45 research outputs found
Difference analogue of the Lemma on the Logarithmic Derivative with applications to difference equations
The Lemma on the Logarithmic Derivative of a meromorphic function has many
applications in the study of meromorphic functions and ordinary differential
equations. In this paper, a difference analogue of the Logarithmic Derivative
Lemma is presented and then applied to prove a number of results on meromorphic
solutions of complex difference equations. These results include a difference
analogue of the Clunie Lemma, as well as other results on the value
distribution of solutions.Comment: 12 pages. To appear in the Journal of Mathematical Analysis and
Application
Finite-order meromorphic solutions and the discrete Painleve equations
Let w(z) be a finite-order meromorphic solution of the second-order
difference equation w(z+1)+w(z-1) = R(z,w(z)) (eqn 1) where R(z,w(z)) is
rational in w(z) and meromorphic in z. Then either w(z) satisfies a difference
linear or Riccati equation or else equation (1) can be transformed to one of a
list of canonical difference equations. This list consists of all known
difference Painleve equation of the form (1), together with their autonomous
versions. This suggests that the existence of finite-order meromorphic
solutions is a good detector of integrable difference equations.Comment: 34 page
Nevanlinna theory for the difference operator
Certain estimates involving the derivative of a meromorphic
function play key roles in the construction and applications of classical
Nevanlinna theory. The purpose of this study is to extend the usual Nevanlinna
theory to a theory for the exact difference .
An -point of a meromorphic function is said to be -paired at
z\in\C if for a fixed constant c\in\C. In this paper the
distribution of paired points of finite-order meromorphic functions is studied.
One of the main results is an analogue of the second main theorem of Nevanlinna
theory, where the usual ramification term is replaced by a quantity expressed
in terms of the number of paired points of . Corollaries of the theorem
include analogues of the Nevanlinna defect relation, Picard's theorem and
Nevanlinna's five value theorem. Applications to difference equations are
discussed, and a number of examples illustrating the use and sharpness of the
results are given.Comment: 19 page
The C-metric as a colliding plane wave space-time
It is explicitly shown that part of the C-metric space-time inside the black
hole horizon may be interpreted as the interaction region of two colliding
plane waves with aligned linear polarization, provided the rotational
coordinate is replaced by a linear one. This is a one-parameter generalization
of the degenerate Ferrari-Ibanez solution in which the focussing singularity is
a Cauchy horizon rather than a curvature singularity.Comment: 6 pages. To appear in Classical and Quantum Gravit
A nonlocal connection between certain linear and nonlinear ordinary differential equations/oscillators
We explore a nonlocal connection between certain linear and nonlinear
ordinary differential equations (ODEs), representing physically important
oscillator systems, and identify a class of integrable nonlinear ODEs of any
order. We also devise a method to derive explicit general solutions of the
nonlinear ODEs. Interestingly, many well known integrable models can be
accommodated into our scheme and our procedure thereby provides further
understanding of these models.Comment: 12 pages. J. Phys. A: Math. Gen. 39 (2006) in pres
From nothing to something: discrete integrable systems
Chinese ancient sage Laozi said that everything comes from `nothing'.
Einstein believes the principle of nature is simple. Quantum physics proves
that the world is discrete. And computer science takes continuous systems as
discrete ones. This report is devoted to deriving a number of discrete models,
including well-known integrable systems such as the KdV, KP, Toda, BKP, CKP,
and special Viallet equations, from `nothing' via simple principles. It is
conjectured that the discrete models generated from nothing may be integrable
because they are identities of simple algebra, model-independent nonlinear
superpositions of a trivial integrable system (Riccati equation), index
homogeneous decompositions of the simplest geometric theorem (the angle
bisector theorem), as well as the M\"obious transformation invariants.Comment: 11 pages, side 10 repor
Some integrable maps and their Hirota bilinear forms
We introduce a two-parameter family of birational maps, which reduces to a family previously found by Demskoi, Tran, van der Kamp and Quispel (DTKQ) when one of the parameters is set to zero. The study of the singularity confinement pattern for these maps leads to the introduction of a tau function satisfying a homogeneous recurrence which has the Laurent property, and the tropical (or ultradiscrete) analogue of this homogeneous recurrence confirms the quadratic degree growth found empirically by Demskoi et al. We prove that the tau function also satisfies two different bilinear equations, each of which is a reduction of the Hirota-Miwa equation (also known as the discrete KP equation, or the octahedron recurrence). Furthermore, these bilinear equations are related to reductions of particular two-dimensional integrable lattice equations, of discrete KdV or discrete Toda type. These connections, as well as the cluster algebra structure of the bilinear equations, allow a direct construction of Poisson brackets, Lax pairs and first integrals for the birational maps. As a consequence of the latter results, we show how each member of the family can be lifted to a system that is integrable in the Liouville sense, clarifying observations made previously in the original DTKQ case