Excitation Thresholds for Nonlinear Localized Modes on Lattices

We consider spatially localized and time periodic solutions to discrete extended Hamiltonian dynamical systems (coupled systems of infinitely many "oscillations" which conserve total energy). These play a central role as carriers of energy in models of a variety of physical phenomena. Such phenomena include nonlinear waves in crystals, biological molecules and arrays of coupled optical waveguides. (See the recent experimental work.) In this paper, we study excitation thresholds for (nonlinearly dynamically stable) ground state localized modes, sometimes referred to as "breathers", for networks of coupled nonlinear oscillators and wave equations of nonlinear Schrodinger (NLS) type. Excitation thresholds are rigorously characterized by variational methods. The excitation threshold is related to the optimal (best) constant in a class of discrete interpolation inequalities related to the Hamiltonian energy. We establish a precise connection among d, the dimensionality of the lattice, 2 sigma + 1, the degree of the nonlinearity and the existence of an excitation threshold for discrete nonlinear Schrodinger systems (DNLS).