Rigorous Numerics for ill-posed PDEs: Periodic Orbits in the Boussinesq Equation

Autor(es)

Castelli, Roberto ^[1] ; Gameiro, Marcio ^[2] ; Lessard, Jean-Philippe ^[3]

Afiliação do(s) autor(es)

^[1] Vrije Univ Amsterdam, De Boelelaan 1081, NL-1081 HV Amsterdam - Netherlands

^[2] Univ Sao Paulo, Inst Ciencias Matemat & Comp, Caixa Postal 668, BR-13560970 Sao Carlos, SP - Brazil

^[3] McGill Univ, Dept Math & Stat, 805 Sherbrooke St W, Montreal, PQ H3A 0B9 - Canada

• Tipo de documento: Artigo Científico
• Fonte: Archive for Rational Mechanics and Analysis; v. 228, n. 1, p. 129-157, APR 2018.
• Citações Web of Science: 5

Resumo

In this paper, we develop 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 proofs of existence of the periodic orbits in a weighted a{''}{''}(1) Banach space of space-time Fourier coefficients with exponential decay. We present several computer-assisted proofs of the existence of periodic orbits at different parameter values. (AU)

• Processo FAPESP: 16/08704-5 - Computações rigorosas em sistemas dinâmicos
• Beneficiário: Marcio Fuzeto Gameiro
• Modalidade de apoio: Auxílio à Pesquisa - Regular
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• Beneficiário: Marcio Fuzeto Gameiro
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• Processo FAPESP: 13/50382-7 - Rigorous computations for nonlinear partial differential equations. (fapesp-caldo)
• Beneficiário: Marcio Fuzeto Gameiro
• Modalidade de apoio: Auxílio à Pesquisa - Regular
• Processo FAPESP: 13/07460-7 - Computações rigorosas para EDPs
• Beneficiário: Marcio Fuzeto Gameiro
• Modalidade de apoio: Auxílio à Pesquisa - Regular