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International Mathematics Research Notices Advance Access published online on September 3, 2009
International Mathematics Research Notices, doi:10.1093/imrn/rnp134
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© The Author 2009. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: journals.permissions@oxfordjournals.org

Integrable Equations of the Dispersionless Hirota type and Hypersurfaces in the Lagrangian Grassmannian

Evgeny Vladimirovich Ferapontov, Lenos Hadjikos and Karima Robertovna Khusnutdinova

Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire LE11 3TU, United Kingdom

Correspondence: Correspondence to be sent to: e.v.ferapontov{at}lboro.ac.uk

We investigate integrable second-order equations of the form


Formula

which typically arise as the Hirota-type relations for various (2 + 1)-dimensional dispersionless hierarchies. Familiar examples include the Boyer–Finley equation Formula , the potential form of the dispersionless Kadomtsev–Petviashvili (dKP) equation Formula , the dispersionless Hirota equation Formula , etc. The integrability is understood as the existence of an infinity of hydrodynamic reductions. We demonstrate that the natural equivalence group of the problem is isomorphic to Sp(6), revealing a remarkable correspondence between differential equations of the above type and hypersurfaces of the Lagrangian Grassmannian. We prove that the moduli space of integrable equations of the dispersionless Hirota type is 21-dimensional, and the action of the equivalence group Sp(6) on the moduli space has an open orbit.

Communicated by Prof. Igor Krichever


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