations of an integrable system known as the EulerâÂÂArnoldâÂÂManakov top, a generalization of the Euler top, and they state that the Kovalevsaya top is also a reduction of the self-dual YangâÂÂMills equations.
Via the PenroseâÂÂWard transform these solutions give the holomorphic vector bundles often seen in the context of algebraic integrable systems.