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

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 $\epsilon$. 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 $\epsilon$ are Miura-trivial, while all those of order two in $\epsilon$ 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.

💡 Research Summary

The paper investigates integrable deformations of systems of conservation laws by extending concepts originally developed for scalar equations to the multi‑component case. The authors first introduce a matrix built from the quasilinear part of a general evolutionary PDE system
(u^i_t = A^i_j(u) u^j_x + \varepsilon B^i_j(u) u^j_{xx} + \varepsilon^2 C^i_j(u) u^j_{xxx} + \dots).
From this matrix they define the “Miura matrix” (M^i_j(u,p)=A^i_j(u)+B^i_j(u)p+C^i_j(u)p^2+\dots) and call its eigenvalues (\lambda_i(u,p)) Miura invariants. Theorem 2.2 proves that under Miura transformations of the form (w^i = u^i + \sum_{k\ge1}\varepsilon^k F^i_k(u,u_x,\dots)) (with the leading part being the identity) the (\lambda_i) transform as scalars, i.e. they are unchanged. Consequently, the dispersion relation of the linearized system around a constant state (u_0) is simply (\omega_j(k) = -k,\lambda_j(u_0, i k)). This establishes a direct link between the newly defined invariants and the physical wave propagation properties of the system.

The second part of the work focuses on concrete families of two‑field integrable dispersionless systems that arise from flat and bi‑flat (F)-manifold structures on the orbit spaces of the Coxeter groups (B_2) and (I_2(m)). These families depend on a single real parameter (c). For generic values of (c) the systems are non‑Hamiltonian (they possess a semi‑Hamiltonian structure in the sense of Tsarev but no local Poisson bracket), while for special values ((c=-3/4) for (B_2) and (c=0) for (I_2(m))) they become bi‑Hamiltonian, corresponding to the principal hierarchy of a Frobenius manifold.

The authors then perform a systematic perturbative analysis up to order (\varepsilon^2). Their main findings are:

1. First‑order deformations ((\varepsilon)) – For almost all values of (c) the first‑order deformation is Miura‑trivial; it can be eliminated by an appropriate Miura transformation. Exceptions occur at (c=-1) and (c=-\tfrac12) for the (B_2) family (and at (c=\pm2) for the (I_2(m)) family). In these exceptional cases the first‑order deformation cannot be removed and is governed by a single arbitrary function of one of the Riemann invariants. This reflects a degeneracy of one of the primary flows of the principal hierarchy.

2. Second‑order deformations ((\varepsilon^2)) – For generic (c) the second‑order deformation is parametrised by two arbitrary functions of a single variable, each depending on one of the Riemann invariants. In the exceptional cases mentioned above, the additional functional freedom present at first order survives, leading to three independent functional parameters at second order.

3. Miura invariants and dispersion – The eigenvalues (\lambda_i(u,p)) computed from the quasilinear part coincide with the coefficients that appear in the dispersion relation, confirming that Miura invariants encode the linear wave speeds of the deformed system.

4. Geometric interpretation – When (c) takes the special bi‑Hamiltonian values, the underlying flat (F)-manifold is actually a Frobenius manifold; for generic (c) the structure is a bi‑flat (F)-manifold. The number of functional parameters in the deformations mirrors the richness of the underlying geometry: the bi‑Hamiltonian (Frobenius) case behaves like the well‑studied central invariants of Dubrovin‑Zhang, while the non‑Hamiltonian bi‑flat case still admits a complete classification in terms of Miura invariants.

Overall, the paper provides a substantial extension of the theory of Miura invariants from scalar to multi‑component systems, demonstrates their physical relevance via dispersion relations, and delivers a thorough classification of integrable deformations up to second order for a non‑trivial family of two‑field conservation laws. The results suggest that even in the absence of a local Poisson structure, a robust set of invariants governs the integrability and deformation theory, opening avenues for further exploration of higher‑order deformations and more general Coxeter‑type (F)-manifolds.