Rigorous computation of solutions of semi-linear PDEs on unbounded domains via spectral methods

2023/02/24 by Matthieu Cadiot, Jean‐Philippe Lessard, Cadiot, Matthieu +3 · 4 citations
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #Dynamical Systems (math.DS) #FOS: Mathematics #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2302.12877

openalex publication_date 2023/02/24 · openalex created_date 2023/03/03 · openalex updated_date 2026/07/28

Abstract

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 H^l⊂ H^s(ℝ^m) (s≥1) via computer-assisted proofs. Our approach is fully spectral and uses Fourier series to approximate functions in H^l as well as bounded linear operators from L² to H^l. 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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• Proving symmetry of localized solutions and application to dihedral patterns in the planar Swift-Hohenberg PDE
• Stability analysis for localized solutions in PDEs and nonlocal equations on ℝ^m

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