11,771 research outputs found
Integral representation of solutions to Fuchsian system and Heun's equation
We obtain integral representations of solutions to special cases of the
Fuchsian system of differential equations and Heun's differential equation. In
particular, we calculate the monodromy of solutions to the Fuchsian equation
that corresponds to Picard's solution of the sixth Painlev\'e equation, and to
Heun's equation.Comment: 21 page
The Heun equation and the Calogero-Moser-Sutherland system IV: the Hermite-Krichever Ansatz
We develop a theory for the Hermite-Krichever Ansatz on the Heun equation. As
a byproduct, we find formulae which reduce hyperelliptic integrals to elliptic
ones.Comment: 34 page
Integral transformation and Darboux transformation
We review Darboux-Crum transformation of Heun's differential equation. By
rewriting an integral transformation of Heun's differential equation into a
form of elliptic functions, we see that the integral representation is a
generalization of Darboux-Crum transformation. We also consider conservation of
monodromy with respect to the transformations.Comment: 7 pages, based on the talk presented at International Workshop on
Nonlinear and Modern Mathematical Physics, China, 19th July 200
Integral transformation of Heun's equation and some applications
It is known that the Fuchsian differential equation which produces the sixth
Painlev\'e equation corresponds to the Fuchsian differential equation with
different parameters via Euler's integral transformation, and Heun's equation
also corresponds to Heun's equation with different parameters, again via
Euler's integral transformation. In this paper we study the correspondences in
detail. After investigating correspondences with respect to monodromy, it is
demonstrated that the existence of polynomial-type solutions corresponds to
apparency (non-branching) of a singularity. For the elliptical representation
of Heun's equation, correspondence with respect to monodromy implies
isospectral symmetry. We apply the symmetry to finite-gap potentials and
express the monodromy of Heun's equation with parameters which have not yet
been studied.Comment: 43 page
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