Rigorous computation of solutions of semi-linear PDEs on unbounded domains via spectral methods
arxiv(2023)
摘要
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 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(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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关键词
rigorous computation,methods,domains,semi-linear
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