Introduction

Kuehn, Christian

doi:10.1007/978-3-319-12316-5_1

Cited by 1 publication

(2 citation statements)

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“…Equation (2.20) simply means that the derivative of the traveling wave is an eigenvector of the frozen-wave operator, which actually arises due to translation invariance [59]…”

Section: 2mentioning

confidence: 99%

“…A singular perturbation argument in combination with Melnikov's method [44,58,85] yields the existence of a slow pulse with wave speed s ≈ 0. Yet, an application of the Sturm-Liouville theory [52,59] shows that the slow pulse is unstable. As it is deterministically already unstable, considering this pulse under the influence of noise is not expected to be biophysically relevant as the noisy small perturbations will be amplified exponentially near the slow pulse.…”

Section: Introductionmentioning

confidence: 99%

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Multiscale analysis for traveling-pulse solutions to the stochastic FitzHugh–Nagumo equations

2022

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We investigate the stability of traveling-pulse solutions to the stochastic FitzHugh-Nagumo equations with additive noise. Special attention is given to the effect of small noise on the classical deterministically stable fast traveling pulse. Our method is based on adapting the velocity of the traveling wave by solving a scalar stochastic ordinary differential equation (SODE) and tracking perturbations to the wave meeting a system of a scalar stochastic partial differential equation (SPDE) coupled to a scalar ordinary differential equation (ODE). This approach has been recently employed by Krüger and Stannat (Nonlinear Anal. 162 (2017) 197-223) for scalar stochastic bistable reaction-diffusion equations such as the Nagumo equation. A main difference in our situation of an SPDE coupled to an ODE is that the linearization has essential spectrum parallel to the imaginary axis and thus only generates a strongly continuous semigroup. Furthermore, the linearization around the traveling wave is not self-adjoint anymore, so that fluctuations around the wave cannot be expected to be orthogonal in a corresponding inner product. We demonstrate that this problem can be overcome by making use of Riesz instead of orthogonal spectral projections as recently employed in a series of papers by Hamster and Hupkes in case of analytic semigroups. We expect that our approach can also be applied to traveling waves and other patterns in more general situations such as systems of SPDEs with linearizations only generating a strongly continuous semigroup. This provides a relevant generalization as these systems are prevalent in many applications. CONTENTS

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“…Equation (2.20) simply means that the derivative of the traveling wave is an eigenvector of the frozen-wave operator, which actually arises due to translation invariance [59]…”

Section: 2mentioning

confidence: 99%