On travelling waves for the stochastic Fisher–Kolmogorov–Petrovsky–Piscunov equation (original) (raw)
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Journal of Differential Equations, 2013
We prove that the solutions to the stochastic wave equation in O ⊂ R d , dẊ t − ∆X t dt + g(X t)dt = σ(X t)dW t , for d = 1, 2, 3, where g is a C 1-function with polynomial growth less than 3 and σ is Lipschitz with σ(0) = 0, propagate with finite speed. This result resembles the classical finite speed of propagation result for the solution to the Klein-Gordon equation and extends to equations with dissipative damping. A similar result follows for the equation with additive noise of the form F (t)dW t where F (t) = F (t, ξ) has compact support (in ξ) for each t > 0.
Two properties of stochastic KPP equations: ergodicity and pathwise property
Nonlinearity, 2001
In this paper we study the random approximate travelling wave solutions of the stochastic KPP equations. Two new properties of the stochastic KPP equations are obtained. We prove the ergodicity that for almost all sample paths, behind the wave front x = γt, the lower limit of 1 t t 0 u(s, x)ds as t → ∞ is positive, and ahead of the wave front, the limit is zero. In some cases, behind the wave front, the limit of 1 t t 0 u(s, x)ds as t → ∞ exists and is positive almost surely. We also prove that behind the wave front, for almost each ω, the solution of some special stochastic KPP equations converges to a stationary trajectory of the corresponding stochastic differential equation. In front of wave front, the solution converges to 0 which is another stationary trajectory of the corresponding SDE. We also study the space derivative of the solution for large time. We show that away from the wave front, for almost all large t the solution is flat in the x-direction for almost all sample paths.
Persistence of travelling waves in a generalized Fisher equation
Physics Letters A, 2009
Travelling waves of the Fisher equation with arbitrary power of nonlinearity are studied in the presence of long-range diffusion. Using analogy between travelling waves and heteroclinic solutions of corresponding ODEs, we employ the geometric singular perturbation theory to prove the persistence of these waves when the influence of long-range effects is small. When the long-range diffusion coefficient becomes larger, the behaviour of travelling waves can only be studied numerically. In this case we find that starting with some values, solutions of the model lose monotonicity and become oscillatory.
Travelling waves for discrete stochastic bistable equations
arXiv: Probability, 2020
Many physical, chemical and biological systems have an inherent discrete spatial structure that strongly influences their dynamical behaviour. Similar remarks apply to internal or external noise, as well as to nonlocal coupling. In this paper we study the combined effect of nonlocal spatial discretization and stochastic perturbations on travelling waves in the Nagumo equation, which is a prototypical model for bistable reaction-diffusion partial differential equations (PDEs). We prove that under suitable parameter conditions, various discrete-stochastic variants of the Nagumo equation have solutions, which stay close on long time scales to the classical monotone Nagumo front with high probability if the noise level and spatial discretization are sufficiently small.
Fisher Waves in the Strong Noise Limit
Physical Review Letters, 2009
We investigate the effects of strong number fluctuations on traveling waves in the Fisher-Kolmogorov reaction-diffusion system. Our findings are in stark contrast to the commonly used deterministic and weak-noise approximations. We compute the wave velocity in one and two spatial dimensions, for which we find a linear and a square-root dependence of the speed on the particle density. Instead of smooth sigmoidal wave profiles, we observe fronts composed of a few rugged kinks that diffuse, annihilate, and rarely branch; this dynamics leads to power-law tails in the distribution of the front sizes.
Travelling Waves in the Discrete Stochastic Nagumo Equation
2018
Many physical, chemical and biological systems have an inherent discrete spatial structure that strongly influences their dynamical behaviour. Similar remarks apply to internal or external noise, as well as to nonlocal coupling. In this paper we study the combined effect of nonlocal spatial discretization and stochastic perturbations on travelling waves in the Nagumo equation. We prove that under suitable parameter conditions, various discrete-stochastic variants of the Nagumo equation have solutions, which stay close on long time scales to the classical monotone Nagumo front with high probability if the noise level and spatial discretization are sufficiently small. Preliminary preprint version as of September 10, 2018
Traveling waves and long-time behavior in a doubly nonlocal Fisher-KPP equation
2015
We consider a Fisher‐KPP-type equation, where both diusion and nonlinear part are nonlocal, with anisotropic probability kernels. Under minimal conditions on the coecients, we prove existence, uniqueness, and uniform space-time boundedness of the positive solution. We investigate existence, uniqueness, and asymptotic behavior of monotone traveling waves for the equation. We also describe the existence and main properties of the front of propagation.