Nonlinear waves in Newton's cradle and the discrete p-Schroedinger equation

We study nonlinear waves in Newton's cradle, a classical mechanical system consisting of a chain of beads attached to linear pendula and interacting nonlinearly via Hertz's contact forces. We formally derive a spatially discrete modulation equation, for small amplitude nonlinear waves consisting of slow modulations of time-periodic linear oscillations. The fully-nonlinear and unilateral interactions between beads yield a nonstandard modulation equation that we call the discrete p-Schroedinger (DpS) equation. It consists of a spatial discretization of a generalized Schroedinger equation with p-Laplacian, with fractional p>2 depending on the exponent of Hertz's contact force. We show that the DpS equation admits explicit periodic travelling wave solutions, and numerically find a plethora of standing wave solutions given by the orbits of a discrete map, in particular spatially localized breather solutions. Using a modified Lyapunov-Schmidt technique, we prove the existence of exact periodic travelling waves in the chain of beads, close to the small amplitude modulated waves given by the DpS equation. Using numerical simulations, we show that the DpS equation captures several other important features of the dynamics in the weakly nonlinear regime, namely modulational instabilities, the existence of static and travelling breathers, and repulsive or attractive interactions of these localized structures

Gaussian solitary waves and compactons in Fermi-Pasta-Ulam lattices with Hertzian potentials

We consider a class of fully-nonlinear Fermi-Pasta-Ulam (FPU) lattices, consisting of a chain of particles coupled by fractional power nonlinearities of order $\alpha >1$. This class of systems incorporates a classical Hertzian model describing acoustic wave propagation in chains of touching beads in the absence of precompression. We analyze the propagation of localized waves when $\alpha$ is close to unity. Solutions varying slowly in space and time are searched with an appropriate scaling, and two asymptotic models of the chain of particles are derived consistently. The first one is a logarithmic KdV equation, and possesses linearly orbitally stable Gaussian solitary wave solutions. The second model consists of a generalized KdV equation with H\"older-continuous fractional power nonlinearity and admits compacton solutions, i.e. solitary waves with compact support. When $\alpha \rightarrow 1^+$, we numerically establish the asymptotically Gaussian shape of exact FPU solitary waves with near-sonic speed, and analytically check the pointwise convergence of compactons towards the limiting Gaussian profile

Breathers in oscillator chains with Hertzian interactions

We prove nonexistence of breathers (spatially localized and time-periodic oscillations) for a class of Fermi-Pasta-Ulam lattices representing an uncompressed chain of beads interacting via Hertz's contact forces. We then consider the setting in which an additional on-site potential is present, motivated by the Newton's cradle under the effect of gravity. Using both direct numerical computations and a simplified asymptotic model of the oscillator chain, the so-called discrete p-Schr\"odinger (DpS) equation, we show the existence of discrete breathers and study their spectral properties and mobility. Due to the fully nonlinear character of Hertzian interactions, breathers are found to be much more localized than in classical nonlinear lattices and their motion occurs with less dispersion. In addition, we study numerically the excitation of a traveling breather after an impact at one end of a semi-infinite chain. This case is well described by the DpS equation when local oscillations are faster than binary collisions, a situation occuring e.g. in chains of stiff cantilevers decorated by spherical beads. When a hard anharmonic part is added to the local potential, a new type of traveling breather emerges, showing spontaneous direction-reversing in a spatially homogeneous system. Finally, the interaction of a moving breather with a point defect is also considered in the cradle system. Almost total breather reflections are observed at sufficiently high defect sizes, suggesting potential applications of such systems as shock wave reflectors

From Newton's cradle to the discrete p-Schr\"odinger equation

We investigate the dynamics of a chain of oscillators coupled by fully-nonlinear interaction potentials. This class of models includes Newton's cradle with Hertzian contact interactions between neighbors. By means of multiple-scale analysis, we give a rigorous asymptotic description of small amplitude solutions over large times. The envelope equation leading to approximate solutions is a discrete p-Schr\"odinger equation. Our results include the existence of long-lived breather solutions to the original model. For a large class of localized initial conditions, we also estimate the maximal decay of small amplitude solutions over long times

Electroweak Phase Transition and LHC Signatures in the Singlet Majoron Model

We reconsider the strength of the electroweak phase transition in the singlet Majoron extension of the Standard Model, with a low (~TeV) scale of the singlet VEV. A strongly first order phase transition, of interest for electroweak baryogenesis, is found in sizeable regions of the parameter space, especially when the cross-coupling lambda_{hs}|S|^2|H|^2 between the singlet and the doublet Higgs is significant. Large Majorana Yukawa couplings of the singlet neutrinos, y_i S nu_i^c nu_i, are also important for strengthening the transition. We incorporate the LEP and Tevatron constraints on the Higgs masses, and electroweak precision constraints, in our search for allowed parameters; successful examples include singlet masses ranging from 5 GeV to several TeV. Models with a strong phase transition typically predict a nonstandard Higgs with mass in the range 113 GeV < m_H < 200 GeV and production cross sections reduced by mixing with the singlet, with cos^2(theta) significantly less than 1. We also find examples where the singlet is light and the decay H -> SS can modify the Higgs branching ratios relative to Standard Model expectations.Comment: 36 pages, 18 figure

An academic odyssey: Writing over time

In this paper we present and discuss the results of six enquiries into the first author's academic writing over the last fifty years. Our aim is to assess whether or not his academic writing style has changed with age, experience, and cognitive decline. The results of these studies suggest that the readability of textbook chapters written by Hartley has remained fairly stable for over 50 years, with the later chapters becoming easier to read. The format of the titles used for chapters and papers has also remained much the same, with an increase in the use of titles written in the form of questions. It also appears that the format of the chosen titles had no effect on citation rates, but that papers that obtained the highest citation rates were written with colleagues rather by Hartley alone. Finally it is observed that Hartley's publication rate has remained much the same for over fifty years but that this has been achieved at the expense of other academic activities

Do men and women differ in their use of tables and graphs in academic publications?

International audienceIn psychological research there is huge literature on differences between the sexes. Typically it used to be thought that women were more verbally and men more spatially oriented. These differences now seem to be waning. In this article we present three studies on sex differences in the use of tables and graphs in academic articles. These studies are based on data mining from approximately 2,000 articles published in over 200 peer-reviewed journals in the sciences and social sciences. In Study 1 we found that, in the sciences, men used 26% more graphs and figures than women, but that there were no significant differences between them in their use of tables. In Study 2 we found no significant differences between men and women in their use of graphs and figures or tables in social science articles. In Study 3 we found no significant differences between men and women in their use of what we termed 'data' and 'text' tables in social science articles. It is possible that these findings indicate that academic writing is now becoming a genre that is equally undertaken by men and women

Intrinsic localized modes in nonlinear models inspired by DNA

International audienceWe discuss nonlinear dynamic models for the fluctuational opening of the base pairs in DNA and show that a standard model which is satisfactory for time-independent properties has to be improved to properly describe the time scales of the fluctuations. The existence of an energy barrier for the closing of the base pairs has to be taken into account. This introduces a model which sustains a new class of Intrinsically Localized Modes (ILMs). We investigate their properties numerically, and then consider two simplified versions of the improved DNA model allowing an analytical study of some properties of those ILMs. The models are different because the effective barrier necessary for the existence of this new class of ILMs is obtained either through the on-site potential or through the nonlinear stacking interaction, but they nevertheless sustain similar nonlinear localized excitations. An extension of the usual anti--continuum has to be introduced for the analysis, and relies on a continuation of localized equilibria from infinity