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Rigorous numerics for
ill-posed PDEs:
periodic orbits in the Boussinesq equation
Roberto Castelli, Marcio
Gameiro and Jean-Philippe Lessard
In this paper, we develop some
computer-assisted techniques for the analysis of periodic
orbits of ill-posed partial differential equations. As a
case study, our proposed method is applied to the
Boussinesq equation, which has been investigated
extensively because of its role in the theory of shallow
water waves. The idea is to use the symmetry of the
solutions and a Newton-Kantorovich type argument (the
radii polynomial approach), to obtain rigorous proof of
existence of periodic orbits in a weighted ell-one Banach
space of space-time Fourier coefficients with geometric
decay. We present several computer-assisted proofs of
existence of periodic orbits at different parameter
values.
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