As a result, their solutions and properties lie at the heart of the field of exact nonlinearity and integrable systems.
Inspired by results in statistical mechanics, a seminal contribution to the theory was made by Michio Jimbo, Tetsuji Miwa, and Kimio Ueno, who studied cases involving irregular singularities.
(It is thus most natural to regard a Fuchsian system geometrically, as a connection with simple poles on a trivial rank n vector bundle over the Riemann sphere).
[2] However, the proof only holds for generic data, and it was shown in 1989 by Andrei Bolibrukh that there are certain 'degenerate' cases when the answer is 'no'.
One therefore is led to study families of Fuchsian systems, where the matrices Ai depend on the positions of the poles.
In 1912 Ludwig Schlesinger proved that in general, the deformations which preserve the monodromy data of a generic Fuchsian system are governed by the integrable holonomic system of partial differential equations which now bear his name:[4] The last equation is often written equivalently as
For non-generic isomonodromic deformations, there will still be an integrable isomonodromy equation, but it will no longer be Schlesinger.
If one limits attention to the case when the Ai take values in the Lie algebra
Motivated by the appearance of Painlevé transcendents in correlation functions in the theory of Bose gases, Michio Jimbo, Tetsuji Miwa and Kimio Ueno extended the notion of isomonodromic deformation to the case of irregular singularities with any order poles, under the following assumption: the leading coefficient at each pole is generic, i.e. it is a diagonalisable matrix with simple spectrum.
Jimbo, Miwa and Ueno showed that this approach nevertheless provides canonical solutions near the singularities, and can therefore be used to define extended monodromy data.
The extended monodromy data consists of As before, one now considers families of systems of linear differential equations, all with the same (generic) singularity structure.
Jimbo, Miwa and Ueno proved that if one defines a one-form on the 'deformation parameter space' by (where D denotes exterior differentiation with respect to the components of the
As before, these equations can be interpreted as the flatness of a natural connection on the deformation parameter space.
The isomonodromy equations enjoy a number of properties that justify their status as nonlinear special functions.
This means that all essential singularities of the solutions are fixed, although the positions of poles may move.
It was proved by Bernard Malgrange for the case of Fuchsian systems, and by Tetsuji Miwa in the general setting.
Then, 'possessing a reduction to an isomonodromy equation' is more or less equivalent to the Painlevé property, and can therefore be used as a test for integrability.
The study of precisely what this transcendence means has been largely carried out by the invention of 'nonlinear differential Galois theory' by Hiroshi Umemura and Bernard Malgrange.
The study of such algebraic solutions involves examining the topology of the deformation parameter space (and in particular, its mapping class group); for the case of simple poles, this amounts to the study of the action of braid groups.
For the particularly important case of the sixth Painlevé equation, there has been a notable contribution by Boris Dubrovin and Marta Mazzocco, which has been recently extended to larger classes of monodromy data by Philip Boalch.
This viewpoint was extensively pursued by Kazuo Okamoto in a series of papers on the Painlevé equations in the 1980s.
They can also be regarding as a natural extension of the Atiyah–Bott symplectic structure on spaces of flat connections on Riemann surfaces to the world of meromorphic geometry - a perspective pursued by Philip Boalch.
Indeed, if one fixes the positions of the poles, one can even obtain complete hyperkähler manifolds; a result proved by Olivier Biquard and Philip Boalch.
[6] There is another description in terms of moment maps to (central extensions of) loop algebras - a viewpoint introduced by John Harnad and extended to the case of general singularity structure by Nick Woodhouse.
By the Penrose–Ward transform they can therefore be interpreted in terms of holomorphic vector bundles on complex manifolds called twistor spaces.
This leads to alternative descriptions of the isomonodromy equations in terms of abelian functions - an approach known to Fuchs and Painlevé, but lost until rediscovery by Yuri Manin in 1996.
The study of such behaviour goes back to the early days of isomonodromy, in work by Pierre Boutroux and others.
Their universality as some of the simplest nonlinear integrable systems means that the isomonodromy equations have a diverse range of applications.
The initial impetus for the resurgence of interest in isomonodromy in the 1970s was the appearance of transcendents in correlation functions in Bose gases.
They can also easily be adapted to take values in any Lie group, by replacing the diagonal matrices by the maximal torus, and other similar modifications.