Hamiltonian Models for the Propagation of Long Gravity Waves, Higher-Order KdV-Type Equations and Integrability

A single incompressible, inviscid, irrotational fluid medium bounded above by a free surface is considered. The Hamiltonian of the system is expressed in terms of the so-called Dirichlet–Neumann operators. The equations for the surface waves are presented in Hamiltonian form. Specific scaling of the variables is selected, which leads to a KdV approximation with higher-order nonlinearities and dispersion (higher-order KdV-type equation, or HKdV). The HKdV is related to the known integrable PDEs with an explicit nonlinear and nonlocal transformation.