Existence and Stability of Localized Oscillations in 1-Dimensional Lattices With Soft-Spring and Hard-Spring Potentials

In this paper, we use the method of homoclinic orbits to study the existence and stability of discrete breathers, i.e., spatially localized and time-periodic oscillations of a class of one-dimensional (1D) nonlinear lattices. The localization can be at one or several sites and the 1D lattices we investigate here have linear interaction between nearest neighbors and a quartic on-site potential $Vu=12Ku2±14u4,$ where the (+) sign corresponds to “hard spring” and (−) to “soft spring” interactions. These localized oscillations—when they are stable under small perturbations—are very important for physical systems because they seriously affect the energy transport properties of the lattice. Discrete breathers have recently been created and observed in many experiments, as, e.g., in the Josephson junction arrays, optical waveguides, and low-dimensional surfaces. After showing how to construct them, we use Floquet theory to analyze their linear (local) stability, along certain curves in parameter space (α,ω), where α is the coupling constant and ω the frequency of the breather. We then apply the Smaller Alignment Index method (SALI) to investigate more globally their stability properties in phase space. Comparing our results for the ± cases of $Vu,$ we find that the regions of existence and stability of breathers of the “hard spring” lattice are considerably larger than those of the “soft spring” system. This is mainly due to the fact that the conditions for resonances between breathers and linear modes are much less restrictive in the former than the latter case. Furthermore, the bifurcation properties are quite different in the two cases: For example, the phenomenon of complex instability, observed only for the “soft spring” system, destabilizes breathers without giving rise to new ones, while the system with “hard springs” exhibits curves in parameter space along which the number of monodromy matrix eigenvalues on the unit circle is constant and hence breather solutions preserve their stability character.