Isomonodromic

Isomonodromic describes a class of deformations of linear differential equations in the complex plane in which the monodromy representation remains unchanged. The monodromy is the action on the vector of solutions obtained by analytic continuation around the singularities; it encodes how solutions transform when transported along loops in the punctured domain.

Typical setting: a system dY/dz = A(z) Y with A meromorphic, often decomposed as sum_i A_i/(z - t_i) with moving poles t_i. As the poles t_i vary, the matrices A_i(t) are adjusted so that the monodromy of the fundamental solution Y around the t_i stays the same. The compatibility conditions are given by the Schlesinger equations, a system of nonlinear differential equations governing A_i(t).

Isomonodromic deformations are central to the theory of integrable systems. They connect to Painlevé equations: many Painlevé equations arise as the nonlinear differential equations governing the positions or accessory parameters in isomonodromic families of 2x2 linear systems with a fixed monodromy.

Historically, the concept traces to the study of linear ODEs and their monodromy (Fuchs, Schlesinger) and was developed in modern form by Jimbo, Miwa, and Ueno in the 1980s. The term isomonodromic (or isomonodromy) is used for the condition of fixed monodromy under deformations.

Applications include mathematical physics, random matrix theory, and algebraic geometry.