Travelling corners for spatially discrete reaction-diffusion systems

H. J. Hupkes, ,Mathematisch Instituut - Universiteit Leiden, P.O. Box 9512; 2300 RA Leiden; The Netherlands, L. Morelli
2020 Communications on Pure and Applied Analysis  
We consider reaction-diffusion equations on the planar square lattice that admit spectrally stable planar travelling wave solutions. We show that these solutions can be continued into a branch of travelling corners. As an example, we consider the monochromatic and bichromatic Nagumo lattice differential equation and show that both systems exhibit interior and exterior corners. Our result is valid in the setting where the group velocity is zero. In this case, the equations for the corner can be
more » ... the corner can be written as a difference equation posed on an appropriate Hilbert space. Using a non-standard global center manifold reduction, we recover a two-component difference equation that describes the behaviour of solutions that bifurcate off the planar travelling wave. The main technical complication is the lack of regularity caused by the spatial discreteness, which prevents the symmetry group from being factored out in a standard fashion. 2000 Mathematics Subject Classification. 34A33, 35B36.
doi:10.3934/cpaa.2020058 fatcat:2fkmlmx4yvdsphotz5b3gqcetm