Flat F-manifolds, Miura invariants, and integrable systems of conservation laws

Abstract

We extend some of the results proved for scalar equations in [3,4], to the case of systems of integrable conservation laws. In particular, for such systems we prove that the eigenvalues of a matrix obtained from the quasilinear part of the system are invariants under Miura transformations and we show how these invariants are related to dispersion relations. Furthermore, focusing on one-parameter families of dispersionless systems of integrable conservation laws associated to the Coxeter groups of rank 2 found in [1], we study the corresponding integrable deformations up to order 2 in the deformation parameter ǫ. Each family contains both bi-Hamiltonian and non-Hamiltonian systems of conservation laws and therefore we use it to probe to which extent the properties of the dispersionless limit impact the nature and the existence of integrable deformations. It turns out that a part two values of the parameter all deformations of order one in ǫ are Miura-trivial, while all those of order two in ǫ are essentially parameterized by two arbitrary functions of single variables (the Riemann invariants) both in the bi-Hamiltonian and in the non-Hamiltonian case. In the two remaining cases, due to the existence of non-trivial first order deformations, there is an additional functional parameter.

Flat F-Manifolds, F-CohFTs, and Integrable Hierarchies

“…Unfortunately, for generic choices of the functional parameters the existence of a full dispersive hierarchy is an open problem. Concerning this we point out that, in [AL18], it was conjectured that, up to equivalence, integrable deformations for systems of any rank are labelled by a simple λ i = v i + λ i 1 p + λ i 2 p 2 + . .…”

“…Here we recall the notion of a flat F-manifold ( [Get04,Man05], see also [AL18,LPR09]) and its main properties.…”

“…In other words, we require only integrability, i.e., pairwise commutativity of the flows. In the table below, we summarize the results of computations of such deformations at the approximation up to ε 2 (the results for Case I were already obtained in [AL18]). When we refer to a functional parameter relative to an integrable deformation, we mean that at a specified order the equivalence classes of deformations depend on an arbitrary function.…”

“…3.3, we briefly mention the problem of computing general integrable deformations of principal hierarchies of flat F-manifolds. It was conjectured in [AL18] that the equivalence classes of such deformations are labeled by certain functional parameters called Miura invariants. In the case of Dubrovin-Frobenius manifolds and bihamiltonian deformations, these…”

Riemannian F-manifolds, bi-flat F-manifolds, and flat pencils of metrics

“…Many of the constructions appearing in the theory of Dubrovin-Frobenius manifolds have been generalized to the non-Egorov set-up, where Dubrovin-Frobenius manifolds are replaced by flat and bi-flat F-manifolds (in the conformal case). We refer to the papers [AL13, Lor14, KMS20, AL19, KM19] for relations with Painlevé trascendents, to the papers [AL17, KMS20,KMS18] for relations with reflection groups, to the papers [ABLR20a,BB19] for relations with F-cohomological field theories, and to [AL18,ABLR20b] for relations with integrable systems. In particular, the results of [ABLR20a] combined with the results of [BR18] allow one to construct a generalization of double ramification hierarchy [Bur15,BR16] for any semisimple flat F-manifold.…”