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-diffe-ren-tial equation $partial_tC_j=-C_j+C_{j-1^2$ from initial conditions $0le C_j(0)le1$ such that $C_j(0)to1$ as $jtoinfty$ sufficiently fast. We show that the velocity $v(t)$ of the front converges to a constant value $v^*$ according to $v(t)=v^*-3/(2lambda^*t) +(3sqrt{pi/2);Dlambda^*/({lambda^*^2Dt)^{3/2+{cal O(1/t^2)$. Here $v^*$, $lambda^*$ and $D$ are determined by the properties of the equation linearized around $C_j=1$. Ebert and Van Saarloos recently derived the same asymptotic expression for fronts in the nonlinear diffusion equation where the values of the parameters $lambda^*$, $v^*$ and $D$ are specific to the equation. The identity of methods and results for both equations is due to a common propagation mechanism of pulled fronts. This gives reasons to believe, that this universal algebraic convergence actually occurs in an even larger class of equations.