Discrete breathers in a two-dimensional hexagonal Fermi–Pasta–Ulam lattice

Abstract

We consider a two-dimensional Fermi–Pasta–Ulam (FPU) lattice with hexagonal symmetry. Using asymptotic methods based on small amplitude ansatz, at third order we obtain a reduction to a cubic nonlinear Schrödinger equation (NLS) for the breather envelope. However, this does not support stable soliton solutions, so we pursue a higher order analysis yielding a generalized NLS, which includes known stabilizing terms. We present numerical results which suggest that long-lived stationary and moving breathers are supported by the lattice. We find breather solutions which move in an arbitrary direction, an ellipticity criterion for the wavenumbers of the carrier wave, asymptotic estimates for the breather energy, and a minimum threshold energy below which breathers cannot be found. This energy threshold is maximized for stationary breathers and becomes vanishingly small near the boundary of the elliptic domain where breathers attain a maximum speed. Several of the results obtained are similar to those obtained for the square FPU lattice (Butt and Wattis 2006 J. Phys. A: Math. Gen. 39 4955), though we find that the square and hexagonal lattices exhibit different properties in regard to the generation of harmonics, and the isotropy of the generalized NLS equation.

Discrete breathers in honeycomb Fermi–Pasta–Ulam lattices

“…Our aim is to investigate the combined leading-order effects of nonlinear nearest-neighbour interactions and the honeycomb geometry, by finding leading-order asymptotic forms of discrete breathers in this lattice. This complements previous studies of square and hexagonal lattices [6,7]. Numerical studies of Marin et al [23,24] required the use of an onsite potential as well as nonlinear nearest-neighbour interactions to general breathers in two-dimensional lattices.…”

“…We use Q m,n for a general node, in practice, this will be either one of the left-facing ( Q m,n ) or the right-facing (Q m,n ) nodes. A derivation from Kirchoff's laws has been given in [7]. Here we simply quote the Hamiltonian H = m,n;s.t.…”

“…and so, in the optical case, the frequency lies above the frequency of linear waves. From (6)- (7) we find Q m,n = 2εAφ(r) cos(km + lhn + ωt),…”

“…Square [6] Hexagonal [7] Honeycomb Second harmonic The absence of any second harmonic is a property shared with the square lattice. Whilst the hexagonal lattice generates no third harmonic, it does generate a second harmonic.…”

“…One aim of the current work is to provide parameter regimes and initial conditions where breathers may exist in a system with only nonlinear nearest-neighbour interactions. We follow a similar analytic procedure to that of [6,7], using the method of multiple scales to obtain a system of equations from which we derive a nonlinear Schrödinger (NLS) equation that allows us to determine approximate small amplitude breathers. However, the analysis of the honeycomb lattice is significantly more complicated than the square or hexagonal cases due to the geometry of the lattice's interconnections which mean that there are two distinct types of node, which we call left-facing and right-facing.…”