35 research outputs found
Coverings and the fundamental group for partial differential equations
Following I. S. Krasilshchik and A. M. Vinogradov, we regard systems of PDEs as manifolds with involutive distributions and consider their special morphisms called differential coverings, which include constructions like Lax pairs and B\"acklund transformations in soliton theory. We show that, similarly to usual coverings in topology, at least for some PDEs differential coverings are determined by actions of a sort of fundamental group. This is not a discrete group, but a certain system of Lie groups. From this we deduce an algebraic necessary condition for two PDEs to be connected by a B\"acklund transformation. For the KdV equation and the nonsingular Krichever-Novikov equation these systems of Lie groups are determined by certain infinite-dimensional Lie algebras of Kac-Moody type. We prove that these two equations are not connected by any B\"acklund transformation. To achieve this, for a wide class of Lie algebras we prove that any subalgebra of of finite codimension contains an ideal of of finite codimension
Nonlocal conservation laws of PDEs possessing differential coverings
In his 1892 paper [L. Bianchi, Sulla trasformazione di B\"{a}cklund per le
superfici pseudosferiche, Rend. Mat. Acc. Lincei, s. 5, v. 1 (1892) 2, 3--12],
L. Bianchi noticed, among other things, that quite simple transformations of
the formulas that describe the B\"{a}cklund transformation of the sine-Gordon
equation lead to what is called a nonlocal conservation law in modern language.
Using the techniques of differential coverings [I.S. Krasil'shchik, A.M.
Vinogradov, Nonlocal trends in the geometry of differential equations:
symmetries, conservation laws, and B\"{a}cklund transformations, Acta Appl.
Math. v. 15 (1989) 1-2, 161--209], we show that this observation is of a quite
general nature. We describe the procedures to construct such conservation laws
and present a number of illustrative examples
Homogeneous Hamiltonian operators and the theory of coverings
A new method (by Kersten, Krasil'shchik and Verbovetsky), based on the theory
of differential coverings, allows to relate a system of PDEs with a
differential operator in such a way that the operator maps symmetries/conserved
quantities into symmetries/conserved quantities of the system of PDEs. When
applied to a quasilinear first-order system of PDEs and a Dubrovin-Novikov
homogeneous Hamiltonian operator the method yields conditions on the operator
and the system that have interesting differential and projective geometric
interpretations
Recursion Operators and Nonlocal Symmetries for Integrable rmdKP and rdDym Equations
We find direct and inverse recursion operators for integrable cases of the
rmdKP and rdDym equations. Also, we study actions of these operators on the
contact symmetries and find shadows of nonlocal symmetries of these equations