Existence of monostable fronts for a KPP infinite-difference numerical scheme

TL;DR

This paper proves the existence of traveling wave solutions for a numerical scheme approximating the KPP equation, demonstrating monostable fronts for all super-critical speeds when the spatial step size is small, using spectral convergence techniques.

Contribution

It introduces a novel spectral convergence approach to establish the existence of monostable fronts in a discretized KPP equation, including cases with infinite range discretizations and sign-changing coefficients.

Findings

Existence of monostable fronts for all super-critical speeds.

Uniform resolvent bounds with respect to step size.

Applicability to infinite range discretizations with sign-changing coefficients.

Abstract

We study the existence of traveling wave solutions for a numerical counterpart of the KPP equation. We obtain the existence of monostable fronts for all super-critical speeds in the regime where the spatial step size is small. The key strategy is to transfer the invertibility of certain linear operators related to the front solutions from the continuous setting to the discrete case we are interested in. We rely on resolvent bounds which are uniform with respect to the step size, a procedure which is also known as spectral convergence. The approach is also able to handle infinite range discretizations with geometrically decaying coefficients that are allowed to have both signs, which prevents the use of the comparison principle.