Discrete Breathers

Nonlinear classical Hamiltonian lattices exhibit generic solutions in the form of discrete breathers. These solutions are time-periodic and (typically exponentially) localized in space. The lattices exhibit discrete translational symmetry. Discrete breathers are not confined to certain lattice dimensions. Necessary ingredients for their occurence are the existence of upper bounds on the phonon spectrum (of small fluctuations around the groundstate) of the system as well as the nonlinearity in the differential equations. We will present existence proofs, formulate necessary existence conditions, and discuss structural stability of discrete breathers. The following results will be also discussed: the creation of breathers through tangent bifurcation of band edge plane waves; dynamical stability; details of the spatial decay; numerical methods of obtaining breathers; interaction of breathers with phonons and electrons; movability; influence of the lattice dimension on discrete breather properties; quantum lattices - quantum breathers. Finally we will formulate a new conceptual aproach capable of predicting whether discrete breather exist for a given system or not, without actually solving for the breather. We discuss potential applications in lattice dynamics of solids (especially molecular crystals), selective bond excitations in large molecules, dynamical properties of coupled arrays of Josephson junctions, and localization of electromagnetic waves in photonic crystals with nonlinear response.Comment: 62 pages, LaTeX, 14 ps figures. Physics Reports, to be published; see also at http://www.mpipks-dresden.mpg.de/~flach/html/preprints.htm

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 $a\approx 1.85$

Field exposed water in a nanopore: liquid or vapour?

We study the behavior of ambient temperature water under the combined effects of nanoscale confinement and applied electric field. Using molecular simulations we analyze the thermodynamic causes of field-induced expansion at some, and contraction at other conditions. Repulsion among parallel water dipoles and mild weakening of interactions between partially aligned water molecules prove sufficient to destabilize the aqueous liquid phase in isobaric systems in which all water molecules are permanently exposed to a uniform electric field. At the same time, simulations reveal comparatively weak field-induced perturbations of water structure upheld by flexible hydrogen bonding. In open systems with fixed chemical potential, these perturbations do not suffice to offset attraction of water into the field; additional water is typically driven from unperturbed bulk phase to the field-exposed region. In contrast to recent theoretical predictions in the literature, our analysis and simulations confirm that classical electrostriction characterizes usual electrowetting behavior in nanoscale channels and nanoporous materials.Comment: 20 pages, 6 figures + T.O.C. figure, in press in PCC

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

From Single Agent to Multi-Agent: Improving Traffic Signal Control

Due to accelerating urbanization, the importance of solving the signal control problem increases. This paper analyzes various existing methods and suggests options for increasing the number of agents to reduce the average travel time. Experiments were carried out with 2 datasets. The results show that in some cases, the implementation of multiple agents can improve existing methods. For a fine-tuned large language model approach there is small enhancement on all metrics.Comment: 13 page

Blind source separation via multinode sparse representation

We consider a problem of blind source separation from a set of instan taneous linear mixtures, where the mixing matrix is unknown. It was discovered recently, that exploiting the sparsity of sources in an appro priate representation according to some signal dictionary, dramatically improves the quality of separation. In this work we use the property of multiscale transforms, such as wavelet or wavelet packets, to decompose signals into sets of local features with various degrees of sparsity. We use this intrinsic property for selecting the best (most sparse) subsets of features for further separation. The performance of the algorithm is verified on noise-free and noisy data. Experiments with simulated signals, musical sounds and images demonstrate significant improvement of separation quality over previously reported results

Representations of solutions of the wave equation based on relativistic wavelets

A representation of solutions of the wave equation with two spatial coordinates in terms of localized elementary ones is presented. Elementary solutions are constructed from four solutions with the help of transformations of the affine Poincar\'e group, i.e., with the help of translations, dilations in space and time and Lorentz transformations. The representation can be interpreted in terms of the initial-boundary value problem for the wave equation in a half-plane. It gives the solution as an integral representation of two types of solutions: propagating localized solutions running away from the boundary under different angles and packet-like surface waves running along the boundary and exponentially decreasing away from the boundary. Properties of elementary solutions are discussed. A numerical investigation of coefficients of the decomposition is carried out. An example of the field created by sources moving along a line with different speeds is considered, and the dependence of coefficients on speeds of sources is discussed.Comment: submitted to J. Phys. A: Math. Theor., 20 pages, 6 figure