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

Christian Kuehn; Pascal Sedlmeier

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arXiv · 2502.08349

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

Christian Kuehn·Pascal Sedlmeier

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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.

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Christian Kuehn, Pascal Sedlmeier. 2025-02-12. Stability of N-front and N-back solutions in the Barkley model. https://arxiv.org/abs/2502.08349

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