Somonodromic Deformations Along the Caustic of a Dubrovin-Frobenius Manifold

TL;DR

This paper investigates the differential equations linked to Dubrovin-Frobenius manifolds along their caustic, establishing a normal form and analyzing the monodromy properties under specific conditions.

Contribution

It introduces a normal form for the differential equations near the caustic and proves strong isomonodromic behavior of solutions, connecting monodromy exponents to the manifold's structure.

Findings

Normal form for the family of differential equations along the caustic.

Proof of strong isomonodromic property of solutions.

Relation between monodromy exponents and the manifold's multiplication structure.

Abstract

We study the family of ordinary differential equations associated to a Dubrovin-Frobenius manifold along its caustic. Upon just loosing an idempotent at the caustic and under a non-degeneracy condition, we write down a normal form for this family and prove that the corresponding fundamental matrix solutions are strongly isomonodromic. It is shown that the exponent of formal monodromy is related to the multiplication structure of the Dubrovin-Frobenius manifold along its caustic.