In mathematics, Ward's conjecture is the conjecture made by Ward ( 1985, 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".
Ablowitz, Chakravarty, and Halburd ( 2003) 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.
Via the Penrose–Ward transform these solutions give the holomorphic vector bundles often seen in the context of algebraic integrable systems.
- Ablowitz, M. J.; Chakravarty, S.; R. G., Halburd (2003), "Integrable systems and reductions of the self-dual Yang–Mills equations", Journal of Mathematical Physics, 44 (8): 3147–3173, Bibcode: 2003JMP....44.3147A, doi: 10.1063/1.1586967 http://www.ucl.ac.uk/~ucahrha/Publications/sdym-03.pdf
- Ward, R. S. (1985), "Integrable and solvable systems, and relations among them", Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences, 315 (1533): 451–457, Bibcode: 1985RSPTA.315..451W, doi: 10.1098/rsta.1985.0051, ISSN 0080-4614, MR 0836745, S2CID 123659512
- Mason, L. J.; Woodhouse, N. M. J. (1996), Integrability, Self-duality, and Twistor Theory, Clarendon
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In mathematics, Ward's conjecture is the conjecture made by Ward ( 1985, 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".
Ablowitz, Chakravarty, and Halburd ( 2003) 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.
Via the Penrose–Ward transform these solutions give the holomorphic vector bundles often seen in the context of algebraic integrable systems.
- Ablowitz, M. J.; Chakravarty, S.; R. G., Halburd (2003), "Integrable systems and reductions of the self-dual Yang–Mills equations", Journal of Mathematical Physics, 44 (8): 3147–3173, Bibcode: 2003JMP....44.3147A, doi: 10.1063/1.1586967 http://www.ucl.ac.uk/~ucahrha/Publications/sdym-03.pdf
- Ward, R. S. (1985), "Integrable and solvable systems, and relations among them", Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences, 315 (1533): 451–457, Bibcode: 1985RSPTA.315..451W, doi: 10.1098/rsta.1985.0051, ISSN 0080-4614, MR 0836745, S2CID 123659512
- Mason, L. J.; Woodhouse, N. M. J. (1996), Integrability, Self-duality, and Twistor Theory, Clarendon
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