Rigorous Computation of Solutions of Semilinear PDEs on Unbounded Domains via Spectral Methods

SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 1966-2017, September 2024.
Abstract.In this article we present a general method to rigorously prove existence of strong solutions to a large class of autonomous semilinear PDEs in a Hilbert space [math] ([math]) via computer-assisted proofs. Our approach is fully spectral and uses Fourier series to approximate functions in [math] as well as bounded linear operators from [math] to [math]. 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 [math] as well as eigenpairs of the linearization about the soliton. These results allow us to prove the stability of the aforementioned traveling wave.