Snakes on Lieb lattice
arXiv:2206.09180 · doi:10.1007/s00332-022-09810-z
Abstract
We consider the discrete Allen-Cahn equation with cubic and quintic nonlinearity on the Lieb lattice. We study localized nonlinear solutions of the system that have linear multistability and hysteresis in their bifurcation diagram. In this work, we investigate the system's homoclinic snaking, i.e. snaking-like structure of the bifurcation diagram, particularly the effect of the lattice type. Numerical continuation using a pseudoarclength method is used to obtain localized solutions along the bifurcation diagram. We then develop an active-cell approximation to classify the type of solution at the turning points, which gives good agreement with the numerical results when the sites are weakly coupled. Time dynamics of localized solutions inside and outside the pinning region is also discussed.
published
References in corpus (9)
- Observation of a localized flat-band state in a photonic Lieb lattice
- Advances in Shell Buckling: Theory and Experiments
- Snaking and isolas of localised states in bistable discrete lattices
- Defect-like structures and localized patterns in SH357
- Localized patterns in planar bistable weakly coupled lattice systems
- Homoclinic snaking in the discrete Swift-Hohenberg equation
- Gapped vegetation patterns: crown/root allometry and snaking bifurcation
- Snakes in square, honeycomb, and triangular lattices
- Snaking bifurcations of localized patterns on ring lattices