Snakes on Lieb lattice

TL;DR

This paper investigates localized solutions and homoclinic snaking phenomena in the discrete Allen-Cahn equation on the Lieb lattice, revealing effects of lattice structure on bifurcation behavior through numerical and analytical methods.

Contribution

It introduces an active-cell approximation to classify solutions at bifurcation points, enhancing understanding of snaking in lattice systems.

Findings

Localized solutions exhibit multistability and hysteresis.

Active-cell approximation agrees with numerical results for weak coupling.

Time dynamics differ inside and outside the pinning region.

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.