Dispersionless integrable systems in 3D and Einstein-Weyl geometry

Ferapontov, E. V.; Kruglikov, Boris

doi:10.4310/jdg/1405447805

Cited by 74 publications

(134 citation statements)

References 55 publications

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“…• For every solution of the equation, the symbol of formal linearisation defines an EinsteinWeyl structure [10].…”

Section: Characteristic Integrals Of Second Order Pdes In 3dmentioning

confidence: 99%

Characteristic integrals in 3D and linear degeneracy

Ferapontov¹,

Moss²

2021

JNMP

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Conservation laws vanishing along characteristic directions of a given system of PDEs are known as characteristic conservation laws, or characteristic integrals. In 2D, they play an important role in the theory of Darboux-integrable equations. In this paper we discuss characteristic integrals in 3D and demonstrate that, for a class of second order linearly degenerate dispersionless integrable PDEs, the corresponding characteristic integrals are parametrised by points on the Veronese variety.

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“…• For every solution of the equation, the symbol of formal linearisation defines an EinsteinWeyl structure [10].…”

Section: Characteristic Integrals Of Second Order Pdes In 3dmentioning

confidence: 99%

Characteristic integrals in 3D and linear degeneracy

Ferapontov¹,

Moss²

2021

JNMP

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“…where D x k denotes total derivative with respect to the independent variable x k (note that g depends on first-order jets of the solution u, v). Formula (4) appeared in [24] in geometric approach to the dispersionless integrability in 3D. It is invariant under the gauge transformation g → λg, ω → ω + d ln λ, the property characteristic of Einstein-Weyl geometry.…”

Section: Einstein-weyl Geometry In 3dmentioning

confidence: 99%

On integrability in Grassmann geometries: integrable systems associated with fourfolds in Gr(3,5)

Doubrov

Ferapontov

Kruglikov

et al. 2018

Proc. London Math. Soc.

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Let 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 Σ(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 well-known examples as the dispersionless Kadomtsev-Petviashvili equation, the Boyer-Finley equation, Plebańsky'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 six-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 P 4 Gr(3, 5)). The fourfolds corresponding to 'generic' integrable systems are not algebraic, and can be parametrised by generalised hypergeometric functions.

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“…It was shown that the moduli space of integrable equations is 20-dimensional. The recent result of [15], stating that the integrability of a PDE of the form (1) is equivalent to the Einstein-Weyl property of the symbol of its formal linearisation, provides an efficient geometric test of integrability. It was demonstrated in [41] that the coefficients of 'generic' integrable equations (1) can be parametrised by generalized hypergeometric functions.…”

Section: Theoremmentioning

confidence: 99%

“…Although the subject is classical, the following result is new: Since conformal flatness is the necessary condition for integrability, a complete list of linearly degenerate integrable PDEs can be obtained by going through the list of Theorem 3 and either calculating the integrability conditions as derived in [7], or verifying the existence of a Lax pair. Another possibility is to utilise the recent result of [15] …”

Section: Normal Forms Of Quadratic Line Complexes and Linearly Degenementioning

confidence: 99%

“…This constitutes eleven canonical forms which are analysed case-by-case below. In each case we calculate the conditions of vanishing of the Cotton tensor (responsible for conformal flatness in three dimensions), as well as the integrability conditions as derived in [7], see also [15] for their geometric reformulation. Recall that conformal flatness is a necessary condition for integrability: this requirement already leads to a compact list of conformally flat subcases which can be checked for integrability by calculating the Lax pair.…”

Section: Proof Of Theorems 2-4mentioning

confidence: 99%

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Linearly degenerate partial differential equations and quadratic line complexes

Ferapontov

Mos

2015

Communications in Analysis and Geometry

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A quadratic line complex is a three-parameter family of lines in projective space P 3 specified by a single quadratic relation in the Plücker coordinates. Fixing a point p in P 3 and taking all lines of the complex passing through p we obtain a quadratic cone with vertex at p. This family of cones supplies P 3 with a conformal structure, which can be represented in the form f ij (p)dp i dp j in a system of affine coordinates p = (p 1 , p 2 , p 3 ). With this conformal structure we associate a three-dimensional second-order quasilinear wave equation,whose coefficients can be obtained from f ij (p) by setting p 1 = u x1 , p 2 = u x2 , p 3 = u x3 . We show that any PDE arising in this way is linearly degenerate, furthermore, any linearly degenerate PDE can be obtained by this construction. This provides a classification of linearly degenerate wave equations into eleven types, labelled by Segre symbols of the associated quadratic complexes. We classify Segre types for which the structure f ij (p)dp i dp j is conformally flat, as well as Segre types for which the corresponding PDE is integrable.MSC: 14J81, 35A30, 35L10, 37K10, 37K25, 53B50, 53C80.

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Dispersionless integrable systems in 3D and Einstein-Weyl geometry

Cited by 74 publications

References 55 publications

Characteristic integrals in 3D and linear degeneracy

Characteristic integrals in 3D and linear degeneracy

On integrability in Grassmann geometries: integrable systems associated with fourfolds in Gr(3,5)

Linearly degenerate partial differential equations and quadratic line complexes

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