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 , with early pioneering contributions from Lazarus Fuchs, Paul Painlevé, , 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 with arbitrary singularity structure.

We consider the Fuchsian system of linear differential equations

where the independent variable x takes values in the complex projective line P¹(C), the solution Y takes values in Cⁿ and the A_i are constant n×n matrices. By placing n independent column solutions into a fundamental matrix we can regard Y as taking values in GL(n, C). Solutions to this equation have simple poles at x = λ_i. For simplicity, we shall assume that there is no further pole at infinity which amounts to the condition that

Now, fix a basepoint b on the Riemann sphere away from the poles. Analytic continuation of the solution Y around any pole λ_i and back to the basepoint will produce a new solution Y′ defined near b. The new and old solutions are linked by the monodromy matrix M_i as follows:

...
Wikipedia