154 research outputs found
Small amplitude breathers in 1D and 2D Klein-Gordon lattices
No full textWe construct small amplitude breathers in one-dimensional (1D) and two-dimensional (2D) Klein-Gordon (KG) infinite lattices. We also show that the breathers are well-approximated by the ground state of the nonlinear Schrdinger equation. The result is obtained by exploiting the relation between the KG lattice and the discrete nonlinear Schrdinger model. The proof is based on a Lyapunov-Schmidt decomposition and continuum approximation techniques introduced in [Bambusi and Penati, Continuous approximation of ground states in DNLS lattices, Nonlinearity 23 (2010), pp. 143-157], actually using its main result as an important lemma
Equipartition times in a Fermi-Pasta-Ulam system
No full textWe investigate with numerical methods the celebrated Fermi--Pasta--Ulam model, a chain of non--linearly coupled oscillators with identical masses. We are interested in the evolution towards equipartition when energy is initially given to one or a few modes.
In previous works we considered the initial energy being given on the lower part of the spectrum. Using the spectral entropy as a numerical indicator we obtained a strong indication that the relaxation time to equipartition increases exponentially with an inverse power of the specific energy. Such a scaling appears to remain valid in the
thermodynamic limit.
In this paper we explore the dynamics obtained with the initial excitation on the high frequency modes, and we obtain also in this case indication of exponentially long times to equipartition
Hamiltonian lattice dynamics
No full textHamiltonian lattice dynamics is a very active and relevant field of research. In this Special Issue, by means of some recent results by leading experts in the field, we tried to illustrate how broad and rich it can be, and how it can be seen as excellent playground for Mathematics in Engineering
An extensive adiabatic invariant for the Klein-Gordon model in the thermodynamic limit
No full textIn the quest for a mathematically rigorous foundation of Statistical Physics in general, and Statistical Mechanics in particular, despite many efforts and recent successes, a lot of work is still to be done. More specifically, if one considers an Hamiltonian system, instead of some ad hoc model, for the microscopic description of large systems, the behaviour over different long time scales is often still a challenge. One of the possible, and natural strategies, is to apply the techniques and results of Hamiltonian perturbation theory to large systems, with particular attention to the thermodynamic limit, i.e. when the number of degrees of freedom grows very large, at fixed, non vanishing, specific energy. The present report (based on paper [10]) is concerned with the existence of an adiabatic invariant for an arbitrarily large one dimensional Klein-Gordon chain, with estimates uniform in the size of the system
Breathers and Q-Breathers: two sides of the same coin
No full textWe construct, and approximate from the continuum, two-parameter families of time periodic, small amplitude, localized solutions, for both the focusing and defocusing finite discrete nonlinear Schrödinger models, with Dirichlet boundary conditions. Within such families, depending on the parameters, both real space localization (breathers) and Fourier space localization (Q-breathers) are present. For the former type of solutions, convergence to the ground state of the focusing infinite chain is also proved; for the latter, a description of the localization properties is given, and some numerical results on the difference between the focusing and defocusing cases are explained. The proofs are based on continuation tools, ideas from the finite element methods, and techniques of convergence of variational problems
Long time stability of small-amplitude Breathers in a mixed FPU-KG model
No full textIn the limit of small couplings in the nearest neighbor interaction, and small total energy, we apply the resonant normal form result of a previous paper of ours to a finite but arbitrarily large mixed Fermi–Pasta–Ulam Klein–Gordon chain, i.e., with both linear and nonlinear terms in both the on-site and interaction potential, with periodic boundary conditions. An existence and orbital stability result for Breathers of such a normal form, which turns out to be a generalized discrete nonlinear Schrödinger model with exponentially decaying all neighbor interactions, is first proved. Exploiting such a result as an intermediate step, a long time stability theorem for the true Breathers of the KG and FPU–KG models, in the anti-continuous limit, is proven
Continuous approximation of breathers in one and two dimensional DNLS lattices
No full textIn this paper we construct and approximate breathers in the DNLS model starting from the continuous limit: such periodic solutions are obtained as perturbations of the ground state of the NLS model in , with n = 1, 2. In both the dimensions we recover the Sievers–Takeno and the Page (P) modes; furthermore, in also the two hybrid (H) modes are constructed. The proof is based on the interpolation of the lattice using the finite element method (FEM)
An extensive resonant normal form for an arbitrary large Klein-Gordon model
No full textWe consider a finite but arbitrarily large Klein-Gordon chain, with periodic boundary conditions. In the limit of small couplings in the nearest neighbor interaction, and small (total or specific) energy, a high order resonant normal form is constructed with estimates uniform in the number of degrees of freedom.
In particular, the first order normal form is a generalized discrete nonlinear Schroedinger model, characterized by all-to-all sites coupling with exponentially decaying strength
Tail resonances of FPU q-breathers and their impact on the pathway to equipartition
No full textUpon initial excitation of a few normal modes the energy distribution among all modes of a nonlinear atomic chain (the Fermi-Pasta-Ulam model) exhibits exponential localization on large time scales. At the same time resonant anomalies (peaks) are observed in its weakly excited tail for long times preceding equipartition. We observe a similar resonant tail structure also for exact time-periodic Lyapunov orbits, coined q-breathers due to their exponential localization in modal space. We give a simple explanation for this structure in terms of superharmonic resonances. The resonance analysis agrees very well with numerical results and has predictive power. We extend a previously developed perturbation method, based essentially on a Poincar\'e-Lindstedt scheme, in order to account for these resonances, and in order to treat more general model cases, including truncated Toda potentials. Our results give qualitative and semiquantitative account for the superharmonic resonances of q-breathers and natural packets. ©2007 American Institute of Physic
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