Integrable Dispersionless PDEs in 4D, Their Symmetry Pseudogroups and Deformations

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.

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“…In all cases we present dispersionless Lax pairs in λ-dependent vector fields. We refer to [16] for further examples of this kind.…”

Section: Examples and Classification Resultsmentioning

confidence: 99%

Second-order PDEs in four dimensions with half-flat conformal structure

et al. 2020

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“…Additionally; we were strongly influenced both by the works of Pavlov; Bogdanov; Dryuma; Konopelchenko and Manakov [12,[32][33][34]; as well as by the work of Ferapontov and Moss [35]; in which they devised new effective differential-geometric and analytical methods for studying an integrable degenerate multi-dimensional dispersionless heavenly type hierarchy of equations; the mathematical importance of which is still far from being properly appreciated. Concerning other Lie-algebraic approaches to constructing integrable heavenly equations; we mention work by Szablikowski and Sergyeyev [36,37]; Ovsienko [17,18] and by Kruglikov and Morozov [38].…”

Section: Aψ=0mentioning

confidence: 99%

“…where 1 Moreover; the Lie algebra   can be naturally split [18,19,38] with respect to the the pairing (169) and the Lie bracket (174) into two subalgebras …”

Section: Remark 42mentioning

confidence: 99%

“…The rest of this Section is devoted to describing the Lie algebraic structure and integrability properties of a generalized hierarchy of the Lax-Sato type compatible systems of vector field equations and related nonlinear dynamical systems; based on the AKS-algebraic and related -structure schemes [22][23][24][25][26]; recently devised in B. Szablikowski and A. Sergyeyev [37,38]; V. Ovsienko [18,19] and also in refs. [28,29].…”

Section: Remark 52mentioning

confidence: 99%

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The Novel Lie-Algebraic Approach to Studying Integrable Heavenly Type Multi-Dimensional Dynamical Systems

Blackmore¹,

Eo²,

Ak³

2017

J Generalized Lie Theory Appl

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The review is devoted to a novel Lie-algebraic approach to studying integrable heavenly type multi-dimensional dynamical systems and its relationships to old and recent investigations of the classical Buhl problem of describing compatible linear vector field equations, its general Pfeiffer and modern Lax-Sato type special solutions. Eespecially we analyze the related Lie-algebra structures and integrability properties of a very interesting class of nonlinear dynamical systems called the dispersionless heavenly type equations, which were initiated by Pleban′ski and later analyzed in a series of articles. The AKS-algebraic and related -structure schemes are used to study the orbits of the corresponding co-adjoint actions, which are intimately related to the classical Lie-Poisson structures on them. It is demonstrated that their compatibility condition coincides with the corresponding heavenly type equations under consideration. It is shown that all these equations originate in this way and can be represented as a Lax compatibility condition for specially constructed loop vector fields on the torus. The infinite hierarchy of conservations laws related to the heavenly equations is described, and its analytical structure connected with the Casimir invariants, is mentioned. In addition, typical examples of such equations, demonstrating in detail their integrability via the scheme devised herein, are presented. The relationship of a very interesting Lagrange-d'Alembert type mechanical interpretation of the devised integrability scheme with the Lax-Sato equations is also discussed.

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“…In [3] we constructed symmetric integrable deformations of some heavenly type equations. These also exist for the equations (1), (2), (4) and (6) considered in this paper.…”

mentioning

confidence: 99%

A Bäcklund transformation between the four-dimensional Martínez Alonso–Shabat and Ferapontov–Khusnutdinova equations

Kruglikov

Morozov²

2016

Theor Math Phys

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The aim of this note is to construct a Bäcklund transformation between the Lax-integrable 4-dimensional equationsandintroduced in [4] and [1], respectively. Equation (2) has the following Lax pair [1]:with non-removable parameter λ. Excluding u from (3) and normalizing λ = 1 in the resulting equation by re-scaling we get the equationThus the differential covering (3) (in terms of [2]) defines a Bäcklund transformation between equations ( 2) and (4).Proposition. Equations ( 1) and ( 4) are point equivalent.Proof. Prolongation of the transformation ψ :maps equation (4) to equation (1). q.e.d.Corollary. The superposition of ( 5) and (3) defines a Bäcklund transformation between equations (1) and (2).

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Integrable Dispersionless PDEs in 4D, Their Symmetry Pseudogroups and Deformations

Cited by 42 publications

References 27 publications

Second-order PDEs in four dimensions with half-flat conformal structure

Second-order PDEs in four dimensions with half-flat conformal structure

The Novel Lie-Algebraic Approach to Studying Integrable Heavenly Type Multi-Dimensional Dynamical Systems

A Bäcklund transformation between the four-dimensional Martínez Alonso–Shabat and Ferapontov–Khusnutdinova equations

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