Long time stability of small amplitude Breathers in a mixed FPU-KG model

In 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 Schr\"odinger 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.Comment: Substantial revision in the presentation. Stability time scale slightly modifie

Excited States of Open Strings From N=4\mathcal{N}=4 SYM

We continue the analysis of building open strings stretched between giant gravitons from $\mathcal{N}=4$ SYM by going to second order in perturbation theory using the three-loop dilatation generator from the field theory. In the process we build a Fock-like space of states using Cuntz oscillators which can be used to access the excited open string states. We find a remarkable cancellation among the excited states that shows the ground state energy is consistent with a fully relativistic dispersion relation.Comment: 33 pages. Typos fixe

A Pendant for Polya: The One-Loop Partition Function of N=4 SYM on R x S^3

We study weakly coupled SU(N) N = 4 super Yang-Mills theory on R x S^3 at infinite N, which has interesting thermodynamics, including a Hagedorn transition, even at zero Yang-Mills coupling. We calculate the exact one-loop partition function below the Hagedorn temperature. Our calculation employs the representation of the one-loop dilatation operator as a spin chain Hamiltonian acting on neighboring sites and a generalization of Polya's counting of `necklaces' (gauge-invariant operators) to include necklaces with a `pendant' (an operator which acts on neighboring beads). We find that the one-loop correction to the Hagedorn temperature is delta ln T_H = + lambda/8 pi^2.Comment: 39 pages, harvmac. v2: references and some clarifications added, v3: proof of (3.28) correcte

Delayed Dynamical Systems: Networks, Chimeras and Reservoir Computing

We present a systematic approach to reveal the correspondence between time delay dynamics and networks of coupled oscillators. After early demonstrations of the usefulness of spatio-temporal representations of time-delay system dynamics, extensive research on optoelectronic feedback loops has revealed their immense potential for realizing complex system dynamics such as chimeras in rings of coupled oscillators and applications to reservoir computing. Delayed dynamical systems have been enriched in recent years through the application of digital signal processing techniques. Very recently, we have showed that one can significantly extend the capabilities and implement networks with arbitrary topologies through the use of field programmable gate arrays (FPGAs). This architecture allows the design of appropriate filters and multiple time delays which greatly extend the possibilities for exploring synchronization patterns in arbitrary topological networks. This has enabled us to explore complex dynamics on networks with nodes that can be perfectly identical, introduce parameter heterogeneities and multiple time delays, as well as change network topologies to control the formation and evolution of patterns of synchrony

Higher Spin Gravity Amplitudes From Zero-form Charges

We examine zero-form charges in Vasiliev's four-dimensional bosonic higher spin gravities. These are classical observables given by integrals over noncommutative twistor space of adjoint combinations of the zero-form master fields, including insertions of delta functions in the deformed oscillators serving as gauge invariant regulators. The regularized charges admit perturbative expansions in terms of multi-linear functionals in the Weyl zero-form, which are Bose symmetric and higher spin invariant by construction, and that can be interpreted as basic building blocks for higher spin gravity amplitudes. We compute two- and three-point functions by attaching external legs given by unfolded bulk-to-boundary propagators, and identify the result with the two- and three-current correlation functions in theories of free conformal scalars and fermions in three dimensions. Modulo assumptions on the structure of the sub-leading corrections, and relying on the generalized Hamiltonian off-shell formulation, we are thus led to propose an expression for the free energy as a sum of suitably normalized zero-form chargesComment: V2: Typos corrected, references added, footnote and note added, discussion section improve

An exact solution method for 1D polynomial Schr\"odinger equations

Stationary 1D Schr\"odinger equations with polynomial potentials are reduced to explicit countable closed systems of exact quantization conditions, which are selfconsistent constraints upon the zeros of zeta-regularized spectral determinants, complementing the usual asymptotic (Bohr--Sommerfeld) constraints. (This reduction is currently completed under a certain vanishing condition.) In particular, the symmetric quartic oscillators are admissible systems, and the formalism is tested upon them. Enforcing the exact and asymptotic constraints by suitable iterative schemes, we numerically observe geometric convergence to the correct eigenvalues/functions in some test cases, suggesting that the output of the reduction should define a contractive fixed-point problem (at least in some vicinity of the pure $q^4$ case).Comment: flatex text.tex, 4 file

Multi-cluster dynamics in coupled phase oscillator networks

In this paper we examine robust clustering behaviour with multiple nontrivial clusters for identically and globally coupled phase oscillators. These systems are such that the dynamics is completely determined by the number of oscillators N and a single scalar function $g(\varphi)$ (the coupling function). Previous work has shown that (a) any clustering can stably appear via choice of a suitable coupling function and (b) open sets of coupling functions can generate heteroclinic network attractors between cluster states of saddle type, though there seem to be no examples where saddles with more than two nontrivial clusters are involved. In this work we clarify the relationship between the coupling function and the dynamics. We focus on cases where the clusters are inequivalent in the sense of not being related by a temporal symmetry, and demonstrate that there are coupling functions that give robust heteroclinic networks between periodic states involving three or more nontrivial clusters. We consider an example for N=6 oscillators where the clustering is into three inequivalent clusters. We also discuss some aspects of the bifurcation structure for periodic multi-cluster states and show that the transverse stability of inequivalent clusters can, to a large extent, be varied independently of the tangential stability

Non-Markovian entanglement dynamics of quantum continuous variable systems in thermal environments

We study two continuous variable systems (or two harmonic oscillators) and investigate their entanglement evolution under the influence of non-Markovian thermal environments. The continuous variable systems could be two modes of electromagnetic fields or two nanomechanical oscillators in the quantum domain. We use quantum open system method to derive the non-Markovian master equations of the reduced density matrix for two different but related models of the continuous variable systems. The two models both consist of two interacting harmonic oscillators. In model A, each of the two oscillators is coupled to its own independent thermal reservoir, while in model B the two oscillators are coupled to a common reservoir. To quantify the degrees of entanglement for the bipartite continuous variable systems in Gaussian states, logarithmic negativity is used. We find that the dynamics of the quantum entanglement is sensitive to the initial states, the oscillator-oscillator interaction, the oscillator-environment interaction and the coupling to a common bath or to different, independent baths.Comment: 10 two-column pages, 8 figures, to appear in Phys. Rev.

Quantum Gravity and Higher Curvature Actions

Effective equations are often useful to extract physical information from quantum theories without having to face all technical and conceptual difficulties. One can then describe aspects of the quantum system by equations of classical type, which correct the classical equations by modified coefficients and higher derivative terms. In gravity, for instance, one expects terms with higher powers of curvature. Such higher derivative formulations are discussed here with an emphasis on the role of degrees of freedom and on differences between Lagrangian and Hamiltonian treatments. A general scheme is then provided which allows one to compute effective equations perturbatively in a Hamiltonian formalism. Here, one can expand effective equations around any quantum state and not just a perturbative vacuum. This is particularly useful in situations of quantum gravity or cosmology where perturbations only around vacuum states would be too restrictive. The discussion also demonstrates the number of free parameters expected in effective equations, used to determine the physical situation being approximated, as well as the role of classical symmetries such as Lorentz transformation properties in effective equations. An appendix collects information on effective correction terms expected from loop quantum gravity and string theory.Comment: 28 pages, based on a lecture course at the 42nd Karpacz Winter School of Theoretical Physics ``Current Mathematical Topics in Gravitation and Cosmology,'' Ladek, Poland, February 6-11, 200