29 research outputs found
A Reciprocal Transformation for the Constant Astigmatism Equation
We introduce a nonlocal transformation to generate exact solutions of the
constant astigmatism equation . The transformation
is related to the special case of the famous B\"acklund transformation of the
sine-Gordon equation with the B\"acklund parameter . It is also
a nonlocal symmetry
On construction of symmetries and recursion operators from zero-curvature representations and the Darboux-Egoroff system
The Darboux-Egoroff system of PDEs with any number of independent
variables plays an essential role in the problems of describing -dimensional
flat diagonal metrics of Egoroff type and Frobenius manifolds. We construct a
recursion operator and its inverse for symmetries of the Darboux-Egoroff system
and describe some symmetries generated by these operators.
The constructed recursion operators are not pseudodifferential, but are
Backlund autotransformations for the linearized system whose solutions
correspond to symmetries of the Darboux-Egoroff system. For some other PDEs,
recursion operators of similar types were considered previously by
Papachristou, Guthrie, Marvan, Poboril, and Sergyeyev.
In the structure of the obtained third and fifth order symmetries of the
Darboux-Egoroff system, one finds the third and fifth order flows of an
-component vector modified KdV hierarchy.
The constructed recursion operators generate also an infinite number of
nonlocal symmetries. In particular, we obtain a simple construction of nonlocal
symmetries that were studied by Buryak and Shadrin in the context of the
infinitesimal version of the Givental-van de Leur twisted loop group action on
the space of semisimple Frobenius manifolds.
We obtain these results by means of rather general methods, using only the
zero-curvature representation of the considered PDEs.Comment: 20 pages; v2: minor change
Matching van Stockum dust to Papapetrou vacuum
Addressing a long-standing problem, we show that every van Stockum dust can
be matched to a 1-parametric family of non-static Papapetrou vacuum metrics,
and the converse. The boundary, if existing, is determined by vanishing of
certain first-order invariant on the vacuum side. Moreover, we establish a
relation to Ehlers and Kramer--Neugebauer transformations, which allows us to
look for dust clouds with a prescribed boundary. Explicit examples include the
Bonnor metric and a new vacuum exterior to the Lanczos--van Stockum dust
metric, as well as dust clouds with nontrivial topology.Comment: 13 pages, 1 figure. New in version 2: Sections 4, 8, 9, 1
On the horizontal cohomology with general coefficients
summary:[For the entire collection see Zbl 0699.00032.] \par A new cohomology theory suitable for understanding of nonlinear partial differential equations is presented. This paper is a continuation of the following paper of the author [Differ. geometry and its appl., Proc. Conf., Brno/Czech. 1986, Commun., 235-244 (1987; Zbl 0629.58033)]