Regular non-semisimple Dubrovin–Frobenius manifolds

We study regular non-semisimple Dubrovin–Frobenius manifolds in dimensions 2, 3, and 4. Our results rely on the existence of special local coordinates introduced by David and Hertling [Ann. Sc. Norm. Super. Pisa, Cl. Sci. 17(5), 1121–1152 (2017)] for regular flat F-manifolds endowed with an Euler vector field. In such coordinates, the invariant metric of the Dubrovin–Frobenius manifold takes a special form, which is the starting point of our construction. We give a complete classification in the case where the Jordan canonical form of the operator of multiplication by the Euler vector field has a single Jordan block, and we reduce the classification problem to a third-order ordinary differential equation and to a system of third-order PDEs in the remaining three-dimensional and four-dimensional cases. In all the cases, we provide explicit examples of Dubrovin–Frobenius potentials.