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

Louis Garénaux; Hermen Jan Hupkes

doi:10.3934/dcds.2025134

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2026, Volume 48: 30-47. Doi: 10.3934/dcds.2025134

This volume Previous Article Extremal values of $ L^2 $-Pohozaev manifolds and their applications Next Article Poincaré-Treshchëv mechanism in integrable Hamiltonian systems under almost-periodic perturbations

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

Louis Garénaux^{1, 2, ,} and
Hermen Jan Hupkes^1,

1.
Niels Bohrweg 1, 2333 CA Leiden, The Netherlands
2.
Englerstraße 2, 76131 Karlsruhe, Germany

^*Corresponding author: Louis Garénaux
^*Corresponding author: Louis Garénaux

Received: January 2025

Early access: August 25, 2025

Published online: August 25, 2025

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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.

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Mathematics Subject Classification: Primary: 39A14, 39A12; Secondary: 39A70, 47A10, 47N40, 47A12.

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