Nonlocal Hamiltonian structures of the kinetic equation for soliton gas under polychromatic reductions
Pierandrea Vergallo
TL;DR
The paper develops a nonlocal Hamiltonian formulation for El's kinetic equation of soliton gas under polychromatic reductions, linking the nonlocal operators to semi-Riemannian geometries with constant curvature and to conformally flat metrics. Building on the Dubrovin–Novikov framework and Tsarev compatibility, it shows that local structures for the reduced system are augmented by nonlocal tails that satisfy Ferapontov-type conditions, enabling a bi-Hamiltonian (or multi-Hamiltonian) description in several kernels. It provides explicit constructions for constant-curvature and conformally flat cases, with detailed examples for kernels including KdV, Lieb–Liniger, and separable interactions, and demonstrates how nonlocality yields richer geometric and algebraic structures, particularly in low-dimensional reductions. The work paves the way for extending nonlocal Hamiltonian formulations to the full kinetic equation and invites further exploration of geometric interpretations via submanifold theory and the Gauss–Codazzi framework, as well as broader classes of nonlocal operators.
Abstract
We deepen the existence of a nonlocal Hamiltonian formalism for the El's kinetic equation for soliton gas under the polychromatic reduction for a class of interaction kernels. The nonlocality presented is related to semi-Riemannian metrics of constant curvature, conformally flat metrics and hypersurfaces in a pseudo-Euclidean space. These results generalise a previous one that Vergallo and Ferapontov obtained with local Hamiltonian operators. Some examples as the Korteweg-de Vries, the Lieb-Liniger and the separable cases are analysed.