Papers › Rigorous computation of solutions of semi-linear PDEs on unbounded domains via spectral methods
Rigorous computation of solutions of semi-linear PDEs on unbounded domains via spectral methods
Matthieu Cadiot, Jean-Philippe Lessard, Jean-Christophe Nave
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In this article we present a general method to rigorously prove existence of strong solutions to a large class of autonomous semi-linear PDEs in a Hilbert space Hˡ⊂Hˢ(ℝᵐ) (s≥1) via computer-assisted proofs. Our approach is fully spectral and uses Fourier series to approximate functions in Hˡ as well as bounded linear operators from L² to Hˡ. In particular, we construct approximate inverses of differential operators via Fourier series approximations. Combining this construction with a Newton-Kantorovich approach, we develop a numerical method to prove existence of strong solutions. To do so, we introduce a finite-dimensional trace theorem from which we build smooth functions with support on a hypercube. The method is then generalized to systems of PDEs with extra equations/parameters such as eigenvalue problems. As an application, we prove the existence of a traveling wave (soliton) in the Kawahara equation in H⁴(ℝ) as well as eigenpairs of the linearization about the soliton. These results allow us to prove the stability of the aforementioned traveling wave.
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