On Boussinesq's equation for water waves

A century and a half ago, J. Boussinesq derived an equation for the propagation of shallow water waves in a channel. Despite the fundamental importance of this equation for a number of physical phenomena, mathematical results on it remain scarce. One reason for this is that the equation is ill-posed. In this paper, we establish several results on the Boussinesq equation. First, by solving the direct and inverse problems for an associated third-order spectral problem, we develop an Inverse Scattering Transform (IST) approach to the initial value problem. Using this approach, we establish a number of existence, uniqueness, and blow-up results. For example, the IST approach allows us to identify physically meaningful global solutions and to construct, for each $T > 0$, solutions that blow up exactly at time $T$. Our approach also yields an expression for the solution of the initial value problem for the Boussinesq equation in terms of the solution of a Riemann--Hilbert problem. By analyzing this Riemann--Hilbert problem, we arrive at asymptotic formulas for the solution. We identify ten main asymptotic sectors in the $(x,t)$-plane; in each of these sectors, we compute an exact expression for the leading asymptotic term together with a precise error estimate. The asymptotic picture that emerges consists, roughly speaking, of two nonlinearly coupled copies of the corresponding picture for the (unidirectional) KdV equation, one copy for right-moving and one for left-moving waves. Of particular interest are the sectors which describe the interaction of right and left moving waves, which present qualitatively new phenomena.Comment: 111 pages, 23 figure

Room temperature Peierls distortion in small radius nanotubes

By means of {\it ab initio} simulations, we investigate the phonon band structure and electron-phonon coupling in small 4-\AA diameter nanotubes. We show that both the C(5,0) and C(3,3) tubes undergo above room temperature a Peierls transition mediated by an acoustical long-wavelength and an optical $q=2k_F$ phonons respectively. In the armchair geometry, we verify that the electron-phonon coupling parameter $\lambda$ originates mainly from phonons at $q=2k_F$ and is strongly enhanced when the diameter decreases. These results question the origin of superconductivity in small diameter nanotubes.Comment: submitted 21oct2004 accepted 6jan2005 (Phys.Rev.Lett.

Probing the electron-phonon coupling in ozone-doped graphene by Raman spectroscopy

We have investigated the effects of ozone treatment on graphene by Raman scattering. Sequential ozone short-exposure cycles resulted in increasing the $p$ doping levels as inferred from the blue shift of the 2$D$ and $G$ peak frequencies, without introducing significant disorder. The two-phonon 2$D$ and 2$D'$ Raman peak intensities show a significant decrease, while, on the contrary, the one-phonon G Raman peak intensity remains constant for the whole exposure process. The former reflects the dynamics of the photoexcited electrons (holes) and, specifically, the increase of the electron-electron scattering rate with doping. From the ratio of 2$D$ to 2$D$ intensities, which remains constant with doping, we could extract the ratio of electron-phonon coupling parameters. This ratio is found independent on the number of layers up to ten layers. Moreover, the rate of decrease of 2$D$ and 2$D'$ intensities with doping was found to slowdown inversely proportional to the number of graphene layers, revealing the increase of the electron-electron collision probability

Thermal rectification in carbon nanotube intramolecular junctions: Molecular dynamics calculations

We study heat conduction in (n, 0)/(2n, 0) intramolecular junctions by using molecular dynamics method. It is found that the heat conduction is asymmetric, namely, heat transports preferably in one direction. This phenomenon is also called thermal rectification. The rectification is weakly dependent on the detailed structure of connection part, but is strongly dependent on the temperature gradient. We also study the effect of the tube radius and intramolecular junction length on the rectification. Our study shows that the tensile stress can increase rectification. The physical mechanism of the rectification is explained

Determination of the Intershell Conductance in Multiwalled Carbon Nanotubes

We report on the intershell electron transport in multiwalled carbon nanotubes (MWNT). To do this, local and nonlocal four-point measurements are used to study the current path through the different shells of a MWNT. For short electrode separations $\lesssim$ 1 $\mu$m the current mainly flows through the two outer shells, described by a resistive transmission line with an intershell conductance per length of ~(10 k\Omega)^{-1}/$\mu$m. The intershell transport is tunnel-type and the transmission is consistent with the estimate based on the overlap between $\pi$-orbitals of neighboring shells.Comment: 5 pages, 4 figure

The "good'' Boussinesq equation: long-time asymptotics

We consider the initial-value problem for the ``good'' Boussinesq equation on the line. Using inverse scattering techniques, the solution can be expressed in terms of the solution of a $3 \times 3$-matrix Riemann-Hilbert problem. We establish formulas for the long-time asymptotics of the solution by performing a Deift-Zhou steepest descent analysis of a regularized version of this Riemann-Hilbert problem.Comment: 34 pages, 13 figure

Electrical conductivity measured in atomic carbon chains

The first electrical conductivity measurements of monoatomic carbon chains are reported in this study. The chains were obtained by unraveling carbon atoms from graphene ribbons while an electrical current flowed through the ribbon and, successively, through the chain. The formation of the chains was accompanied by a characteristic drop in the electrical conductivity. The conductivity of carbon chains was much lower than previously predicted for ideal chains. First-principles calculations using both density functional and many-body perturbation theory show that strain in the chains determines the conductivity in a decisive way. Indeed, carbon chains are always under varying non-zero strain that transforms its atomic structure from cumulene to polyyne configuration, thus inducing a tunable band gap. The modified electronic structure and the characteristics of the contact to the graphitic periphery explain the low conductivity of the locally constrained carbon chain.Comment: 21 pages, 9 figure

First-Principle Description of Correlation Effects in Layered Materials

We present a first-principles description of anisotropic materials characterized by having both weak (dispersion-like) and strong covalent bonds, based on the Adiabatic--Connection Fluctuation--Dissipation Theorem within Density Functional Theory. For hexagonal boron nitride the in-plane and out of plane bonding as well as vibrational dynamics are well described both at equilibrium and when the layers are pulled apart. Also bonding in covalent and ionic solids is described. The formalism allows to ping-down the deficiencies of common exchange-correlation functionals and provides insight towards the inclusion of dispersion interactions into the correlation functional.Comment: Accepted for publication in Physical Review Letter