13 research outputs found
Algebraic entropy for differential-delay equations
We extend the definition of algebraic entropy to a class of
differential-delay equations. The vanishing of the entropy, as a structural
property of an equation, signals its integrability. We suggest a simple way to
produce differential-delay equations with vanishing entropy from known
integrable differential-difference equations
On the algebraic structure of rational discrete dynamical systems
We show how singularities shape the evolution of rational discrete dynamical
systems. The stabilisation of the form of the iterates suggests a description
providing among other things generalised Hirota form, exact evaluation of the
algebraic entropy as well as remarkable polynomial factorisation properties. We
illustrate the phenomenon explicitly with examples covering a wide range of
models
Scattering of cosmic strings by black holes: loop formation
We study the deformation of a long cosmic string by a nearby rotating black
hole. We examine whether the deformation of a cosmic string, induced by the
gravitational field of a Kerr black hole, may lead to the formation of a loop
of cosmic string. The segment of the string which enters the ergosphere of a
rotating black hole gets deformed and, if it is sufficiently twisted, it can
self-intersect chopping off a loop of cosmic string. We find that the formation
of a loop, via this mechanism, is a rare event. It will only arise in a small
region of the collision phase space, which depends on the string velocity, the
impact parameter and the black hole angular momentum. We conclude that
generically, the cosmic string is simply scattered or captured by the rotating
black hole.Comment: 11 pages, 2 figures, RevTe
Integrable lattice equations with vertex and bond variables
We present integrable lattice equations on a two dimensional square lattice
with coupled vertex and bond variables. In some of the models the vertex
dynamics is independent of the evolution of the bond variables, and one can
write the equations as non-autonomous "Yang-Baxter maps". We also present a
model in which the vertex and bond variables are fully coupled. Integrability
is tested with algebraic entropy as well as multidimensional consistencyComment: 15 pages, remarks added, other minor change
How to detect the integrability of discrete systems
International audienceSeveral integrability tests for discrete equations will be reviewed. All tests considered can be applied directly to a given discrete equation and do not rely on the a priori knowledge of the existence of related structures such as Lax pairs. Specifically, singularity confinement, algebraic entropy, Nevanlinna theory, Diophantine integrability and discrete systems over finite fields will be described