Integrability of Dispersionless Hirota-Type Equations and the Symplectic Monge–Ampère Property

2020 ◽  
Author(s):  
E V Ferapontov ◽  
B Kruglikov ◽  
V Novikov
Keyword(s):  
Type Equation ◽  
Lax Pair ◽  
Complete Classification ◽  
Conformal Structure ◽  
Characteristic Variety ◽  
Self Duality ◽  
Symmetry Properties ◽  
Monge Ampere ◽  
Involutive System

Abstract We prove that integrability of a dispersionless Hirota-type equation implies the symplectic Monge–Ampère property in any dimension $\geq 4$. In 4D, this yields a complete classification of integrable dispersionless partial differential equations (PDEs) of Hirota type through a list of heavenly type equations arising in self-dual gravity. As a by-product of our approach, we derive an involutive system of relations characterizing symplectic Monge–Ampère equations in any dimension. Moreover, we demonstrate that in 4D the requirement of integrability is equivalent to self-duality of the conformal structure defined by the characteristic variety of the equation on every solution, which is in turn equivalent to the existence of a dispersionless Lax pair. We also give a criterion of linearizability of a Hirota-type equation via flatness of the corresponding conformal structure and study symmetry properties of integrable equations.

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

2020 ◽  
Vol 476 (2233) ◽  
pp. 20190642
Author(s):  
S. Berjawi ◽  
E. V. Ferapontov ◽  
B. Kruglikov ◽  
V. Novikov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Lax Pair ◽  
Conformal Structure ◽  
Second Order ◽  
Partial Differential ◽  
Characteristic Variety ◽  
Four Dimensions ◽  
Partial Classification ◽  
Monge Ampere

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.

scholarly journals PROJECTIVE THREEFOLDS WITH HOLOMORPHIC CONFORMAL STRUCTURE

2005 ◽  
Vol 16 (06) ◽  
pp. 595-607 ◽  
Author(s):  
PRISKA JAHNKE ◽  
IVO RADLOFF
Keyword(s):  
Tangent Bundle ◽  
Complete Classification ◽  
Conformal Structure ◽  
Complete List ◽  
Projective Threefolds ◽  
Symmetric Square

The authors give a complete classification of projective threefolds admitting a holomorphic conformal structure. A corollary is the complete list of projective threefolds, whose tangent bundle is a symmetric square.

scholarly journals Second-Order PDEs in 3D with Einstein–Weyl Conformal Structure

2021 ◽  
Author(s):  
S. Berjawi ◽  
E. V. Ferapontov ◽  
B. S. Kruglikov ◽  
V. S. Novikov
Keyword(s):  
Three Dimensional ◽  
Lax Pair ◽  
Conformal Structure ◽  
Second Order ◽  
Contact Transformation ◽  
Conformal Class ◽  
Weyl Geometry ◽  
Symmetric Connection ◽  
Integrable Pdes ◽  
Monge Ampere

AbstractEinstein–Weyl geometry is a triple $$({\mathbb {D}},g,\omega )$$ ( D , g , ω ) where $${\mathbb {D}}$$ D is a symmetric connection, [g] is a conformal structure and $$\omega $$ ω is a covector such that $$\bullet $$ ∙ connection $${\mathbb {D}}$$ D preserves the conformal class [g], that is, $${\mathbb {D}}g=\omega g$$ D g = ω g ; $$\bullet $$ ∙ trace-free part of the symmetrised Ricci tensor of $${\mathbb {D}}$$ D vanishes. Three-dimensional Einstein–Weyl structures naturally arise on solutions of second-order dispersionless integrable PDEs in 3D. In this context, [g] coincides with the characteristic conformal structure and is therefore uniquely determined by the equation. On the contrary, covector $$\omega $$ ω is a somewhat more mysterious object, recovered from the Einstein–Weyl conditions. We demonstrate that, for generic second-order PDEs (for instance, for all equations not of Monge–Ampère type), the covector $$\omega $$ ω is also expressible in terms of the equation, thus providing an efficient ‘dispersionless integrability test’. The knowledge of g and $$\omega $$ ω provides a dispersionless Lax pair by an explicit formula which is apparently new. Some partial classification results of PDEs with Einstein–Weyl characteristic conformal structure are obtained. A rigidity conjecture is proposed according to which for any generic second-order PDE with Einstein–Weyl property, all dependence on the 1-jet variables can be eliminated via a suitable contact transformation.

scholarly journals Integrability via Geometry: Dispersionless Differential Equations in Three and Four Dimensions

2020 ◽  
Author(s):  
David M. J. Calderbank ◽  
Boris Kruglikov
Keyword(s):  
Spectral Parameter ◽  
Characteristic Property ◽  
Lax Pair ◽  
Conformal Structure ◽  
Order Partial Differential Equation ◽  
Main Ingredient ◽  
Characteristic Variety ◽  
Four Dimensions ◽  
Quadric Hypersurface ◽  
Systems Of Pdes

AbstractWe prove that the existence of a dispersionless Lax pair with spectral parameter for a nondegenerate hyperbolic second order partial differential equation (PDE) is equivalent to the canonical conformal structure defined by the symbol being Einstein–Weyl on any solution in 3D, and self-dual on any solution in 4D. The first main ingredient in the proof is a characteristic property for dispersionless Lax pairs. The second is the projective behaviour of the Lax pair with respect to the spectral parameter. Both are established for nondegenerate determined systems of PDEs of any order. Thus our main result applies more generally to any such PDE system whose characteristic variety is a quadric hypersurface.

