202 research outputs found
Another integrable case in the Lorenz model
A scaling invariance in the Lorenz model allows one to consider the usually
discarded case sigma=0. We integrate it with the third Painlev\'e function.Comment: 3 pages, no figure, to appear in J. Phys.
A mixed methods exploratory study of women’s relationships with and uses of fertility tracking apps
This work was conducted as a BSc (Hons) research project at the University of Aberdeen. No external funding was received.Peer reviewedPublisher PD
Constructing Integrable Third Order Systems:The Gambier Approach
We present a systematic construction of integrable third order systems based
on the coupling of an integrable second order equation and a Riccati equation.
This approach is the extension of the Gambier method that led to the equation
that bears his name. Our study is carried through for both continuous and
discrete systems. In both cases the investigation is based on the study of the
singularities of the system (the Painlev\'e method for ODE's and the
singularity confinement method for mappings).Comment: 14 pages, TEX FIL
Mutation du foncier agricole en frange urbaine : élaboration et mise à l'épreuve d'une politique de régulation territoriale
International audienceLe pôle urbain de Montpellier s'est largement étalé lors des dernières décennies à la faveur de son dynamisme démographique et du retrait de la viticulture : la " maîtrise " communale de l'urbanisme s'est traduite par une urbanisation tous azimuts. Comment mettre fin à ces processus sans compromettre l'économie résidentielle ? C'est une priorité de la nouvelle intercommunalité dont la politique d'aménagement est analysée. La genèse de cette politique est retracée ainsi que sa traduction dans le schéma de cohérence territoriale, dont les objectifs d'économie d'espace et les outils - identification des limites, densification, maîtrise foncière - sont détaillés. Sa mise en ½uvre au niveau communal révèle les forces et les faiblesses de cette politique. Cette étude empirique soutient une réflexion plus générale sur les dynamiques foncières périurbaines et les outils politiques de régulation, à la croisée des problématiques de gouvernance territoriale et de préservation des ressources
Bilinear Discrete Painleve-II and its Particular Solutions
By analogy to the continuous Painlev\'e II equation, we present particular
solutions of the discrete Painlev\'e II (d-P) equation. These
solutions are of rational and special function (Airy) type. Our analysis is
based on the bilinear formalism that allows us to obtain the function
for d-P. Two different forms of bilinear d-P are obtained
and we show that they can be related by a simple gauge transformation.Comment: 9 pages in plain Te
Integrable systems without the Painlev\'e property
We examine whether the Painlev\'e property is a necessary condition for the
integrability of nonlinear ordinary differential equations. We show that for a
large class of linearisable systems this is not the case. In the discrete
domain, we investigate whether the singularity confinement property is
satisfied for the discrete analogues of the non-Painlev\'e continuous
linearisable systems. We find that while these discrete systems are themselves
linearisable, they possess nonconfined singularities
Singularity confinement and algebraic integrability
Two important notions of integrability for discrete mappings are algebraic
integrability and singularity confinement, have been used for discrete
mappings. Algebraic integrability is related to the existence of sufficiently
many conserved quantities whereas singularity confinement is associated with
the local analysis of singularities. In this paper, the relationship between
these two notions is explored for birational autonomous mappings. Two types of
results are obtained: first, algebraically integrable mappings are shown to
have the singularity confinement property. Second, a proof of the non-existence
of algebraic conserved quantities of discrete systems based on the lack of
confinement property is given.Comment: 18 pages, no figur
Mappings preserving locations of movable poles: a new extension of the truncation method to ordinary differential equations
The truncation method is a collective name for techniques that arise from
truncating a Laurent series expansion (with leading term) of generic solutions
of nonlinear partial differential equations (PDEs). Despite its utility in
finding Backlund transformations and other remarkable properties of integrable
PDEs, it has not been generally extended to ordinary differential equations
(ODEs). Here we give a new general method that provides such an extension and
show how to apply it to the classical nonlinear ODEs called the Painleve
equations. Our main new idea is to consider mappings that preserve the
locations of a natural subset of the movable poles admitted by the equation. In
this way we are able to recover all known fundamental Backlund transformations
for the equations considered. We are also able to derive Backlund
transformations onto other ODEs in the Painleve classification.Comment: To appear in Nonlinearity (22 pages
B\"acklund transformations for the second Painlev\'e hierarchy: a modified truncation approach
The second Painlev\'e hierarchy is defined as the hierarchy of ordinary
differential equations obtained by similarity reduction from the modified
Korteweg-de Vries hierarchy. Its first member is the well-known second
Painlev\'e equation, P2.
In this paper we use this hierarchy in order to illustrate our application of
the truncation procedure in Painlev\'e analysis to ordinary differential
equations. We extend these techniques in order to derive auto-B\"acklund
transformations for the second Painlev\'e hierarchy. We also derive a number of
other B\"acklund transformations, including a B\"acklund transformation onto a
hierarchy of P34 equations, and a little known B\"acklund transformation for P2
itself.
We then use our results on B\"acklund transformations to obtain, for each
member of the P2 hierarchy, a sequence of special integrals.Comment: 12 pages in LaTeX 2.09 (uses ioplppt.sty), to appear in Inverse
Problem
Form factor expansions in the 2D Ising model and Painlev\'e VI
We derive a Toda-type recurrence relation, in both high and low temperature
regimes, for the - extended diagonal correlation functions
of the two-dimensional Ising model, using an earlier
connection between diagonal form factor expansions and tau-functions within
Painlev\'e VI (PVI) theory, originally discovered by Jimbo and Miwa. This
greatly simplifies the calculation of the diagonal correlation functions,
particularly their -extended counterparts. We also conjecture a closed
form expression for the simplest off-diagonal case where
a connection to PVI theory is not known. Combined with the results for diagonal
correlations these give all the initial conditions required for the
\l-extended version of quadratic difference equations for the correlation
functions discovered by McCoy, Perk and Wu. The results obtained here should
provide a further potential algorithmic improvement in the \l-extended case,
and facilitate other developments.Comment: 23 pages, references added, introduction extended, abstract modified,
misprints correcte
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