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Multibreathers and homoclinic orbits in 1-dimensional nonlinear lattices. (English) Zbl 0940.82018

Summary: 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, the authors view these solutions as orbits of a (non-integrable) \(2N\)-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 an \(N\)-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 Schrödinger equation with only one Fourier mode \((N=1)\), represented by a 2-D map. The authors then construct the \(2N\)-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.

MSC:

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
37K60 Lattice dynamics; integrable lattice equations
39A12 Discrete version of topics in analysis
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