scholarly journals On the blow-up and positive entire solutions of semilinear elliptic equations

2000 ◽  
Vol 43 (3) ◽  
pp. 625-631
Author(s):  
Jann-Long Chern
Keyword(s):  
Elliptic Equations ◽  
Blow Up ◽  
Type Equation ◽  
Semilinear Elliptic Equation ◽  
Complete Classification ◽  
Entire Solutions ◽  
Solution Structures ◽  
Semilinear Elliptic ◽  
Image Position

AbstractIn this paper we consider the following semilinear elliptic equationwhere n ≥ 3, and β ≥ 0, γ ≥ 0, q > p ≥ 1, μ and ν are real constants. We note that if γ = 0, β > 0 and ν ≥ 2, then the equation above is called the Matukuma-type equation. If β = 0, γ > 0 and ν > 2, then the complete classification of all possible positive solutions had been conducted by Cheng and Ni. If β > 0, γ > 0 and μ ≥ ν ≥ 2, then some results about the maximal solution and positive solution structures can be found in Chern. The purpose of this paper is to discuss and investigate the blow-up and positive entire solutions of the equation above for the μ ≥ 2 ≥ ν case.

scholarly journals Integrable Systems in Four Dimensions Associated with Six-Folds in Gr(4, 6)

2018 ◽  
Vol 2019 (21) ◽  
pp. 6585-6613 ◽  
Author(s):  
Boris Doubrov ◽  
Evgeny V Ferapontov ◽  
Boris Kruglikov ◽  
Vladimir S Novikov
Keyword(s):  
Integrable Systems ◽  
Conformal Structure ◽  
Characteristic Variety ◽  
Four Dimensions ◽  
Quadratic Maps ◽  
All Solutions ◽  
Hermitian Geometry ◽  
Linear Sections ◽  
Plücker Embedding ◽  
Monge Ampere

Abstract Let Gr(d, n) be the Grassmannian of d-dimensional linear subspaces of an n-dimensional vector space V. A submanifold X ⊂ Gr(d, n) gives rise to a differential system Σ(X) that governs d-dimensional submanifolds of V whose Gaussian image is contained in X. We investigate a special case of this construction where X is a six-fold in Gr(4, 6). The corresponding system Σ(X) reduces to a pair of first-order PDEs for 2 functions of 4 independent variables. Equations of this type arise in self-dual Ricci-flat geometry. Our main result is a complete description of integrable systems Σ(X). These naturally fall into two subclasses. • Systems of Monge–Ampère type. The corresponding six-folds X are codimension 2 linear sections of the Plücker embedding Gr(4, 6)$ \hookrightarrow \mathbb{P}^{14}$. • General linearly degenerate systems. The corresponding six-folds X are the images of quadratic maps $\mathbb{P}^{6}\dashrightarrow \ $Gr(4, 6) given by a version of the classical construction of Chasles. We prove that integrability is equivalent to the requirement that the characteristic variety of system Σ(X) gives rise to a conformal structure which is self-dual on every solution. In fact, all solutions carry hyper-Hermitian geometry.

scholarly journals Multiplicative automatic sequences

2021 ◽  
Author(s):  
Jakub Konieczny ◽  
Mariusz Lemańczyk ◽  
Clemens Müllner
Keyword(s):  
Complete Classification ◽  
Automatic Sequences ◽  
Complex Valued

AbstractWe obtain a complete classification of complex-valued sequences which are both multiplicative and automatic.

scholarly journals Characteristic cohomology and observables in higher spin gravity

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Alexey Sharapov ◽  
Evgeny Skvortsov
Keyword(s):  
Higher Spin Gravity ◽  
Complete Classification ◽  
Gravity Models ◽  
Simons Theory ◽  
Surface Charges ◽  
Higher Spin ◽  
Dynamical Invariants ◽  
Chern Simons ◽  
Conserved Currents

Abstract We give a complete classification of dynamical invariants in 3d and 4d Higher Spin Gravity models, with some comments on arbitrary d. These include holographic correlation functions, interaction vertices, on-shell actions, conserved currents, surface charges, and some others. Surprisingly, there are a good many conserved p-form currents with various p. The last fact, being in tension with ‘no nontrivial conserved currents in quantum gravity’ and similar statements, gives an indication of hidden integrability of the models. Our results rely on a systematic computation of Hochschild, cyclic, and Chevalley-Eilenberg cohomology for the corresponding higher spin algebras. A new invariant in Chern-Simons theory with the Weyl algebra as gauge algebra is also presented.

scholarly journals Nil-clean quadratic elements

2017 ◽  
Vol 16 (10) ◽  
pp. 1750197 ◽  
Author(s):  
Janez Šter
Keyword(s):  
Complete Classification ◽  
Strong Condition ◽  
Clean Rings

We provide a strong condition holding for nil-clean quadratic elements in any ring. In particular, our result implies that every nil-clean involution in a ring is unipotent. As a consequence, we give a complete classification of weakly nil-clean rings introduced recently in [Breaz, Danchev and Zhou, Rings in which every element is either a sum or a difference of a nilpotent and an idempotent, J. Algebra Appl. 15 (2016) 1650148, doi: 10.1142/S0219498816501486].

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