Towards computational Morse-Floer homology: forcing results for connecting orbits by computing relative indices of critical points

TL;DR

This paper develops a computer-assisted framework to determine the relative indices of stationary solutions in PDEs, facilitating progress in Morse-Floer homology and enabling proofs of connecting orbits and traveling waves.

Contribution

It introduces a novel computational approach for calculating relative indices of equilibria in PDEs, advancing Morse-Floer homology applications.

Findings

Successfully computed relative indices for multiple stationary points

Demonstrated the use of forcing results to prove connecting orbits

Provided a rigorous implementation with accessible code

Abstract

To make progress towards better computability of Morse-Floer homology, and thus enhance the applicability of Floer theory, it is essential to have tools to determine the relative index of equilibria. Since even the existence of nontrivial stationary points is often difficult to accomplish, extracting their index information is usually out of reach. In this paper we establish a computer-assisted proof approach to determining relative indices of stationary states. We introduce the general framework and then focus on three example problems described by partial differential equations to show how these ideas work in practice. Based on a rigorous implementation, with accompanying code made available, we determine the relative indices of many stationary points. Moreover, we show how forcing results can be then used to prove theorems about connecting orbits and traveling waves in partial differential equations.