Compatible Pairs of Dubrovin–Novikov Poisson Brackets and Lagrangian Representations of Integrable Hierarchies

We consider compatible pairs of local Hamiltonian structures of Dubrovin–Novikov type and the construction of nonlocal Hamiltonian structures of Ferapontov type. We prove that the third and fourth Hamiltonian structures of Ferapontov type cannot contain more terms in the nonlocal part than the number of field variables. We also give local Lagrangian representations for both these nonlocal Hamiltonian structures. Thus, we demonstrate that any hydrodynamic-type system equipped with a compatible pair of local Hamiltonian structures of Dubrovin–Novikov type simultaneously possesses four local Lagrangian representations.