Stability of Traveling Waves for Systems of Reaction-Diffusion Equations with Multiplicative Noise

Hamster, Christian; Hupkes, Hermen Jan

doi:10.48550/arxiv.1808.04283

Cited by 4 publications

(18 citation statements)

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“…Stability Our first contribution is that we establish that the wave (Φ σ , c σ ) is stable, in the sense that the perturbation V (t) remains small over time scales of O(σ −2 ). In particular, we show that the semigroup techniques developed in our earlier work [16,17] are general enough to remain applicable in the present more convoluted setting. The main effort is to verify that certain technical estimates remain valid, which is possible by the powerful theory that has been developed for cylindrical Q-Wiener processes.…”

Section: Previous Resultsmentioning

confidence: 77%

“…Phase tracking Our work here builds on the framework developed in [16,17] to study travelling waves in stochastic reaction-diffusion equations forced by a single Brownian motion. The main idea is to use a phase-tracking approach that is based purely on technical considerations rather than ad hoc geometric intuition.…”

Section: Previous Resultsmentioning

confidence: 99%

“…In particular, (1.15) does not fit into the framework of any previous results in this area. In fact, besides our earlier work [16,17], there do not seem to be many rigorous studies of travelling waves for multi-component SPDEs in the literature.…”

Section: Introductionmentioning

confidence: 88%

“…We show how these equations can be understood as a stochastic partial differential equation (SPDE) forced by a cylindrical Q-Wiener process and subsequently explain how to study stochastic travelling waves in this setting. In particular, we generalize the phase tracking framework that was developed in [16,17] for noise processes driven by a single Brownian motion. The main focus lies on explaining how this framework naturally leads to long term approximations for the stochastic wave profile and speed.…”

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confidence: 99%

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

Hamster

Hupkes

2020

Physica D: Nonlinear Phenomena

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Inspired by applications, we consider reaction-diffusion equations on R that are stochastically forced by a small multiplicative noise term that is white in time, coloured in space and invariant under translations. We show how these equations can be understood as a stochastic partial differential equation (SPDE) forced by a cylindrical Q-Wiener process and subsequently explain how to study stochastic travelling waves in this setting. In particular, we generalize the phase tracking framework that was developed in [16,17] for noise processes driven by a single Brownian motion. The main focus lies on explaining how this framework naturally leads to long term approximations for the stochastic wave profile and speed. We illustrate our approach by two fully worked-out examples, which highlight the predictive power of our expansions.

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Section: Previous Resultsmentioning

confidence: 77%

Section: Previous Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 88%

mentioning

confidence: 99%

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

Hamster

Hupkes

2020

Physica D: Nonlinear Phenomena

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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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A collective coordinate framework to study the dynamics of travelling waves in stochastic partial differential equations

Cartwright

Gottwald

2019

Physica D: Nonlinear Phenomena

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We propose a formal framework based on collective coordinates to reduce infinitedimensional stochastic partial differential equations (SPDEs) with symmetry to a set of finitedimensional stochastic differential equations which describe the shape of the solution and the dynamics along the symmetry group. We study SPDEs arising in population dynamics with multiplicative noise and additive symmetry breaking noise. The collective coordinate approach provides a remarkably good quantitative description of the shape of the travelling front as well as its diffusive behaviour, which would otherwise only be available through costly computational experiments. We corroborate our analytical results with numerical simulations of the full SPDE.

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Stability of Traveling Waves for Systems of Reaction-Diffusion Equations with Multiplicative Noise

Cited by 4 publications

References 0 publications

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

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

Exponential Dichotomies for Nonlocal Differential Operators with Infinite Range Interactions

A collective coordinate framework to study the dynamics of travelling waves in stochastic partial differential equations

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