Exact shock solution of a coupled system of delay differential equations: a car-following model

In this paper, we present exact shock solutions of a coupled system of delay differential equations, which was introduced as a traffic-flow model called {\it the car-following model}. We use the Hirota method, originally developed in order to solve soliton equations. %While, with a periodic boundary condition, this system has % a traveling-wave solution given by elliptic functions. The relevant delay differential equations have been known to allow exact solutions expressed by elliptic functions with a periodic boundary conditions. In the present work, however, shock solutions are obtained with open boundary, representing the stationary propagation of a traffic jam.Comment: 6 pages, 2 figure

Interacting with digital media at home via a second screen

In recent years Interactive Television (iTV) has become a household technology on a global scale. However, iTV is still a new technology in the early stages of its evolution. Our previous research looked at how everyday users of iTV feel about the interactive part of iTV. In a series of studies we investigated how people use iTV services; their likes, dislikes, preferences and opinions. We then developed a second screen-based prototype device in response to these findings and tested it with iTV users in their own homes. This is a work in progress paper that outlines the work carried previously in the area of controlling interactive Television via a second screen. The positive user responses led us to extend the scope of our previous research to look into other related areas such as barriers to digital interactive media and personalisation of digital interactive media at home

Diamagnetic susceptibility obtained from the six-vertex model and its implications for the high-temperature diamagnetic state of cuprate superconductors

We study the diamagnetism of the 6-vertex model with the arrows as directed bond currents. To our knowledge, this is the first study of the diamagnetism of this model. A special version of this model, called F model, describes the thermal disordering transition of an orbital antiferromagnet, known as d-density wave (DDW), a proposed state for the pseudogap phase of the high-Tc cuprates. We find that the F model is strongly diamagnetic and the susceptibility may diverge in the high temperature critical phase with power law arrow correlations. These results may explain the surprising recent observation of a diverging low-field diamagnetic susceptibility seen in some optimally doped cuprates within the DDW model of the pseudogap phase.Comment: 4.5 pages, 2 figures, revised version accepted in Phys. Rev. Let

Identication of linear slow sausage waves in magnetic pores

The analysis of an 11-hour series of high resolution white light observations of a large pore in the sunspot group NOAA 7519, observed on 5 June 1993 with the Swedish Vacuum Solar Telescope at La Palma on Canary Islands, has been recently described by Dorotovič et al. (2002). Special attention was paid to the evolution of a filamentary region attached to the pore, to horizontal motions around the pore, and to small-scale morphological changes. One of the results, relevant to out work here, was the determination of temporal area evolution of the studied pore where the area itself showed a linear trend of decrease with time at an average rate of −0.23 Mm2h−1 during the entire observing period. Analysing the time series of the are of the pore, there is strong evidence that coupling between the solar interior and magnetic atmosphere can occur at various scales and that the referred decrease of the area may be connected with a decrease of the magnetic field strength according to the magnetic field-to-size relation. Periods of global acoustic, e.g. p-mode, driven waves are usually in the range of 5–10 minutes, and are favourite candidates for the coupling of interior oscillations with atmospheric dynamics. However, by assuming that magneto-acoustic gravity waves may be there too, and may act as drivers, the observed periodicities (frequencies) are expected to be much longer (smaller), falling well within the mMHz domain. In this work we determine typical periods of such range in the area evolution of the pore using wavelet analysis. The resulted periods are in the range of 20–70 minutes, suggesting that periodic elements of the temporal evolution of the area of this studied pore could be linked to, and considered as, observational evidence of linear low-frequency slow sausage (magneto-acoustic gravity) waves in magnetic pores. This would give us further evidence on the coupling of global solar oscillations to the overlaying magnetic atmosphere

