Principal hierarchy of Frobenius manifolds with rational and trigonometric superpotentials

In this paper, we construct the principal hierarchies for Frobenius manifolds with rational and trigonometric superpotentials, as well as their almost dualities. We demonstrate that in both cases, submanifolds with even superpotentials form natural Frobenius submanifolds, and their principal hierarchies can be obtained as restrictions of the principal hierarchies for the original Frobenius manifolds. Furthermore, we introduce a natural rank-1 extension for each of these Frobenius manifolds, providing solutions to the associated open WDVV equations. The principal hierarchy for each extension is also explicitly constructed.