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

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 $H^l\subset H^s({\mathbb{R}}^m)$ ($s\ge 1$) 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 $L^2$ 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 $H^4(\mathbb{R})$ as well as eigenpairs of the linearization about the soliton. These results allow us to prove the stability of the aforementioned traveling wave.