Second-order PDEs in four dimensions with half-flat conformal structure
2020 ◽
Vol 476
(2233)
◽
pp. 20190642
Keyword(s):
Lax Pair
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We study second-order partial differential equations (PDEs) in four dimensions for which the conformal structure defined by the characteristic variety of the equation is half-flat (self-dual or anti-self-dual) on every solution. We prove that this requirement implies the Monge–Ampère property. Since half-flatness of the conformal structure is equivalent to the existence of a non-trivial dispersionless Lax pair, our result explains the observation that all known scalar second-order integrable dispersionless PDEs in dimensions four and higher are of Monge–Ampère type. Some partial classification results of Monge–Ampère equations in four dimensions with half-flat conformal structure are also obtained.
2015 ◽
Vol 53
(1)
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pp. 405-420
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2015 ◽
Vol 284
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pp. 807-834
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1961 ◽
Vol 14
(3)
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pp. 171-186
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1992 ◽
Vol 44
(3)
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pp. 371-375
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Keyword(s):
2018 ◽
Vol 61
(11)
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pp. 1947-1962
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Keyword(s):
1973 ◽
Vol 13
(1)
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pp. 57-80
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1992 ◽
Vol 46
(1-2)
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pp. 63-75
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2016 ◽
pp. 165-183