Universal algebraic convergence in time of pulled fronts: the common mechanism for difference-differential and partial differential equations

We analyze the front structures evolving under the difference-differential equation ∂tCj = −Cj + C j−1 from initial conditions 0 6 Cj(0) 6 1 such that Cj(0) → 1 as j → ∞ sufficiently fast. We show that the velocity v(t) of the front converges to a constant value v∗ according to v(t) = v∗ − 3/(2λ∗t) + (3√π/2) Dλ∗/(λ∗Dt)3/2 + O(1/t2). Here v∗, λ∗ and D are determined by the properties of the equation linearized around Cj = 1. The same asymptotic expression is valid for fronts in the nonlinear diffusion equation, where the values of the parameters λ∗, v∗ and D are specific to the equation. The identity of methods and results for both equations is due to a common propagation mechanism of these so-called pulled fronts. This gives reasons to believe that this universal algebraic convergence actually occurs in an even larger class of equations.