R. Ibragimov and D. Pelinovsky

Three-dimensional gravity waves in a channel of variable depth

Communications in Nonlinear Science and Numerical Simulation 13, 2104-2113 (2008)

We consider existence of three-dimensional gravity waves traveling along a channel of variable depth. It is well-known that the long-wave small-amplitude expansion for such waves results in the stationary Korteweg--de Vries (KdV) equation, coefficients of which depend on the transverse topography of the channel. The stationary KdV equation has a single-humped solitary wave localized in the direction of the wave propagation. We show, however, that there exists an infinite set of resonant Fourier modes that travel at the same speed as the solitary wave does. This fact suggests that the solitary wave confined in a channel of variable depth is always surrounded by small-amplitude oscillatory disturbances in the far-field profile.

Navier-Stokes equations, solitary waves, Korteweg-de Vries equation, center manifolds, Hamiltonian systems, linearized equations