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
d y d x = A ( x ) y = ∑ i = 1 n A i x − λ i y {\displaystyle {\frac {dy}{dx}}=A(x)y=\sum _{i=1}^{n}{\frac {A_{i}}{x-\lambda _{i}}}y}
where x takes values in the complex projective line C P 1 {\displaystyle \mathbb {CP} ^{1}} , the y takes values in C n {\displaystyle \mathbb {C} ^{n}} and the Ai are constant n×n matrices. Solutions to this equation have polynomial growth in the limit x = λi. By placing n independent column solutions into a fundamental matrix Y = ( y 1 , . . . , y n ) {\displaystyle Y=(y_{1},...,y_{n})} then d Y d x = A Y {\displaystyle {\frac {dY}{dx}}=AY} and one can regard Y {\displaystyle Y} as taking values in G L ( n , C ) {\displaystyle \mathrm {GL} (n,\mathbb {C} )} . For simplicity, assume that there is no further pole at infinity, which amounts to the condition that
∑ i = 1 n A i = 0. {\displaystyle \sum _{i=1}^{n}A_{i}=0.}
Monodromy data Now, fix a basepoint b on the Riemann sphere away from the poles. Analytic continuation of a fundamental solution Y 1 {\displaystyle Y_{1}} around any pole λi and back to the basepoint will produce a new solution Y 2 {\displaystyle Y_{2}} defined near b. The new and old solutions are linked by the monodromy matrix Mi as follows:
Y 2 = Y 1 M i . {\displaystyle Y_{2}=Y_{1}M_{i}.}
One therefore has the Riemann–Hilbert homomorphism from the fundamental group of the punctured sphere to the monodromy representation:
π 1 ( C P 1 − { λ 1 , … , λ n } ) → G L ( n , C ) . {\displaystyle \pi _{1}\left(\mathbb {CP} ^{1}-\{\lambda _{1},\dots ,\lambda _{n}\}\right)\to \mathrm {GL} (n,\mathbb {C} ).}
A change of basepoint merely results in a (simultaneous) 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 x ^ {\displaystyle {\hat {x}}} which are related by Möbius transformations, and also do not distinguish between gauge equivalent Fuchsian systems - this means that A and
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