Wave propagation for 1-dimensional reaction-diffusion equation with nonzero random drift
Abstract
We consider the wave propagation for a reaction-diffusion equation on the real line, with a random drift and Fisher-Kolmogorov-Petrovskii-Piscounov (FKPP) type nonlinear reaction. We show that when the average drift is positive, the asymptotic wave fronts propagating to the positive and negative directions are both pushed in the negative direction, leading to the possibility that both wave fronts propagate toward negative infinity. Our proof is based on the Large Deviations Principle for diffusion processes in random environments, as well as an analysis of the Feynman-Kac formula. Such probabilistic arguments also reveal the underlying physical mechanism of the wave fronts formation: the drift acts as an external field that shifts the (quenched) free-energy reference level without altering the intrinsic fluctuation structure of the system.
Summary
This paper investigates wave propagation in a 1-dimensional reaction-diffusion equation (RDE) with a random drift and Fisher-Kolmogorov-Petrovskii-Piscounov (FKPP) type nonlinearity. The main research question is to characterize the influence of a non-zero random drift on the long-time behavior of the solution u(t,x) to the RDE, specifically whether it exhibits traveling wave profiles. The authors use a probabilistic approach based on the Large Deviations Principle (LDP) for diffusion processes in random environments and the Feynman-Kac formula to represent the solution. The key finding is that when the average drift is positive, both asymptotic wave fronts propagating to the positive and negative directions are pushed in the negative direction. This can lead to both wave fronts propagating towards negative infinity. They derive full LDP results without parameter restrictions, extending previous work that assumed zero drift or large reaction rates. They also demonstrate that the Scaled Cumulant Generating Functions (SCGF) for backward and forward propagation differ only by an additive constant. This research matters to the field because it provides a complete characterization of wave propagation for general stationary ergodic random drift terms, compensating for previous works that considered only zero drift or specific cases.
Key Insights
- •The paper derives full LDP results for the hitting times of the diffusion process without any parameter restrictions, overcoming limitations of previous approaches that required restrictions on the LDP domain.
- •They prove that the critical values for the Lyapunov functions in the forward and backward directions are equal (η← c = η→ c), indicating a symmetry in the finiteness of the moment generating functions.
- •They demonstrate that the difference between the Lyapunov functions in the forward and backward directions (μ(η) - μ→(η)) is a constant, implying that the drift acts as an external field that shifts the free-energy reference level without altering the intrinsic fluctuation structure.
- •They provide a complete characterization of the wave propagation problem, showing that the regions R0 and R1 are determined by the rate functions S(c) and S→(c) and the reaction rate β.
- •They identify three distinct cases for the wave front shape based on the relationship between the reaction rate β and the critical value ηc: (a) β ∈ (0,ηc), (b) β = ηc, and (c) β ∈ (ηc,+∞), each corresponding to different wave propagation behaviors.
- •The paper discusses the relation sgn(c*1 - c*2) = sgn(Eb(x)), connecting their results to Theorem 1.3 of [21].
- •A truncation argument is used to remove the restriction of the LDP from sets in (0, (v− c)μ ′ (0)) to (0, +∞).
Practical Implications
- •The results have applications in various fields where reaction-diffusion equations with random drift arise, such as turbulent combustion, interacting particle systems, and population biology.
- •Researchers and engineers can use these results to better understand and model the effects of random drift on wave propagation in these systems. Specifically, the analysis provides insights into how the "strength" of the random drift influences the wave front formation and propagation direction.
- •The paper opens up future research directions, including investigating the wave propagation problem in higher dimensions or with more complex random drift models. It also suggests exploring the connections between the results and non-equilibrium statistical mechanics.