| Abstract: |
| We introduce a variational framework that generates a family of shallow-water wave models of increasing order of accuracy, unifying classical asymptotic long-wave theories with nonlocal Hamiltonian formulations. Starting from Luke`s variational principle, we approximate the velocity potential using vertical polynomial ans\atze motivated by the shallow-water structure of the underlying Dirichlet-to-Neumann operator. This construction leads to a hierarchy of models expressed in canonical variables, whose evolution equations preserve the canonical, nonlocal Hamiltonian structure of the full water-wave problem.
In the case of uniform depth, we establish the equivalence with the Isobe--Kakinuma model and clarify connections with several classical and modern asymptotic models. We analyze the linear dispersion and stability properties of the resulting systems and validate the framework numerically through simulations of solitary-wave propagation and interactions. Finally, we extend the formulation to variable bathymetry and present results illustrating nonlinear wave--wave and wave--topography interactions. |
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