  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 $H^{l}\subset H^{s}(\mathbb{R}^{m})$ ($s\geq1$) 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.
