By Magri's theorem, self-dual gravity is completely integrable.
Nutku, Yavuz (2007)
SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]
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Nutku, Yavuz (2007)
SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]
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Chavchanidze, G. (2003)
Georgian Mathematical Journal
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Huang, Xuncheng, Tu, Guizhang (2006)
International Journal of Mathematics and Mathematical Sciences
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Simonetta Abenda, Tamara Grava (2005)
Annales de l’institut Fourier
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We derive the modulation equations (Whitham equations) for the Camassa-Holm (CH) equation. We show that the modulation equations are hyperbolic and admit a bi-Hamiltonian structure. Furthermore they are connected by a reciprocal transformation to the modulation equations of the first negative flow of the Korteweg de Vries (KdV) equation. The reciprocal transformation is generated by the Casimir of the second Poisson bracket of the KdV averaged flow. We show that...
E. Zenhder (1975)
Publications mathématiques et informatique de Rennes
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Boris Khesin (1993)
Recherche Coopérative sur Programme n°25
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Popowicz, Ziemowit (2010)
SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]
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Gupta, V.G., Sharma, P. (2010)
Acta Universitatis Apulensis. Mathematics - Informatics
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Henryk Żołądek (2011)
Banach Center Publications
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The first and the second Painlevé equations are explicitly Hamiltonian with time dependent Hamilton function. By a natural extension of the phase space one gets corresponding autonomous Hamiltonian systems in ℂ⁴. We prove that the latter systems do not have any additional algebraic first integral. In the proof equations in variations with respect to a parameter are used.