Multibreathers and homoclinic orbits in 1-dimensional nonlinear lattices

Spatially localized, time-periodic excitations, known as discrete breathers, have been found to occur in a wide variety of 1-dimensional (1-D) lattices of nonlinear oscillators with nearest-neighbour coupling. Eliminating the time-dependence from the differential-difference equations of motion, and taking into account only the $N$ largest Fourier modes, we view these solutions as orbits of a (non-integrable) 2$N$-D map. For breathers to occur, the trivial rest state of the lattice must be a hyperbolic fixed point of the map, with an $N$-D stable and and $N$-D unstable manifold. The breathers and multibreathers (with one and more spatial oscillations respectively) are then directly related to the intersections of these manifolds, and hence to homoclinic orbits of the 2$N$-D map. This is explicitly shown here on a discretized nonlinear Schr\"odinger quation with only one Fourier mode $(N=1)$, represented by a 2-D map. We then construct the 2$N$-D map for an array of nonlinear oscillators, with nearest neighbour coupling and a quartic on-site potential, and demonstrate how a one-Fourier-mode representation (via a 2-D map) can be used to provide remarkably accurate initial conditions for the breather and multibreather solutions of the system.