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Deformation of big pseudoholomorphic disks and application to the Hanh pseudonorm
(Journal article; Tidsskriftartikkel; Peer reviewed, 2003-04-14)
We simplify proof of the theorem that close to any pseudoholomorphic
disk there passes a pseudoholomorphic disk of arbitrary close size with
any pre-described sufficiently close direction. We apply these results to
the Kobayashi and Hanh pseudodistances. It is shown they coincide in
dimensions higher than four. The result is new even in the complex case.
Spencer δ-cohomology, restrictions, characteristics and involutive symbolic PDEs
(Working paper; Arbeidsnotat, 2005-03-07)
We generalize the notion of involutivity to systems of differential equations
of different orders and show that the classical results relating involutivity,
restrictions, characteristics and characteristicity, known for first
order systems, extend to the general context. This involves, in particular,
a new definition of strong characteristicity. The proof exploits a spectral
sequence relating ...
Symmetry approaches for reductions of PDEs, differential constraints and Lagrange-Charpit method
(Journal article; Tidsskriftartikkel; Peer reviewed, 2007-12-20)
Many methods for reducing and simplifying differential equations are known. They provide
various generalizations of the original symmetry approach of Sophus Lie. Plenty of relations
between them have been noticed and in this note a unifying approach will be discussed.
It is rather close to the differential constraint method, but we make this rigorous basing on
recent advances in compatibility ...
Anomaly of linearization and auxiliary integrals
(Chapter; Bokkapittel, 2007-12-20)
In this note we discuss some formal properties of universal linearization operator, relate this to brackets of non-linear differential operators and discuss application to the calculus of auxiliary integrals, used in compatibility reductions of PDEs.
Invariants of pseudogroup actions: Homological methods and Finiteness theorem
(Working paper; Arbeidsnotat, 2005-12-07)
We study the equivalence problem of submanifolds with respect to a
transitive pseudogroup action. The corresponding differential invariants
are determined via formal theory and lead to the notions of l-variants and
l-covariants, even in the case of non-integrable pseudogroup. Their calculation
is based on the cohomological machinery: We introduce a complex
for covariants, define their cohomology ...
Point classification of 2nd order ODEs: Tresse classification revisited and beyond
(Chapter; Bokkapittel, 2008-09-26)
In 1896 Tresse gave a complete description of relative differential invariants
for the pseudogroup action of point transformations on the 2nd
order ODEs. The purpose of this paper is to review, in light of modern
geometric approach to PDEs, this classification and also discuss the role
of absolute invariants and the equivalence problem.
Invariant characterization of Liouville metrics and polynomial integrals
(Journal article; Tidsskriftartikkel; Peer reviewed, 2007-09-04)
A criterion in terms of differential invariants for a metric on a surface
to be Liouville is established. Moreover, in this paper we completely solve
in invariant terms the local mobility problem of a 2D metric, considered
by Darboux: How many quadratic in momenta integrals does the geodesic
flow of a given metric possess? The method is also applied to recognition
of other polynomial integrals ...
Nijenhuis tensors in pseudoholomorphic curves neighborhoods
(Working paper; Arbeidsnotat, 2000)
In this paper we consider the normal forms of almost complex structures in a neighborhood of pseudoholomorphic curve. We define normal bundles of such curves and study the properties of linear bundle almost
complex structures. We describe 1-jet of the almost complex structure along a curve in terms of its Nijenhuis tensor. For pseudoholomorphic tori we investigate the problem of pseudoholomorphic ...
Integrability via Geometry: Dispersionless Differential Equations in Three and Four Dimensions
(Journal article; Tidsskriftartikkel; Peer reviewed, 2020-11-25)
We 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 ...
Differential invariants of Einstein-Weyl structures in 3D
(Journal article; Tidsskriftartikkel; Peer reviewed, 2018-05-22)
Einstein–Weyl structures on a three-dimensional manifold <i>M</i> are given by a system <i>E</i> of PDEs on sections of a bundle over <i>M</i>. This system is invariant under the Lie pseudogroup <i>G</i> of local diffeomorphisms on <i>M</i>. Two Einstein–Weyl structures are locally equivalent if there exists a local diffeomorphism taking one to the other. Our goal is to describe the quotient equation ...