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R. Ibragimov and D. Pelinovsky

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Three-dimensional gravity waves in a channel of variable depth

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

**Abstract:**

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.

**Keywords**:

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