Discrete nonlinear Schrodinger approximation of a mixed Klein-Gordon/Fermi-Pasta-Ulam chain: Modulational instability and a statistical condition for creation of thermodynamic breathers

We analyze certain aspects of the classical dynamics of a one-dimensional discrete nonlinear Schrodinger model with inter-site as well as on-site nonlinearities. The equation is derived from a mixed Klein-Gordon/Fermi-Pasta-Ulam chain of anharmonic oscillators coupled with anharmonic inter-site potentials, and approximates the slow dynamics of the fundamental harmonic of discrete small-amplitude modulational waves. We give explicit analytical conditions for modulational instability of travelling plane waves, and find in particular that sufficiently strong inter-site nonlinearities may change the nature of the instabilities from long-wavelength to short-wavelength perturbations. Further, we describe thermodynamic properties of the model using the grand-canonical ensemble to account for two conserved quantities: norm and Hamiltonian. The available phase space is divided into two separated parts with qualitatively different properties in thermal equilibrium: one part corresponding to a normal thermalized state with exponentially small probabilities for large-amplitude excitations, and another part typically associated with the formation of high-amplitude localized excitations, interacting with an infinite-temperature phonon bath. A modulationally unstable travelling wave may exhibit a transition from one region to the other as its amplitude is varied, and thus modulational instability is not a sufficient criterion for the creation of persistent localized modes in thermal equilibrium. For pure on-site nonlinearities the created localized excitations are typically pinned to particular lattice sites, while for significant inter-site nonlinearities they become mobile, in agreement with well-known properties of localized excitations in Fermi-Pasta-Ulam-type chains. (c) 2006 Elsevier B.V. All rights reserved.