Travelling waves for reaction–diffusion equations forced by translation invariant noise

Hamster, Christian; Hupkes, Hermen Jan

doi:10.1016/j.physd.2019.132233

Cited by 24 publications

(31 citation statements)

References 43 publications

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“…Now defining σ : {0, 1, 2} → {0, 1, 2} such that σ(0) = 0, σ(1) = 2 and σ(2) = 0 and λ 0 = 0, λ 1 = ω * and λ 2 = −ω * we have (27). We thus find that…”

Section: Rigorous Proof Of the Variational Phase Sdementioning

confidence: 94%

“…Of course, going beyond steady states is important and in the last few decades, the cases of travelling waves and moving interfaces for SPDEs have taken center stage. The research on this topic started in the early 1980s [60] and it has been growing quickly in recent years particularly for the Fisher-KPP equation [48,21,15,11,47], Nagumo-type SPDEs [35,14,27,23,28,44], neural field integro-differential equations [30,34,61,41,45], ecology [16] as well as regarding associated computational tools [42,62,59]. For more detailed surveys including a larger-scale view of the literature on the effect of noise on travelling waves we refer to [50,55,37].…”

Section: Introductionmentioning

confidence: 99%

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Stochastic Rotating Waves

Kuehn¹,

MacLaurin²,

Zucal³

2021

Preprint

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Stochastic dynamics has emerged as one of the key themes ranging from models in applications to theoretical foundations in mathematics. One class of stochastic dynamics problems that has received considerable attention recently are travelling wave patterns occurring in stochastic partial differential equations (SPDEs), i.e., how deterministic travelling waves behave under stochastic perturbations. In this paper, we start the mathematical study of related class of problems: stochastic rotating waves generated by SPDEs. We combine deterministic dynamics PDE techniques with methods from stochastic analysis. We establish two different approaches, the variational phase and the approximated variational phase, for defining stochastic phase variables along the rotating wave, which track the effect of noise on neutral spectral modes associated to the special Euclidean symmetry group of rotating waves. Furthermore, we prove transverse stability results for rotating waves showing that over certain time scales and for small noise, the stochastic rotating wave stays close to its deterministic counterpart.

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“…Now defining σ : {0, 1, 2} → {0, 1, 2} such that σ(0) = 0, σ(1) = 2 and σ(2) = 0 and λ 0 = 0, λ 1 = ω * and λ 2 = −ω * we have (27). We thus find that…”

Section: Rigorous Proof Of the Variational Phase Sdementioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Stochastic Rotating Waves

Kuehn¹,

MacLaurin²,

Zucal³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…formula now easily yields stability type results as we only need to estimate the position and speed, which depend in a simple way on Brownian motion so well understood upper/lower bounds for Brownian motion can be applied [125]. It is far more difficult to obtain general stability results but the bistable case, as illustrated by formula (69), is expected to be quite tame in the small noise regime [75,77,83,100,171,172]; cf. formula (50) for the monostable case.…”

Section: Bistable Stochastic Wavesmentioning

confidence: 99%

Travelling Waves in Monostable and Bistable Stochastic Partial Differential Equations

Kuehn

2019

Jahresber. Dtsch. Math. Ver.

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In this review, we provide a concise summary of several important mathematical results for stochastic travelling waves generated by monostable and bistable reaction-diffusion stochastic partial differential equations (SPDEs). In particular, this survey is intended for readers new to the topic but who have some knowledge in any sub-field of differential equations. The aim is to bridge different backgrounds and to identify the most important common principles and techniques currently applied to the analysis of stochastic travelling wave problems. Monostable and bistable reaction terms are found in prototypical dissipative travelling wave problems, which have already guided the deterministic theory. Hence, we expect that these terms are also crucial in the stochastic setting to understand effects and to develop techniques. The survey also provides an outlook, suggests some open problems, and points out connections to results in physics as well as to other active research directions in SPDEs.

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“…This model is able to reproduce the travelling pulses observed in nature [43] and has been studied extensively as a consequence. These studies have led to the development of many important mathematical techniques in areas such as singular perturbation theory [18-20, 55, 66, 67] variational calculus [22], Maslov index theory [7,23,30,31,57] and stochastic dynamics [50][51][52]. However, as a fully local equation it is unable to incorporate the discrete structure in a direct fashion.…”

Section: Introductionmentioning

confidence: 99%

Exponential Dichotomies for Nonlocal Differential Operators with Infinite Range Interactions

Schouten-Straatman¹,

Hupkes²

2020

Preprint

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We show that MFDEs with infinite range discrete and/or continuous interactions admit exponential dichotomies, building on the Fredholm theory developed by Faye and Scheel for such systems. For the half line, we refine the earlier approach by Hupkes and Verduyn Lunel. For the full line, we construct these splittings by generalizing the finite-range results obtained by Mallet-Paret and Verduyn Lunel. The finite dimensional space that is 'missed' by these splittings can be characterized using the Hale inner product, but the resulting degeneracy issues raise subtle questions that are much harder to resolve than in the finite-range case. Indeed, there is no direct analogue for the standard 'atomicity' condition that is typically used to rule out degeneracies, since it explicitly references the smallest and largest shifts.We construct alternative criteria that exploit finer information on the structure of the MFDE. Our results are optimal when the coefficients are cyclic with respect to appropriate shift semigroups or when the standard positivity conditions typically associated to comparison principles are satisfied. We illustrate these results with explicit examples and counter-examples that involve the Nagumo equation.

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Travelling waves for reaction–diffusion equations forced by translation invariant noise

Cited by 24 publications

References 43 publications

Stochastic Rotating Waves

Stochastic Rotating Waves

Travelling Waves in Monostable and Bistable Stochastic Partial Differential Equations

Exponential Dichotomies for Nonlocal Differential Operators with Infinite Range Interactions

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