On Implicit ODEs with Hexagonal Web of Solutions

  title={On Implicit ODEs with Hexagonal Web of Solutions},
  author={Sergey I. Agafonov},
  journal={Journal of Geometric Analysis},
  • S. Agafonov
  • Published 4 August 2008
  • Mathematics
  • Journal of Geometric Analysis
Solutions of an implicit ODE form a web. Already for a cubic ODE the 3-web of solutions has a nontrivial local invariant, namely the curvature form. Thus, any local classification of implicit ODEs necessarily has functional moduli if no restriction on the class of ODEs is imposed. In this paper the most symmetric case of hexagonal 3-web of solutions is discussed, i.e. the curvature is supposed to vanish identically. A finite list of normal forms is established under some natural regularity… 
On symmetries of singular implicit ODEs
We study implicit ODEs, cubic in derivative, with infinitesimal symmetry at singular points. Cartan showed that even at regular points the existence of nontrivial symmetry imposes restrictions on the
Local classification of singular hexagonal 3-webs with holomorphic Chern connection form and infinitesimal symmetries
Implicit ODE, cubic in derivative, generically has no infinitesimal symmetries even at regular points with distinct roots. Cartan showed that at regular points, ODEs with hexagonal 3-web of solutions
Pairs of foliations on surfaces
We survey in this paper results on a particular set of Implicit Differential Equations (IDEs) on smooth surfaces, called Binary/Quadratic Differential Equations (BDEs). These equations define at most
Webs and singularities
We investigate the singular web structure of first-order PDEs from the viewpoint of singularity theory. Most of the results given have already appeared in papers by others, as well as the author [28,
3-webs with singularities
A 3-web with singularities is an ordered collection of three one-dimensional distributions L1, L2, L3 on a 2-dimensional manifold M. The subset Σ ⊂ M where these distributions are not pairwise
Frobenius 3-Folds via Singular Flat 3-Webs
We give a geometric interpretation of weighted homogeneous solutions to the associativity equation in terms of the web theory and construct a massive Frobenius 3-fold germ via a singular 3-web germ
Flat 3-webs via semi-simple Frobenius 3-manifolds
Note on generic singularities of planar flat 3-webs
We propose a definition of genericity for singular flat planar 3-webs formed by integral curves of implicit ODEs and give a classification of generic singularities of such webs.
A generalization of the Gauss-Bonnet and Hopf-Poincar\'e theorems
We consider a locally trivial fiber bundle $\pi : E \to M$ over a compact oriented two-dimensional manifold $M$, and a section $s$ of this bundle defined over $M \setminus \Sigma$, where $\Sigma$ is


Systems of three differential equations of hydrodynamic type with hexagonal 3-web of characteristics on the solutions
In this article we obtain complete description of the class of hyperbolic systems of three differential equations of hydrodynamic type, for which the characteristics form hexa- gonal 3-web on each
Weakly nonlinear semi-Hamiltonian systems of n differential equations of hydrodynamic type in Riemann invariants are considered, and the geometry of the (n + 2)-web formed by the characteristics and
On binary differential equations
In this paper we give the local classification of solution curves of binary differential equations a(x,y)dy2+2b(x,y)dxdy+c(x,y)dx2=0 at points at which the discriminant function b2-ac has a Morse
Geometrical Methods in the Theory of Ordinary Differential Equations
Since the first edition of this book, geometrical methods in the theory of ordinary differential equations have become very popular and some progress has Since the author explains basic ideas free
Dupin indicatrices and families of curve congruences
We study a number of natural families of binary differential equations (BDE's) on a smooth surface M in R-3. One, introduced by G. J. Fletcher in 1996, interpolates between the asymptotic and
Classification of generic integral diagrams and first order ordinary differential equations
Local normal forms of generic implicit first order ordinary differential equations with independent first integrals with Legendre singularity theory and differential analysis are given.
Invariant description of solutions of hydrodynamic-type systems in hodograph space: hydrodynamic surfaces
Hydrodynamic surfaces are solutions of hydrodynamic-type systems viewed as non-parametrized submanifolds of the hodograph space. We propose an invariant differential-geometric characterization of
Notes on versal deformation offirst order PDE and web structure
Abstract We apply Thom-Mather theory to the diagram of smooth map germs associated to first order partial differential equations. This reduces the problem of function moduli of infinite dimension for
Positive Quadratic Differential Forms: Topological Equivalence Through Newton Polyhedra
Abstract.The purpose of this article is to establish conditions under which a positive quadratic differential form is topologically equivalent to its principal part defined by Newton polyhedra. The