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

arXiv2302.12877MaRDI QIDQ6508947

Jean-Christophe Nave, Jean-Philippe Lessard, Matthieu Cadiot

Abstract: In this article we present a general method to rigorously prove existence of strong solutions to a large class of semi-linear PDEs in a Hilbert space $H^{l}subset{H}^s(mathbb{R}^m)$ ($sgeq1$) via computer-assisted proofs. Our approach is purely spectral and uses approximation of eigenfunctions by periodic functions on large enough cubes. Combining a Newton-Kantorovich approach with explicit bounds for the operator norm of the inverse of differential operators, we develop a numerical method to prove existence of strong solutions to PDEs. 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 travelling wave (soliton) in the Kawahara equation in $H^4(mathbbR)$ as well as eigenpairs of the linearization about the soliton. These results allow us to prove the stability of the aforementioned travelling wave.

Has companion code repository: https://github.com/matthieucadiot/localizedpatternsh

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