Open AccessCurvature-driven front propagation through planar lattices in oblique directions
Author(s) -
Mia Jukić,
Hermen Jan Hupkes
Publication year - 2022
Publication title -
communications on pure and applied analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.077
H-Index - 42
eISSN - 1553-5258
pISSN - 1534-0392
DOI - 10.3934/cpaa.2022055
Subject(s) - curvature , planar , transverse plane , mathematics , physics , discretization , lattice (music) , mathematical analysis , mathematical physics , geometry , computer science , structural engineering , computer graphics (images) , acoustics , engineering
In this paper we investigate the long-term behaviour of solutions to the discrete Allen-Cahn equation posed on a two-dimensional lattice. We show that front-like initial conditions evolve towards a planar travelling wave modulated by a phaseshift \begin{document}$ \gamma_l(t) $\end{document} that depends on the coordinate \begin{document}$ l $\end{document} transverse to the primary direction of propagation. This direction is allowed to be general, but rational, generalizing earlier known results for the horizontal direction. We show that the behaviour of \begin{document}$ \gamma $\end{document} can be asymptotically linked to the behaviour of a suitably discretized mean curvature flow. This allows us to show that travelling waves propagating in rational directions are nonlinearly stable with respect to perturbations that are asymptotically periodic in the transverse direction.