140 research outputs found
Discrete systems related to some equations of the Painlev\'e-Gambier classification
We derive integrable discrete systems which are contiguity relations of two
equations in the Painlev\'e-Gambier classification depending on some parameter.
These studies extend earlier work where the contiguity relations for the six
transcendental Painlev\'e equations were obtained. In the case of the Gambier
equation we give the contiguity relations for both the continuous and the
discrete system.Comment: 10 page
The Gambier Mapping
We propose a discrete form for an equation due to Gambier and which belongs
to the class of the fifty second order equations that possess the Painleve
property. In the continuous case, the solutions of the Gambier equation is
obtained through a system of Riccati equations. The same holds true in the
discrete case also. We use the singularity confinement criterion in order to
study the integrability of this new mapping.Comment: PlainTe
Again, Linearizable Mappings
We examine a family of 3-point mappings that include mappings solvable
through linearization. The different origins of mappings of this type are
examined: projective equations and Gambier systems. The integrable cases are
obtained through the application of the singularity confinement criterion and
are explicitly integrated.Comment: 14 pages, no figures, to be published in Physica
Discrete and Continuous Linearizable Equations
We study the projective systems in both continuous and discrete settings.
These systems are linearizable by construction and thus, obviously, integrable.
We show that in the continuous case it is possible to eliminate all variables
but one and reduce the system to a single differential equation. This equation
is of the form of those singled-out by Painlev\'e in his quest for integrable
forms. In the discrete case, we extend previous results of ours showing that,
again by elimination of variables, the general projective system can be written
as a mapping for a single variable. We show that this mapping is a member of
the family of multilinear systems (which is not integrable in general). The
continuous limit of multilinear mappings is also discussed.Comment: Plain Tex file, 14 pages, no figur
Bilinear structure and Schlesinger transforms of the -P and -P equations
We show that the recently derived (-) discrete form of the Painlev\'e VI
equation can be related to the discrete P, in particular if one
uses the full freedom in the implementation of the singularity confinement
criterion. This observation is used here in order to derive the bilinear forms
and the Schlesinger transformations of both -P and -P.Comment: 10 pages, Plain Te
The Gambier Mapping, Revisited
We examine critically the Gambier equation and show that it is the generic
linearisable equation containing, as reductions, all the second-order equations
which are integrable through linearisation. We then introduce the general
discrete form of this equation, the Gambier mapping, and present conditions for
its integrability. Finally, we obtain the reductions of the Gambier mapping,
identify their integrable forms and compute their continuous limits.Comment: 11 pages, no figures, to be published in Physica
- …