---
tag: KDZF7E66BT
kind: concept
name: Isomonodromic deformation
msc: [34M56, 34M55]
sources:
  - url: https://en.wikipedia.org/wiki/Isomonodromic_deformation
same_as:
  wikidata: Q6086086
  wikipedia: Isomonodromic deformation
---

## Summary

An isomonodromic deformation is a family of linear differential equations whose [[monodromy|C8HD62AB6Q]] 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|8EV1G2Y6CC]], and it links them to random matrices and integrable systems.
See the [[definition|SVDH9C5MCK]].

## Explanations

### L1 human

Take a linear equation with a few singular points and move the singular points a little.
In general the monodromy, the way solutions get mixed around the points, changes.
One can ask to move the points and adjust the coefficients at the same time so that the monodromy does not change.
The coefficients then follow a nonlinear equation in the positions of the points.
Many famous nonlinear equations arise this way.