Hamiltonian and non-Hamiltonian models for water waves. (English) Zbl 0583.76014

Olver, Peter J.

Trends and applications of pure mathematics to mechanics, Symp., Palaiseau/France 1983, Lect. Notes Phys. 195, 273-290 (1984).

[For the entire collection see Zbl 0533.00030.]
A general theory for determining Hamiltonian model equations from noncanonical perturbation expansions of Hamiltonian systems is applied to the Boussinesq expansion for long, small amplitude waves in shallow water, leading to the Korteweg-de-Vries equation. New Hamiltonian model equations, including a natural ”Hamiltonian version” of the KdV equation, are proposed. The method also provides a direct explanation of the complete integrability (soliton property) of the KdV equation. Depth dependence in both the Hamiltonian models and the second-order standard perturbation models is discussed as a possible mechanism for wave breaking.

Reviewer: T.-P.Liu

MSC:

76B15	Water waves, gravity waves; dispersion and scattering, nonlinear interaction
70H25	Hamilton’s principle

Keywords:

Hamiltonian model equations; noncanonical perturbation expansions; Boussinesq expansion; long, small amplitude waves; shallow water; Korteweg-de-Vries equation; complete integrability; soliton property; Depth dependence; second-order standard perturbation models; mechanism for wave breaking

Citations:

Zbl 0533.00030