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On integrability in Grassmann geometries: integrable systems associated with fourfolds in Gr(3, 5)
journal contribution
posted on 2017-12-18, 14:20 authored by B. Doubrov, Evgeny FerapontovEvgeny Ferapontov, B. Kruglikov, Vladimir NovikovVladimir NovikovLet Gr(d; n) be the Grassmannian of d-dimensional linear subspaces of an n-dimensional vector space V n. A submanifold X Gr(d; n) gives rise to a differential system ⊂(X)
that governs d-dimensional submanifolds of V n whose Gaussian image is contained in X.
Systems of the form Σ(X) appear in numerous applications in continuum mechanics, theory
of integrable systems, general relativity and differential geometry. They include such wellknown
examples as the dispersionless Kadomtsev-Petviashvili equation, the Boyer-Finley
equation, Plebansky's heavenly equations, and so on. In this paper we concentrate on the particularly interesting case of this construction
where X is a fourfold in Gr(3; 5). Our main goal is to investigate differential-geometric and
integrability aspects of the corresponding systems Σ(X). We demonstrate the equivalence
of several approaches to dispersionless integrability such as • the method of hydrodynamic reductions, • the method of dispersionless Lax pairs, • integrability on solutions, based on the requirement that the characteristic variety of system Σ(X) defines an Einstein-Weyl geometry on every solution,
• integrability on equation, meaning integrability (in twistor-theoretic sense) of the canonical
GL(2;R) structure induced on a fourfold X ⊂ Gr(3; 5). All these seemingly different approaches lead to one and the same class of integrable systems
Σ(X). We prove that the moduli space of such systems is 6-dimensional. We give a complete
description of linearisable systems (the corresponding fourfold X is a linear section of
Gr(3; 5)) and linearly degenerate systems (the corresponding fourfold X is the image of a
quadratic map P4 99K Gr(3; 5)). The fourfolds corresponding to `generic' integrable systems
are not algebraic, and can be parametrised by generalised hypergeometric functions.
Funding
The research of E Ferapontov was partially supported by the EPSRC grant EP/N031369/1.
History
School
- Science
Department
- Mathematical Sciences
Published in
Proceedings of the London Mathematical SocietyVolume
116Issue
5Pages
1269 - 1300Citation
DOUBROV, B. ...et al., 2018. On integrability in Grassmann geometries: integrable systems associated with fourfolds in Gr(3, 5). Proceedings of the London Mathematical Society, 116(5), pp.1269-1300.Publisher
Wiley (© London Mathematical Society)Version
- AM (Accepted Manuscript)
Publisher statement
This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc/4.0/Acceptance date
2017-12-11Publication date
2018-01-31Notes
This is the accepted version of the following article: DOUBROV, B. ...et al., 2018. On integrability in Grassmann geometries: integrable systems associated with fourfolds in Gr(3; 5). Proceedings of the London Mathematical Society, 116(5), pp.1269-1300, which has been published in final form at https://doi.org/10.1112/plms.12114.ISSN
0024-6115eISSN
1460-244XPublisher version
Language
- en