Time evolution of localized solutions in 1-dimensional inhomogeneous FPU models

Abstract

We study energy localization in a quartic FPU model with spatial inhomogeneity corresponding to a site-dependent number of interacting neighbors. Such lattices can have linear normal modes that are strongly localized in the regions of high connectivity and there is evidence that some of these localized modes persist in the weakly nonlinear regime. The present study shows examples where oscillations can remain localized for long times. Nonlinear normal modes are approximated by periodic orbits that belong to an invariant subspace of a Birkhoff normal form of the system that is spanned by spatially localized modes [F. Martínez-Farías et al., Eur. Phys. J. Special Topics 223, 2943 (2014), F. Martínez-Farías et al., Physica D 335, 10 (2016)]. The invariant subspace is suggested by the dispersion relation and also depends on the overlap between normal modes. Numerical integration from the approximate normal modes suggests that spatial localization persists over a long time in the weakly nonlinear regime and is especially robust in some disordered lattices, where it persists for large, (1), amplitude motions. Large amplitude localization in these examples is seen to be recurrent, i.e. energy flows back and forth between the initial localization region and its vicinity.