176 research outputs found
Solutions for the General, Confluent and Biconfluent Heun equations and their connection with Abel equations
In a recent paper, the canonical forms of a new multi-parameter class of Abel
differential equations, so-called AIR, all of whose members can be mapped into
Riccati equations, were shown to be related to the differential equations for
the hypergeometric 2F1, 1F1 and 0F1 functions. In this paper, a connection
between the AIR canonical forms and the Heun General (GHE), Confluent (CHE) and
Biconfluent (BHE) equations is presented. This connection fixes the value of
one of the Heun parameters, expresses another one in terms of those remaining,
and provides closed form solutions in terms of pFq functions for the resulting
GHE, CHE and BHE, respectively depending on four, three and two irreducible
parameters. This connection also turns evident what is the relation between the
Heun parameters such that the solutions admit Liouvillian form, and suggests a
mechanism for relating linear equations with N and N-1 singularities through
the canonical forms of a non-linear equation of one order less.Comment: Original version submitted to Journal of Physics A: 16 pages, related
to math.GM/0002059 and math-ph/0402040. Revised version according to
referee's comments: 23 pages. Sign corrected (June/17) in formula (79).
Second revised version (July/25): 25 pages. See also
http://lie.uwaterloo.ca/odetools.ht
On the meromorphic solutions to an equation of Hayman
The behaviour of meromorphic solutions to differential
equations has been the subject of much study. Research has
concentrated on the value distribution of meromorphic solutions
and their rates of growth. The purpose of the present paper is
to show that a thorough search will yield a list of all meromorphic
solutions to a multi-parameter ordinary differential equation introduced
by Hayman. This equation does not appear to be integrable
for generic choices of the parameters so we do not find all solutions
—only those that are meromorphic. This is achieved by combining
Wiman-Valiron theory and local series analysis. Hayman conjectured
that all entire solutions of this equation are of finite order.
All meromorphic solutions of this equation are shown to be either
polynomials or entire functions of order one
Quantisation conditions of the quantum Hitchin system and the real geometric Langlands correspondence
Single-valuedness of the eigenfunctions of the quantised Hitchin Hamiltonians
is proposed as a natural quantisation condition. Separation of Variables can be
used to relate the classification of eigenstates to the classification of
projective structures with real holonomy. Using complex Fenchel-Nielsen
coordinates one may reformulate the quantisation conditions in terms of the
generating function for the variety of opers. These results are interpreted as
a variant of the geometric Langlands correspondence.Comment: 30 pages; v2: relevant corrections, close to fina
On the integrability of polynomial fields in the plane by means of Picard-Vessiot theory
We study the integrability of polynomial vector fields using Galois theory of linear differential equations when the associated foliations is reduced to a Riccati type foliation. In particular we obtain integrability results for some families of quadratic vector fields, Li\'enard equations and equations related with special functions such as Hypergeometric and Heun ones. We also study the Poincar\'e problem for some of the families.Preprin
On the integrability of polynomial vector fields in the plane by means of Picard-Vessiot theory
We study the integrability of polynomial vector fields using Galois theory of linear differential equations when the associated foliations is reduced to a Riccati type foliation. In particular we obtain integrability results for some families of quadratic vector fields, Liénard equations and equations related with special functions such as Hypergeometric and Heun ones. The Poincaré problem for some families is also approached
Leonard Euler: addition theorems and superintegrable systems
We consider the Euler approach to construction and to investigation of the
superintegrable systems related to the addition theorems. As an example we
reconstruct Drach systems and get some new two-dimensional superintegrable
Stackel systems.Comment: The text of the talk at International Conference Geometry, Dynamics,
Integrable Systems, September 2-7, 2008, Belgrade, Serbia, LaTeX, 18 page
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