Quasiplanar steep water waves.

A unique description for highly nonlinear potential water waves is suggested, where weak three-dimensional effects are included as small corrections to exact two-dimensional equations written in conformal variables. Contrary to the traditional approach, a small parameter in this theory is not a surface slope, but it is the ratio of a typical wavelength to a large transversal scale along the second horizontal coordinate. A first-order correction for the Hamiltonian functional is calculated, and the corresponding equations of motion are derived for steep water waves over an arbitrary nonuniform quasi-one-dimensional bottom profile.