2009/10/02 by Christopher R. N. Taylor, Taylor, Christopher R. N., Jonathan H. P. Dawes +1
Physics and Astronomy · #FOS: Physical sciences #Pattern Formation and Solitons (nlin.PS) #nlin.PS
paper · pdf · doi:10.48550/arxiv.0910.0294
13 pages, 9 figures, submitted to Physics Letters A
arxiv created 2009/10/02 · arxiv updated 2009/12/01
We consider localized states in a discrete bistable Allen-Cahn equation. This model equation combines bistability and local cell-to-cell coupling in the simplest possible way. The existence of stable localized states is made possible by pinning to the underlying lattice; they do not exist in the equivalent continuum equation. In particular we address the existence of 'isolas': closed curves of solutions in the bifurcation diagram. Isolas appear for some non-periodic boundary conditions in one spatial dimension but seem to appear generically in two dimensions. We point out how features of the bifurcation diagram in 1D help to explain some (unintuitive) features of the bifurcation diagram in 2D.