On a high-order shallow-water wave model with canonical non-local Hamiltonian structure

We derive and study a new family of non-local partial differential equations (PDEs) that model free-surface long gravity waves over a flat bottom. To derive the model equations we approximate the velocity potential as a series of vertical polynomials derived from the shallow-water expansion of the Dirichlet-to-Neumann problem in the Hamiltonian formulation of free-surface potential flow and invoke Luke's variational principle. The resulting evolution equations exhibit a non-local Hamiltonian structure being coupled with a system of linear elliptic spatial PDEs on the horizontal plane. A key advantage of this approach is that it directly yields canonical Hamiltonian equations, which are well-suited for numerical solutions using standard methods. This class of model equations offers high-order shallow-water approximations of the water-wave problem. It contains terms whose spatial derivatives are at most of order two, distinguishing it from asymptotic methods involving higher-order mixed spatio-temporal derivatives. We explore the first non-trivial member of this family, highlighting its connections to other mathematical models and emphasizing its practical utility. We then analyse and discuss its linear dispersive properties and demonstrate that it does not exhibit a specific type of instability known as wave-trough instability. Additionally, we demonstrate its effectiveness in simulating the long-distance steady propagation of strongly non-linear solitary waves and the head-on collision of two counter-propagating solitary waves. In the latter case, comparisons with experimental data confirm the model's ability to capture complex wave dynamics, including wave transformation in the presence of strong non-linearity and dispersion. The extension of this approach to accommodate variable bottom topography is briefly discussed.