Isomonodromic deformation

Summary

An isomonodromic deformation is a family of linear differential equations whose monodromy stays the same. The family depends on parameters, such as the positions of the singular points. Keeping the monodromy fixed forces the coefficients to satisfy nonlinear differential equations in the parameters. For Fuchsian systems these are the Schlesinger equations, and in the simplest nontrivial case they reduce to the sixth Painlevé equation. This gives one explanation of why the Painlevé equations have the Painlevé property, and it links them to random matrices and integrable systems. See the definition.