Asymptotics beyond all orders and Stokes lines in nonlinear differential-difference equations

Abstract

A technique for calculating exponentially small terms beyond all orders in singularly perturbed difference equations is presented. The approach is based on the application of a WKBJ-type ansatz to the late terms in the naive asymptotic expansion and the identification of Stokes lines, and is closely related to the well-known Stokes line smoothing phenomenon in linear ordinary differential equations. The method is illustrated by application to examples and the results extended to time-dependent differential-difference problems.

Depinning Transitions in Discrete Reaction-Diffusion Equations

“…However, the derivation of the local saddle-node normal form and the correct description of the global saddle-node bifurcation involving matching with a fast stage during which the front jumps abruptly one lattice period were apparently omitted by the authors of [26], who used energy arguments. Our picture of the wave front depinning transition has essentially been corroborated in the continuum limit (as an appropriate dimensionless lattice length goes to zero) by King and Chapman, who used asymptotics beyond all orders [27]. An independent confirmation follows from Fáth's calculations for a spatially discrete reaction-diffusion equation with a piecewise linear source term [15] (except that the velocity should scale differently with |F − F c | in this case).…”

“…They found that the wave front velocity scales as the square root of (d − d c ) (d is the diffusivity and d c its critical value at which wave fronts are pinned). Essentially the same results can be found in the appendix of [27]. Kladko, Mitkov, and Bishop [28] introduced an approximation called the single active site theory.…”

“…These arguments can be used for other potentials and suggest that , found in a large class of discrete RD equations [9,10,11,26] and in continuous equations with localized sources [30]. However, exponential asymptotics [27] does not work for A large. We shall therefore follow a different approach.…”

“…In this case, there is a continuous transition between wave fronts moving to the left for F > 0 and moving to the right for F < 0; as for continuous systems, front pinning occurs at only a single field value F = 0 (see [27,16,36,37,38]). Wave front velocity then scales linearly with the field.…”

Wave Front Depinning Transition in Discrete One-Dimensional Reaction-Diffusion Systems

“…This argument can be used for other potentials and it suggests that F c ∼ C e −η/ √ A as A → 0+ (with positive C and η independent of A) holds for a large class of nonlinearities [11]. King and Chapman have obtained an analogous result [13] using exponential asymptotics for the FK potential, F c ∼ C e , found in a large class of discrete RD equations [11,12].…”