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We consider in this paper travelling wave solutions to stochastic partial differential equations and corresponding wave speed. As a particular example we consider the Nagumo equation with multiplicative noise which we mainly consider in the…
A multiscale analysis of 1D stochastic bistable reaction-diffusion equations with additive noise is carried out w.r.t. travelling waves within the variational approach to stochastic partial differential equations. It is shown with explicit…
We consider reaction-diffusion equations that are stochastically forced by a small multiplicative noise term. We show that spectrally stable travelling wave solutions to the deterministic system retain their orbital stability if the…
Travelling wave solutions of reaction-diffusion equations are widely used to model the spatial spread of populations and other phenomena in biology and physics. In this article, we reinterpret the classical variational principle approach…
We consider reaction-diffusion equations that are stochastically forced by a small multiplicative noise term. We show that spectrally stable traveling wave solutions to the deterministic system retain their orbital stability if the…
Inspired by applications, we consider reaction-diffusion equations on $\mathbb{R}$ 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…
Planar travelling waves on $\mathbb R^d,$ with $ d\geq 2,$ are shown to persist in systems of reaction-diffusion equations with multiplicative noise on significantly long timescales with high probability, provided that the wave is orbitally…
We consider reaction-diffusion systems with multiplicative noise on a spatial domain of dimension two or higher. The noise process is white in time, coloured in space, and invariant under translations. In the deterministic setting,…
In this paper we present a general framework in which to rigorously study the effect of spatio-temporal noise on traveling waves and stationary patterns. In particular the framework can incorporate versions of the stochastic neural field…
In this paper we investigate stability of travelling wave solutions to a class of reaction-diffusion equations perturbed by infinite-dimensional additive noise with H\"older continuous paths, covering in particular fractional Brownian…
Planar wave trains are traveling wave solutions whose wave profiles are periodic in one spatial direction and constant in the transverse direction. In this paper, we investigate the stability of planar wave trains in reaction-diffusion…
In order to understand the impact of random influences at physical boundary on the evolution of multiscale systems, a stochastic partial differential equation model under a fast random dynamical boundary condition is investigated. The…
We study travelling-wave solutions for a reaction-diffusion system arising as a model for host-tissue degradation by bacteria. This system consists of a parabolic equation coupled with an ordinary differential equation. For large values of…
We study nonlinear wave equations perturbed by transport noise acting either on the displacement or on the velocity. Such noise models random advection and, under suitable scaling of space covariance, may generate an effective dissipative…
We use a geometric approach to prove the existence of smooth travelling wave solutions of a nonlinear diffusion-reaction equation with logistic kinetics and a convex nonlinear diffusivity function which changes sign twice in our domain of…
We extend the result on the stability of travelling waves for stochastic Nagumo equations in [St] to general bistable reaction-diffusion equations with both additive and multiplicative noise, using a variational approach based on functional…
A modified method of functional constraints is used to construct the exact solutions of nonlinear equations of reaction-diffusion type with delay and which are associated with variable coefficients. This study considers a most generalized…
We study the effect of thermal noise on the propagation speed of a planar flame. We show that this out of equilibrium greatly amplifies the effect of thermal noise to yield macroscopic reductions in the flame speed over what is predicted by…
In this paper we investigate the implementation of the so-called freezing method for second order wave equations in one and several space dimensions. The method converts the given PDE into a partial differential algebraic equation which is…
This paper presents results on the unboundedness and minimal speed of traveling wave solutions for a one-dimensional spatial reaction-diffusion equation with an asymptotically linear reaction term and a saturation parameter. By applying a…