Summary.
We describe traveling waves in a basic model for three-dimensional water-wave dynamics in the weakly nonlinear long-wave regime. Small solutions that are periodic in the direction of translation (or orthogonal to it) form an infinite-dimensional family. We characterize these solutions through spatial dynamics, by reducing a linearly ill-posed mixed-type initial-value problem to a center manifold of infinite dimension and codimension. A unique global solution exists for arbitrary small initial data for the two-component bottom velocity, specified along a single line in the direction of translation (or orthogonal to it). A dispersive, nonlocal, nonlinear wave equation governs the spatial evolution of bottom velocity.
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Received July 20, 2001; accepted November 5, 2001
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Pego, R., Quintero, J. A Host of Traveling Waves in a Model of Three-Dimensional Water-Wave Dynamics. J. Nonlinear Sci. 12, 59–83 (2002). https://doi.org/10.1007/s00332-001-0478-5
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DOI: https://doi.org/10.1007/s00332-001-0478-5