Isomonodromic deformations along a stratum of the coalescence locus

Abstract

We consider deformations of a differential system with Poincaré rank 1 at infinity and Fuchsian singularity at zero along a stratum of a coalescence locus. We give necessary and sufficient conditions for the deformation to be strongly isomonodromic, both as an explicit Pfaffian system (integrable deformation) and as a non linear system of PDEs on the residue matrix A at the Fuchsian singularity. This construction is complementary to that of (Cotti et al 2019 Duke Math. J. 168 967–1108). For the specific system here considered, the results generalize those of (Jimbo et al 1981 Physica D 2 306), by giving up the generic conditions, and those of (Bertola and Mo 2005 Int. Math. Res. Pap. 2005 565–635), by giving up the Lidskii generic assumption. The importance of the case here considered originates form its applications in the study of strata of Dubrovin-Frobenius manifolds and F-manifolds.