Measurement of penetration depths in superconducting films

A closed expression is obtained for the mutual inductance of any arrangement of stacked circular coils between which a thin film of superconducting material is inserted

Charged-particle orbits near a magnetic null point

An approximate analytical expression is obtained for the orbits of a charged particle moving in a cusp magnetic field. The particle orbits pass close to or through a region of zero magnetic field before being reflected in regions where the magnetic field is strong. Comparison with numerically evaluated orbits shows that the analytical formula is surprisingly good and captures all the main features of the particle motion. A map describing the long-time behaviour of such orbits is obtained

Exact analytic solutions for nonlinear waves in cold plasmas

Large amplitude plasma oscillations are studied in a cold electron plasma. Using Lagrangian variables, a new class of exact analytical solutions is found. It turns out that the electric field amplitude is limited either by wave breaking or by the condition that the electron density always has to stay positive. The range of possible amplitudes is determined analytically

Nonlinear magnetoacoustic waves in a cold plasma

The equations describing planar magnetoacoustic waves of permanent form in a cold plasma are rewritten so as to highlight the presence of a naturally small parameter equal to the ratio of the electron and ion masses. If the magnetic field is not nearly perpendicular to the direction of wave propagation, this allows us to use a multiple-scale expansion to demonstrate the existence and nature of nonlinear wave solutions. Such solutions are found to have a rapid oscillation of constant amplitude superimposed on the underlying large-scale variation. The approximate equations for the large-scale variation are obtained by making an adiabatic approximation and in one limit, new explicit solitary pulse solutions are found. In the case of a perpendicular magnetic field, conditions for the existence of solitary pulses are derived. Our results are consistent with earlier studies which were restricted to waves having a velocity close to that of long-wavelength linear magnetoacoustic waves

A solitary-wave solution to a perturbed KdV equation

We derive the approximate form and speed of a solitary-wave solution to a perturbed KdV equation. Using a conventional perturbation expansion, one can derive a first-order correction to the solitary-wave speed, but at the next order, algebraically secular terms appear, which produce divergences that render the solution unphysical. These terms must be treated by a regrouping procedure developed by us previously. In this way, higher-order corrections to the speed are obtained, along with a form of solution that is bounded in space. For this particular perturbed KdV equation, it is found that there is only one possible solitary wave that has a form similar to the unperturbed soliton solution

Coating thickness and elastic modulus measurement using ultrasonic bulk wave resonance

Measurement of the resonant through thickness ultrasonic modes of a homogeneous plate using a fast Fourier transform of the temporal data can be used to calculate plate thickness very accurately. We describe an extension of this principle to two-layer systems, examining a thin coating on a substrate of known properties. The resonant behavior of these systems is predicted and we explain how this approach is used to measure coating thickness and elastic modulus. Noncontact electromagnetic acoustic transducers are used for ultrasonic measurement, as they do not significantly affect the resonant response of the system, unlike alternative contact transducers

Some exact solutions to the Lighthill Whitham Richards Payne traffic flow equations II: moderate congestion

We find a further class of exact solutions to the Lighthill Whitham Richards Payne (LWRP) traffic flow equations. As before, using two consecutive Lagrangian transformations, a linearization is achieved. Next, depending on the initial density, we either obtain exact formulae for the dependence of the car density and velocity on x, t, or else, failing that, the same result in a parametric representation. The calculation always involves two possible factorizations of a consistency condition. Both must be considered. In physical terms, the lineup usually separates into two offshoots at different velocities. Each velocity soon becomes uniform. This outcome in many ways resembles not only Rowlands, Infeld and Skorupski J. Phys. A: Math. Theor. 46 (2013) 365202 (part I) but also the two soliton solution to the Korteweg-de Vries equation. This paper can be read independently of part I. This explains unavoidable repetitions. Possible uses of both papers in checking numerical codes are indicated at the end. Since LWRP, numerous more elaborate models, including multiple lanes, traffic jams, tollgates etc. abound in the literature. However, we present an exact solution. These are few and far between, other then found by inverse scattering. The literature for various models, including ours, is given. The methods used here and in part I may be useful in solving other problems, such as shallow water flow.Comment: 15 pages, 7 figure

