zbMATH — the first resource for mathematics

Isomonodromic deformations of systems of linear differential equations with irregular singularities. (English. Russian original) Zbl 1258.34178
Sb. Math. 203, No. 6, 826-843 (2012); translation from Mat. Sb. 203, No. 6, 63-80 (2012).
The author focuses on the following family of systems of \(d\) linear differential equations \[ \frac{d y}{d z}=A(z, t)\,y,\qquad A(z, t)=\sum_{j=1}^n\,\sum_{k=1}^{r_j+1}\,\frac{A^j_{-k}(t)}{(z-a_j)^k},\qquad \sum_{j=1}^n\,A^j_{-1}(t)=0\, \] with the matrices \(A^j_{-k}(t)\) depending holomorphically on \(t\in D(t^0)\), where \(D(t^0)\) is a neighbourhood of a point \(t^0\) in the parameter space and minimal strictly positive Poincaré rank at the singular points \(a_1, \ldots, a_n\) equal to \(r_1, \ldots, r_n\), respectively.
The author proves that this family of systems defines an isomonodromic deformation with parameters of deformations \(t=(a_1, a_2, \ldots , a_n)\) if and only if there exists a matrix-valued 1-form \(\omega(z, t)\) uniform on \(\mathbb{CP}^1 \times D(t^0)\;\setminus\bigcup_{i=1}^n \{z-a_i=0\}\) and such that \(\omega(z, t)=A(z, t)\,d z\) for each fixed \(t\in D(t^0)\) and \(d\, \omega(z, t)=\omega(z, t) \wedge \omega(z, t)\).
This result is an extension of the classical results of L. Schlesinger [“Über eine Klasse von Differentialsystemen beliebiger Ordnung mit festen kritischen Punkten”, J. für Math. 141, 96–145 (1912; JFM 43.0385.01)] and A. A. Bolibruch [“Differential equations with meromorphic coefficients”, Proc. Steklov Inst. Math. 272, 13–43 (2011)] on the deformations of Fuchsian systems and of M. Jimbo, T. Miwa and K. Ueno [Physica D 2, No. 2, 306–352 (1981; Zbl 1194.34167)] on the deformations of systems with non-resonant irregular singularities.

34M56 Isomonodromic deformations for ordinary differential equations in the complex domain
34M03 Linear ordinary differential equations and systems in the complex domain
34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
Full Text: DOI