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

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

Euler characteristics in relative K-groups

Suppose that M is a finite module under the Galois group of a local or global field. Ever since Tate's papers [17, 18], we have had a simple and explicit formula for the Euler–Poincaré characteristic of the cohomology of M. In this note we are interested in a refinement of this formula when M also carries an action of some algebra [script A], commuting with the Galois action (see Proposition 5.2 and Theorem 5.1 below). This refinement naturally takes the shape of an identity in a relative K-group attached to [script A] (see Section 2). We shall deduce such an identity whenever we have a formula for the ordinary Euler characteristic, the key step in the proof being the representability of certain functors by perfect complexes (see Section 3). This representability may be of independent interest in other contexts. Our formula for the equivariant Euler characteristic over [script A] implies the ‘isogeny invariance’ of the equivariant conjectures on special values of the L-function put forward in [3], and this was our motivation to write this note. Incidentally, isogeny invariance (of the conjectures of Birch and Swinnerton-Dyer) was also a motivation for Tate's original paper [18]. I am very grateful to J-P. Serre for illuminating discussions on the subject of this note, in particular for suggesting that I consider representability. I should also like to thank D. Burns for insisting on a most general version of the results in this paper

Iwasawa Theory and Motivic L-functions

We illustrate the use of Iwasawa theory in proving cases of the (equivariant) Tamagawa number conjecture

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

AC-driven quantum spins: resonant enhancement of transverse DC magnetization

We consider s=1/2 spins in the presence of a constant magnetic field in z-direction and an AC magnetic field in the x-z plane. A nonzero DC magnetization component in y direction is a result of broken symmetries. A pairwise interaction between two spins is shown to resonantly increase the induced magnetization by one order of magnitude. We discuss the mechanism of this enhancement, which is due to additional avoided crossings in the level structure of the system.Comment: 7 pages, 7 figure

Discrete breathers in thermal equilibrium: distributions and energy gaps

We study a discrete two-dimensional nonlinear system that allows for discrete breather solutions. We perform a spectral analysis of the lattice dynamics at thermal equilibrium and use a cooling technique to measure the amount of breathers at thermal equilibrium. Our results confirm the existence of an energy threshold for discrete breathers. The cooling method provides with a novel computational technique of measuring and analyzing discrete breather distribution properties in thermal equilibrium.Comment: 20 pages, 14 figure