37 research outputs found
A systematic method for constructing discrete Painlev\'e equations in the degeneration cascade of the E group
We present a systematic and quite elementary method for constructing discrete
Painlev\'e equations in the degeneration cascade for E. Starting from
the invariant for the autonomous limit of the E equation one wishes
to study, the method relies on choosing simple homographies that will cast this
invariant into certain judiciously chosen canonical forms. These new invariants
lead to mappings the deautonomisations of which allow us to build up the entire
degeneration cascade of the original mapping. We explain the method on three
examples, two symmetric mappings and an asymmetric one, and we discuss the link
between our results and the known geometric structure of these mappings.Comment: 22 pages, 5 figure
Dynamical Studies of Equations from the Gambier Family
We consider the hierarchy of higher-order Riccati equations and establish
their connection with the Gambier equation. Moreover we investigate the
relation of equations of the Gambier family to other nonlinear differential
systems. In particular we explore their connection to the generalized
Ermakov-Pinney and Milne-Pinney equations. In addition we investigate the
consequence of introducing Okamoto's folding transformation which maps the
reduced Gambier equation to a Li\'enard type equation. Finally the conjugate
Hamiltonian aspects of certain equations belonging to this family and their
connection with superintegrability are explored
Do All Integrable Evolution Equations Have the Painlev\'e Property?
We examine whether the Painleve property is necessary for the integrability
of partial differential equations (PDEs). We show that in analogy to what
happens in the case of ordinary differential equations (ODEs) there exists a
class of PDEs, integrable through linearisation, which do not possess the
Painleve property. The same question is addressed in a discrete setting where
we show that there exist linearisable lattice equations which do not possess
the singularity confinement property (again in analogy to the one-dimensional
case).Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Symmetries of Discrete Systems
In this series of lectures presented at the CIMPA Winter School on Discrete
Integrable Systems in Pondicherry, India, in February, 2003 we give a review of
the application of Lie point symmetries, and their generalizations to the study
of difference equations. The overall theme of these lectures could be called
"continuous symmetries of discrete equations".Comment: 58 pages, 5 figures, Lectures presented at the Winter School on
Discrete Integrable Systems in Pondicherry, India, February 200
Multiplicative equations related to the affine Weyl group E
We derive integrable equations starting from autonomous mappings with a
general form inspired by the multiplicative systems associated to the affine
Weyl group E. Five such systems are obtained, three of which turn out
to be linearisable and the remaining two are integrable in terms of elliptic
functions. In the case of the linearisable mappings we derive nonautonomous
forms which contain a free function of the dependent variable and we present
the linearisation in each case. The two remaining systems are deautonomised to
new discrete Painlev\'e equations. We show that these equations are in fact
special forms of much richer systems associated to the affine Weyl groups
E and E respectively.Comment: 9 pages, no figure
Casorati Determinant Solutions for the Discrete Painlev\'e III Equation
The discrete Painlev\'e III equation is investigated based on the bilinear
formalism. It is shown that it admits the solutions expressed by the Casorati
determinant whose entries are given by the discrete Bessel function. Moreover,
based on the observation that these discrete Bessel functions are transformed
to the -Bessel functions by a simple variable transformation, we present a
-difference analogue of the Painlev\'e III equation.Comment: 16 pages in LaTe