  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 Schrodinger 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, asymptotic 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, J Phys A, 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.
