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

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 eduction to a cubic nonlinear Schr{\"o}dinger equation (NLS) for the breather envelope. However, this does not support stable soliton solutions, so we pursue a higher-order analysis yielding a generalised NLS, which includes known stabilising 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, symptotic estimates for the breather energy, and a minimum threshold energy below which breathers cannot be found. This energy threshold is maximised 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 \& Wattis, {\em J Phys A}, {\bf 39}, 4955, (2006)), though we find that the square and hexagonal lattices exhibit different properties in regard to the generation of harmonics, and the isotropy of the generalised NLS equation.

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

Using asymptotic methods, we investigate whether discrete breathers are supported by a two-dimensional Fermi-Pasta-Ulam lattice. A scalar (one-component) two-dimensional Fermi-Pasta-Ulam lattice is shown to model the charge stored within an electrical transmission lattice. A third-order multiple-scale analysis in the semi-discrete limit fails, since at this order, the lattice equations reduce to the (2+1)-dimensional cubic nonlinear Schrödinger (NLS) equation which does not support stable soliton solutions for the breather envelope. We therefore extend the analysis to higher order and find a generalised $(2+1)$-dimensional NLS equation which incorporates higher order dispersive and nonlinear terms as perturbations. We find an ellipticity criterion for the wave numbers of the carrier wave. Numerical simulations suggest that both stationary and moving breathers are supported by the system. Calculations of the energy show the expected threshold behaviour whereby the energy of breathers does {\em not} go to zero with the amplitude; we find that the energy threshold is maximised by stationary breathers, and becomes arbitrarily small as the boundary of the domain of ellipticity is approached.