Self-Similar Blowup Solutions to the 2-Component Camassa-Holm Equations

In this article, we study the self-similar solutions of the 2-component Camassa-Holm equations% \begin{equation} \left\{ \begin{array} [c]{c}% \rho_{t}+u\rho_{x}+\rho u_{x}=0 m_{t}+2u_{x}m+um_{x}+\sigma\rho\rho_{x}=0 \end{array} \right. \end{equation} with \begin{equation} m=u-\alpha^{2}u_{xx}. \end{equation} By the separation method, we can obtain a class of blowup or global solutions for $\sigma=1$ or $-1$. In particular, for the integrable system with $\sigma=1$, we have the global solutions:% \begin{equation} \left\{ \begin{array} [c]{c}% \rho(t,x)=\left\{ \begin{array} [c]{c}% \frac{f\left( \eta\right) }{a(3t)^{1/3}},\text{ for }\eta^{2}<\frac {\alpha^{2}}{\xi} 0,\text{ for }\eta^{2}\geq\frac{\alpha^{2}}{\xi}% \end{array} \right. ,u(t,x)=\frac{\overset{\cdot}{a}(3t)}{a(3t)}x \overset{\cdot\cdot}{a}(s)-\frac{\xi}{3a(s)^{1/3}}=0,\text{ }a(0)=a_{0}% >0,\text{ }\overset{\cdot}{a}(0)=a_{1} f(\eta)=\xi\sqrt{-\frac{1}{\xi}\eta^{2}+\left( \frac{\alpha}{\xi}\right) ^{2}}% \end{array} \right. \end{equation} where $\eta=\frac{x}{a(s)^{1/3}}$ with $s=3t;$ $\xi>0$ and $\alpha\geq0$ are arbitrary constants.\newline Our analytical solutions could provide concrete examples for testing the validation and stabilities of numerical methods for the systems.Comment: 5 more figures can be found in the corresponding journal paper (J. Math. Phys. 51, 093524 (2010) ). Key Words: 2-Component Camassa-Holm Equations, Shallow Water System, Analytical Solutions, Blowup, Global, Self-Similar, Separation Method, Construction of Solutions, Moving Boundar

Example of shock wave in unstaible medium: The focusing nonlinear Schrodinger equation

Dissipationless shock waves in modulational unstable one-dimensional medium are investigated on the simplest example of integrable focusing nonlinear Schr\''odinger (NS) equation. Our approach is based on the construction of special exact solution of the Whitham-NS system, which ''partially saturates'' the modulational instability.Comment: 4 pages, LaTEX, version 2.09, submitted to Phys. Lett.

The Modulation of Multiple Phases Leading to the Modified KdV Equation

This paper seeks to derive the modified KdV (mKdV) equation using a novel approach from systems generated from abstract Lagrangians that possess a two-parameter symmetry group. The method to do uses a modified modulation approach, which results in the mKdV emerging with coefficients related to the conservation laws possessed by the original Lagrangian system. Alongside this, an adaptation of the method of Kuramoto is developed, providing a simpler mechanism to determine the coefficients of the nonlinear term. The theory is illustrated using two examples of physical interest, one in stratified hydrodynamics and another using a coupled Nonlinear Schr\"odinger model, to illustrate how the criterion for the mKdV equation to emerge may be assessed and its coefficients generated.Comment: 35 pages, 5 figure

Weakly versus highly nonlinear dynamics in 1D systems

We analyze the morphological transition of a one-dimensional system described by a scalar field, where a flat state looses its stability. This scalar field may for example account for the position of a crystal growth front, an order parameter, or a concentration profile. We show that two types of dynamics occur around the transition: weakly nonlinear dynamics, or highly nonlinear dynamics. The conditions under which highly nonlinear evolution equations appear are determined, and their generic form is derived. Finally, examples are discussed.Comment: to be published in Europhys. Let

Optical supercavitation in soft-matter

We investigate theoretically, numerically and experimentally nonlinear optical waves in an absorbing out-of-equilibrium colloidal material at the gelification transition. At sufficiently high optical intensity, absorption is frustrated and light propagates into the medium. The process is mediated by the formation of a matter-shock wave due to optically induced thermodiffusion, and largely resembles the mechanism of hydrodynamical supercavitation, as it is accompanied by a dynamic phase-transition region between the beam and the absorbing material.Comment: 4 pages, 5 figures, revised version: corrected typos and reference

Higher-order splitting algorithms for solving the nonlinear Schr\"odinger equation and their instabilities

Since the kinetic and the potential energy term of the real time nonlinear Schr\"odinger equation can each be solved exactly, the entire equation can be solved to any order via splitting algorithms. We verified the fourth-order convergence of some well known algorithms by solving the Gross-Pitaevskii equation numerically. All such splitting algorithms suffer from a latent numerical instability even when the total energy is very well conserved. A detail error analysis reveals that the noise, or elementary excitations of the nonlinear Schr\"odinger, obeys the Bogoliubov spectrum and the instability is due to the exponential growth of high wave number noises caused by the splitting process. For a continuum wave function, this instability is unavoidable no matter how small the time step. For a discrete wave function, the instability can be avoided only for \dt k_{max}^2{<\atop\sim}2 \pi, where $k_{max}=\pi/\Delta x$.Comment: 10 pages, 8 figures, submitted to Phys. Rev.