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Low-dimensional q-Tori in FPU Lattices: Dynamics and Localization Properties

This is a continuation of our study concerning q-tori, i.e. tori of low dimensionality in the phase space of nonlinear lattice models like the Fermi-Pasta-Ulam (FPU) model. In our previous work we focused on the beta FPU system, and we showed that the dynamical features of the q-tori serve as an interpretational tool to understand phenomena of energy localization in the FPU space of linear normal modes. In the present paper i) we employ the method of Poincare - Lindstedt series, for a fixed set of frequencies, in order to compute an explicit quasi-periodic representation of the trajectories lying on q-tori in the alpha model, and ii) we consider more general types of initial excitations in both the alpha and beta models. Furthermore we turn into questions of physical interest related to the dynamical features of the q-tori. We focus on particular q-tori solutions describing low-frequency `packets' of modes, and excitations of a small set of modes with an arbitrary distribution in q-space. In the former case, we find formulae yielding an exponential profile of energy localization, following an analysis of the size of the leading order terms in the Poincare - Lindstedt series. In the latter case, we explain the observed localization patterns on the basis of a rigorous result concerning the propagation of non-zero terms in the Poincare - Lindstedt series from zeroth to subsequent orders. Finally, we discuss the extensive (i.e. independent of the number of degrees of freedom) properties of some q-tori solutions.