Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 081, 17 pages      arXiv:2401.10445      https://doi.org/10.3842/SIGMA.2024.081

Multidimensional Nonhomogeneous Quasi-Linear Systems and Their Hamiltonian Structure

Xin Hu and Matteo Casati
School of Mathematics and Statistics, Ningbo University, Ningbo 315211, P.R. China

Received July 09, 2024, in final form September 04, 2024; Published online September 10, 2024

Abstract
In this paper, we investigate multidimensional first-order quasi-linear systems and find necessary conditions for them to admit Hamiltonian formulation. The insufficiency of the conditions is related to the Poisson cohomology of the admissible Hamiltonian operators. We present in detail the examples of two-dimensional, two-components systems of hydrodynamic type and of a real reduction of the 3-waves system.

Key words: Hamiltonian structures; quasilinear systems; non-homogeneous operators.

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