Ward's conjecture
{{Short description|On self-duality of differential equations}}
In mathematics, Ward's conjecture is the conjecture made by {{harvs|txt|last=Ward|authorlink=Richard S. Ward|year=1985|loc=p. 451}} 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
{{harvs|last1=Ablowitz|last2=Chakravarty|last3=Halburd|year=2003|txt}} 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:
:
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 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
- {{citation|last1=Ablowitz | first1=M. J. | last2=Chakravarty |first2=S.| last3=R. G. | first3= Halburd| title=Integrable systems and reductions of the self-dual Yang–Mills equations | journal=Journal of Mathematical Physics | year=2003 | volume=44 | issue=8 |pages=3147–3173 | doi=10.1063/1.1586967 | bibcode=2003JMP....44.3147A}} http://www.ucl.ac.uk/~ucahrha/Publications/sdym-03.pdf
- {{Citation | last1=Ward | first1=R. S. | title=Integrable and solvable systems, and relations among them | doi=10.1098/rsta.1985.0051 | mr=836745 | year=1985 | journal=Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences | issn=0080-4614 | volume=315 | issue=1533 | pages=451–457|bibcode = 1985RSPTA.315..451W | s2cid=123659512 }}
- {{citation|last1 =Mason| first1= L. J.|last2 = Woodhouse| first2=N. M. J.|title =Integrability, Self-duality, and Twistor Theory | publisher=Clarendon|year =1996}}
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