Abstract
In this paper we analyze the behavior of the solution of the dissipative Boussinesq systems
where α, β, c > 0 are parameters. Those systems model two-dimensional small amplitude long wavelength water waves. For α ≤ 1, this equation is ill-posed and most initial conditions do not lead to solutions. Nevertheless, we show that, for almost every β, c and almost every α ≤ 1, it admits solutions that are quasiperiodic in time. The proof uses the fact that the equation leaves invariant a smooth center manifold and for the restriction of the Boussinesq system to the center manifold, uses arguments of classical perturbation theory by considering the Hamiltonian formulation of the problem and studying the Birkhoff normal form.
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Communicated by P. Constantin
Supported by the Center for Mathematical Analysis, Geometry, and Dynamical Systems, and through Fundação para a Ciência e a Tecnologia by Programs POCTI/FEDER, POSI, and POCI 2010/Fundo Social Europeu, and the grant SFRH/BPD/14404/2003.
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Valls, C. Quasiperiodic Solutions for Dissipative Boussinesq Systems. Commun. Math. Phys. 265, 305–331 (2006). https://doi.org/10.1007/s00220-006-0026-0
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DOI: https://doi.org/10.1007/s00220-006-0026-0