Isomonodromic deformation

In mathematics, the equations governing the isomonodromic deformation of meromorphic linear systems of ordinary differential equations are, in a fairly precise sense, the most fundamental exact nonlinear differential equations. As a result, their solutions and properties lie at the heart of the field of exact nonlinearity and integrable systems.
Isomonodromic deformations were first studied by Richard Fuchs, with early pioneering contributions from Lazarus Fuchs, Paul Painlevé, René Garnier, and Ludwig Schlesinger. 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.

Fuchsian systems and Schlesinger's equations

Fuchsian system

A Fuchsian system is the system of linear differential equations
where x takes values in the complex projective line, the y takes values in and the A_i are constant n×''n matrices. Solutions to this equation have polynomial growth in the limit x'' = λ_i.
By placing n independent column solutions into a fundamental matrix
then and
one can regard as taking values in. For simplicity, assume that there is no further pole at infinity, which amounts to the condition that

Monodromy data

Now, fix a basepoint b on the Riemann sphere away from the poles. Analytic continuation of a fundamental solution around any pole λ_i and back to the basepoint will produce a new solution defined near b. The new and old solutions are linked by the monodromy matrix M_i as follows:
One therefore has the Riemann–Hilbert homomorphism from the fundamental group of the punctured sphere to the monodromy representation:
A change of basepoint merely results in a conjugation of all the monodromy matrices. The monodromy matrices modulo conjugation define the monodromy data of the Fuchsian system.

Hilbert's twenty-first problem

Now, with given monodromy data, can a Fuchsian system be found which exhibits this monodromy? This is one form of Hilbert's twenty-first problem. One does not distinguish between coordinates x and which are related by Möbius transformations, and also do not distinguish between gauge equivalent Fuchsian systems - this means that A and
are regarded as being equivalent for any holomorphic gauge transformation g..
For generic monodromy data, the answer to Hilbert's twenty-first problem is 'yes'. The first proof was given by Josip Plemelj. 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'. Here, the generic case is focused upon entirely.

Schlesinger's equations

There are generically many Fuchsian systems with the same monodromy data. Thus, given any such Fuchsian system with specified monodromy data, isomonodromic deformations can be performed of it. One therefore is led to study families of Fuchsian systems, where the matrices A_i 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:
The last equation is often written equivalently as
These are the isomonodromy equations for generic Fuchsian systems. The natural interpretation of these equations is as the flatness of a natural connection on a vector bundle over the 'deformation parameter space' which consists of the possible pole positions. 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 A_i take values in the Lie algebra, the Garnier integrable systems are obtained. If one specializes further to the case when there are only four poles, then the Schlesinger/Garnier equations can be reduced to the famous sixth Painlevé equation.

Irregular singularities

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.
The linear system under study is now of the form
with n poles, with the pole at λ_i of order. The are constant matrices.

Extended monodromy data

As well as the monodromy representation described in the Fuchsian setting, deformations of irregular systems of linear ordinary differential equations are required to preserve extended monodromy data. Roughly speaking, monodromy data is now regarded as data which glues together canonical solutions near the singularities. If one takes as a local coordinate near a pole λ_iof order, one can then solve term-by-term for a holomorphic gauge transformation g such that locally, the system looks like
where and the are diagonal matrices. If this were valid, it would be extremely useful, because then, one has decoupled the system into n scalar differential equations which one can easily solve to find that :
However, this does not work - because the power series solved term-for-term for g will not, in general, converge.
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. This is due to a theorem of George Birkhoff which states that given such a formal series, there is a unique convergent function G_i such that in any sufficiently large sector around the pole, G_i is asymptotic to g_i, and
is a true solution of the differential equation. A canonical solution therefore appears in each such sector near each pole. The extended monodromy data consists of

the data from the monodromy representation as for the Fuchsian case;
Stokes' matrices which connect canonical solutions between adjacent sectors at the same pole;
connection matrices that connect canonical solutions between sectors at different poles.
Jimbo–Miwa–Ueno isomonodromic deformations

As before, one now considers families of systems of linear differential equations, all with the same singularity structure. One therefore allows the matrices to depend on parameters. One is allowed to vary the positions of the poles λ_i, but now, in addition, one also varies the entries of the diagonal matrices which appear in the canonical solution near each pole.
Jimbo, Miwa and Ueno proved that if one defines a one-form on the 'deformation parameter space' by
then deformations of the meromorphic linear system specified by A are isomonodromic if and only if
These are the Jimbo—Miwa—Ueno isomonodromy equations. As before, these equations can be interpreted as the flatness of a natural connection on the deformation parameter space.

Properties

The isomonodromy equations enjoy a number of properties that justify their status as nonlinear special functions.

Painlevé property

This is perhaps the most important property of a solution to the isomonodromic deformation equations. 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.
Indeed, suppose that one is given a partial differential equation. 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.

Transcendence

In general, solutions of the isomonodromy equations cannot be expressed in terms of simpler functions such as solutions of linear differential equations. However, for particular choices of extended monodromy data, solutions can be expressed in terms of such functions. 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.
There are also very special solutions which are algebraic. The study of such algebraic solutions involves examining the topology of the deformation parameter space ; 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.
Rational solutions are often associated with special polynomials. Sometimes, as in the case of the sixth Painlevé equation, these are well-known orthogonal polynomials, but there are new classes of polynomials with an extremely interesting distribution of zeros and interlacing properties. The study of such polynomials has largely been carried out by Peter Clarkson and collaborators.

Symplectic structure

The isomonodromy equations can be rewritten using Hamiltonian formulations. 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.
There is another description in terms of moment maps to loop algebras - a viewpoint introduced by John Harnad and extended to the case of general singularity structure by Nick Woodhouse. This latter perspective is intimately related to a curious Laplace transform between isomonodromy equations with different pole structure and rank for the underlying equations.