A Framework for the Numerical Computation and a Posteriori Verification of Invariant Objects of Evolution Equations

TL;DR

This paper presents a comprehensive framework for the numerical computation and rigorous verification of invariant objects in semilinear PDEs, enabling computer-assisted proofs of their existence.

Contribution

It introduces a systematic approach for formulating invariance conditions as operator zeros, including preconditioning techniques and function space considerations, to facilitate rigorous computational proofs.

Findings

Developed methods for preconditioning invariance operators.

Outlined procedures for working with nonlinear terms in PDEs.

Provided guidelines for computationally rigorous verification of invariant objects.

Abstract

We develop a theoretical framework for computer-assisted proofs of the existence of invariant objects in semilinear PDEs. The invariant objects considered in this paper are equilibrium points, traveling waves, periodic orbits and invariant manifolds attached to fixed points or periodic orbits. The core of the study is writing down the invariance condition as a zero of an operator. These operators are in general not continuous, so one needs to smooth them by means of preconditioners before classical fixed point theorems can be applied. We develop in detail all the aspects of how to work with these objects: how to precondition the equations, how to work with the nonlinear terms, which function spaces can be useful, and how to work with them in a computationally rigorous way. In two companion papers, we present two different implementations of the tools developed in this paper to study periodic orbits.