Stability analysis of three-dimensional breather solitons in a Bose-Einstein Condensate

We investigate the stability properties of breather soliton trains in a three-dimensional Bose-Einstein Condensate with Feshbach Resonance Management of the scattering length. This is done so as to generate both attractive and repulsive interaction. The condensate is con ned only by a one dimensional optical lattice and we consider both strong, moderate, and weak con nement. By strong con nement we mean a situation in which a quasi two dimensional soliton is created. Moderate con nement admits a fully three dimensional soliton. Weak con nement allows individual solitons to interact. Stability properties are investigated by several theoretical methods such as a variational analysis, treatment of motion in e ective potential wells, and collapse dynamics. Armed with all the information forthcoming from these methods, we then undertake a numerical calculation. Our theoretical predictions are fully con rmed, perhaps to a higher degree than expected. We compare regions of stability in parameter space obtained from a fully 3D analysis with those from a quasi two-dimensional treatment, when the dynamics in one direction are frozen. We nd that in the 3D case the stability region splits into two parts. However, as we tighten the con nement, one of the islands of stability moves toward higher frequencies and the lower frequency region becomes more and more like that for quasi 2D. We demonstrate these solutions in direct numerical simulations and, importantly, suggest a way of creating robust 3D solitons in experiments in a Bose Einstein Condensate in a one-dimensional lattice.Comment: 14 pages, 6 figures; accepted to Proc. Roy. Soc. London

Weakly nonlinear waves in magnetized plasma with a slightly non-Maxwellian electron distribution. Part 2, Stability of cnoidal waves

We determine the growth rate of linear instabilities resulting from long-wavelength transverse perturbations applied to periodic nonlinear wave solutions to the Schamel–Korteweg–de Vries–Zakharov–Kuznetsov (SKdVZK) equation which governs weakly nonlinear waves in a strongly magnetized cold-ion plasma whose electron distribution is given by two Maxwellians at slightly different temperatures. To obtain the growth rate it is necessary to evaluate non-trivial integrals whose number is kept to a minimum by using recursion relations. It is shown that a key instance of one such relation cannot be used for classes of solution whose minimum value is zero, and an additional integral must be evaluated explicitly instead. The SKdVZK equation contains two nonlinear terms whose ratio b increases as the electron distribution becomes increasingly flat-topped. As b and hence the deviation from electron isothermality increases, it is found that for cnoidal wave solutions that travel faster than long-wavelength linear waves, there is a more pronounced variation of the growth rate with the angle θ at which the perturbation is applied. Solutions whose minimum values are zero and which travel slower than long-wavelength linear waves are found, at first order, to be stable to perpendicular perturbations and have a relatively narrow range of θ for which the first-order growth rate is not zero

Weakly nonlinear waves in magnetized plasma with a slightly non-Maxwellian electron distribution. Part 1, Stability of solitary waves

Weakly nonlinear waves in strongly magnetized plasma with slightly non-isothermal electrons are governed by a modified Zakharov–Kuznetsov (ZK) equation, containing both quadratic and half-order nonlinear terms, which we refer to as the Schamel–Korteweg–de Vries–Zakharov–Kuznetsov (SKdVZK) equation. We present a method to obtain an approximation for the growth rate, γ, of sinusoidal perpendicular perturbations of wavenumber, k, to SKdVZK solitary waves over the entire range of instability. Unlike for (modified) ZK equations with one nonlinear term, in this method there is no analytical expression for kc, the cut-off wavenumber (at which the growth rate is zero) or its corresponding eigenfunction. We therefore obtain approximate expressions for these using an expansion parameter, a, related to the ratio of the nonlinear terms. The expressions are then used to find γ for k near kc as a function of a. The approximant derived from combining these analytical results with the ones for small k agrees very well with the values of γ obtained numerically. It is found that both kc and the maximum growth rate decrease as the electron distribution becomes progressively less peaked than the Maxwellian. We also present new algebraic and rarefactive solitary wave solutions to the equation