Obtaining Breathers in Nonlinear Hamiltonian Lattices

We present a numerical method for obtaining high-accuracy numerical solutions of spatially localized time-periodic excitations on a nonlinear Hamiltonian lattice. We compare these results with analytical considerations of the spatial decay. We show that nonlinear contributions have to be considered, and obtain very good agreement between the latter and the numerical results. We discuss further applications of the method and results.Comment: 21 pages (LaTeX), 8 figures in ps-files, tar-compressed uuencoded file, Physical Review E, in pres

Fermionic bound states on a one-dimensional lattice

We study bound states of two fermions with opposite spins in an extended Hubbard chain. The particles interact when located both on a site or on adjacent sites. We find three different types of bound states. Type U is predominantly formed of basis states with both fermions on the same site, while two states of type V originate from both fermions occupying neighbouring sites. Type U, and one of the states from type V, are symmetric with respect to spin flips. The remaining one from type V is antisymmetric. V-states are characterized by a diverging localization length below some critical wave number. All bound states become compact for wave numbers at the edge of the Brilloin zone.Comment: 4 pages, 3 figure

Isochronism and tangent bifurcation of band edge modes in Hamiltonian lattices

In {\em Physica D} {\bf 91}, 223 (1996), results were obtained regarding the tangent bifurcation of the band edge modes ($q=0,\pi$) of nonlinear Hamiltonian lattices made of $N$ coupled oscillators. Introducing the concept of {\em partial isochronism} which characterises the way the frequency of a mode, $\omega$, depends on its energy, $\epsilon$, we generalize these results and show how the bifurcation energies of these modes are intimately connected to their degree of isochronism. In particular we prove that in a lattice of coupled purely isochronous oscillators ($\omega(\epsilon)$ strictly constant), the in-phase mode ($q=0$) never undergoes a tangent bifurcation whereas the out-of-phase mode ($q=\pi$) does, provided the strength of the nonlinearity in the coupling is sufficient. We derive a discrete nonlinear Schr\"odinger equation governing the slow modulations of small-amplitude band edge modes and show that its nonlinear exponent is proportional to the degree of isochronism of the corresponding orbits. This equation may be seen as a link between the tangent bifurcation of band edge modes and the possible emergence of localized modes such as discrete breathers.Comment: 23 pages, 1 figur

On the Existence of Localized Excitations in Nonlinear Hamiltonian Lattices

We consider time-periodic nonlinear localized excitations (NLEs) on one-dimensional translationally invariant Hamiltonian lattices with arbitrary finite interaction range and arbitrary finite number of degrees of freedom per unit cell. We analyse a mapping of the Fourier coefficients of the NLE solution. NLEs correspond to homoclinic points in the phase space of this map. Using dimensionality properties of separatrix manifolds of the mapping we show the persistence of NLE solutions under perturbations of the system, provided NLEs exist for the given system. For a class of nonintegrable Fermi-Pasta-Ulam chains we rigorously prove the existence of NLE solutions.Comment: 13 pages, LaTeX, 2 figures will be mailed upon request (Phys. Rev. E, in press

Breathers on lattices with long range interaction

We analyze the properties of breathers (time periodic spatially localized solutions) on chains in the presence of algebraically decaying interactions $1/r^s$. We find that the spatial decay of a breather shows a crossover from exponential (short distances) to algebraic (large distances) decay. We calculate the crossover distance as a function of $s$ and the energy of the breather. Next we show that the results on energy thresholds obtained for short range interactions remain valid for $s>3$ and that for $s < 3$ (anomalous dispersion at the band edge) nonzero thresholds occur for cases where the short range interaction system would yield zero threshold values.Comment: 4 pages, 2 figures, PRB Rapid Comm. October 199

Slow Relaxation and Phase Space Properties of a Conservative System with Many Degrees of Freedom

We study the one-dimensional discrete $\Phi^4$ model. We compare two equilibrium properties by use of molecular dynamics simulations: the Lyapunov spectrum and the time dependence of local correlation functions. Both properties imply the existence of a dynamical crossover of the system at the same temperature. This correlation holds for two rather different regimes of the system - the displacive and intermediate coupling regimes. Our results imply a deep connection between slowing down of relaxations and phase space properties of complex systems.Comment: 14 pages, LaTeX, 10 Figures available upon request (SF), Phys. Rev. E, accepted for publicatio

Acoustic breathers in two-dimensional lattices

The existence of breathers (time-periodic and spatially localized lattice vibrations) is well established for i) systems without acoustic phonon branches and ii) systems with acoustic phonons, but also with additional symmetries preventing the occurence of strains (dc terms) in the breather solution. The case of coexistence of strains and acoustic phonon branches is solved (for simple models) only for one-dimensional lattices. We calculate breather solutions for a two-dimensional lattice with one acoustic phonon branch. We start from the easy-to-handle case of a system with homogeneous (anharmonic) interaction potentials. We then easily continue the zero-strain breather solution into the model sector with additional quadratic and cubic potential terms with the help of a generalized Newton method. The lattice size is $70\times 70$. The breather continues to exist, but is dressed with a strain field. In contrast to the ac breather components, which decay exponentially in space, the strain field (which has dipole symmetry) should decay like $1/r^a, a=2$. On our rather small lattice we find an exponent $a\approx 1.85$