Stability of N-front and N-back solutions in the Barkley model

TL;DR

This paper proves the stability of N-front and N-back traveling wave solutions in a model for pipe flow transition to turbulence, based on heteroclinic loop analysis and abstract stability hypotheses.

Contribution

It provides the first detailed stability analysis of these solutions within the heteroclinic loop framework for the Barkley model.

Findings

N-front and N-back solutions are stable for N > 1 in the specified Reynolds number domain.

The stability proof verifies abstract hypotheses for traveling waves derived from heteroclinic loops.

Completes the existence and stability characterization of these solutions in the model.

Abstract

In this paper we establish for an intermediate Reynolds number domain the stability of N-front and N-back solutions for each N > 1 corresponding to traveling waves, in an experimentally validated model for the transition to turbulence in pipe flow proposed in [Barkley et al., Nature 526(7574):550-553, 2015]. We base our work on the existence analysis of a heteroclinic loop between a turbulent and a laminar equilibrium proved by Engel, Kuehn and de Rijk in [Engel, Kuehn, de Rijk, Nonlinearity 35:5903, 2022], as well as some results from this work. The stability proof follows the verification of a set of abstract stability hypotheses stated by Sandstede in [SIAM Journal on Mathematical Analysis 29.1 (1998), pp. 183-207] for traveling waves motivated by the FitzHugh-Nagumo equations. In particular, this completes the first detailed analysis of Engel, Kuehn and de Rijk in [Engel, Kuehn, de Rijk, Nonlinearity 35:5903, 2022] leading to a complete existence and stability statement that nicely fits within the abstract framework of waves generated by twisted heteroclinic loops.