Movability of localized excitations in nonlinear discrete systems: A separatrix problem

Abstract

We analyze the effect of internal degrees of freedom on the movability properties of localized excitations on nonlinear Hamiltonian lattices by means of properties of a local phase space which is at least of dimension six. We formulate generic properties of a movability separatrix in this local phase space. We prove that due to the presence of internal degrees of freedom of the localized excitation it is generically impossible to define a Peierls-Nabarro potential in order to describe the motion of the excitation through the lattice. The results are verified analytically and numerically for Fermi-Pasta-Ulam chains.PACS number(s): 03.20.+i ; 63.20.Pw ; 63.20.Ry Physical Review Letters, in press. Date: 11/19/93 Recently localized breatherlike excitations were discovered to exist in several different Hamiltonian lattices in one and two dimensions [1],[2],[3],[4], [5],[6]. They are self-localized (no disorder) and appear in nonlinear lattices -thus we name them nonlinear localized excitations (NLEs). For certain systems it was possible to create moving NLEs [7],[8]. Consequently the idea arose to describe their motion in a Peierls-Nabarro-Potential (PNP) [9],[10], [11],[12] related to the PNP of kinks [13], [14]. Numerical simulations strongly support the existence of a PNP-related phenomenon in Fermi-Pasta-Ulam systems [15] and Klein-Gordon systems [16]. However as we show below it is generically impossible to define a PNP for NLEs.The NLE solutions are nontopological, i.e. no special structure of the underlying many-particle potential is required. The only condition is to have nonlinear terms in the potential. One can perform stability analysis and show that if the NLE is localized enough (in practice it will contain only a few particles which are involved in the motion) then generically all Hamiltonian lattices will exhibit families of stable time-periodic NLE solutions [6], [17], [18]. Hereafter we will call these stable periodic NLEs elliptic NLEs to emphasize their stability property (in a Poincare mapping they would appear as stable elliptic fixed points [6]). One can view the NLE as a solution of a reduced problem where only M particles are involved in the motion, the rest of the lattice members are held at their groundstate positions. We showed that many frequency NLEs can be excited by perturbing the elliptic NLEs and that thus NLE solutions are motions on M-dimensional tori in the phase space of the reduced problem and in the corresponding local subspace of the phase space of the full system [6], [17]. Besides these stable NLE solutions unstable periodic NLEs exist. Their feature is that certain local perturbations destroy the unstable NLE or cause it to move [9], [15], [19]. Hereafter we will call them hyperbolic NLEs. If one calculates the energy density distribution e l for the NLE solutions, one can define the position of the energy center of the distribution by X E = l l · e l /( l e l ). For a given system the elliptic NLE solution yields X E = l 0 (i.e. centered on a lattice site l 0 ) and the hyper...

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