Traveling Plane Wave Research Articles

Traveling waves in Hall-magnetohydrodynamics and the ion-acoustic shock structure

Hall-MHD is a mixed hyperbolic-parabolic partial differential equation that describes the dynamics of an ideal two fluid plasma with massless electrons. We study the only shock wave family that exists in this system (the other discontinuities being contact discontinuities and not shocks). We study planar travelling wave solutions and we find solutions with discontinuities in the hydrodynamic variables, which arise due to the presence of real characteristics in Hall-MHD. We introduce a small viscosity into the equations and use the method of matched asymptotic expansions to show that solutions with a discontinuity satisfying the Rankine-Hugoniot conditions and also an entropy condition have continuous shock structures. The lowest order inner equations reduce to the compressible Navier-Stokes equations, plus an equation which implies the constancy of the magnetic field inside the shock structure. We are able to show that the current is discontinuous across the shock, even as the magnetic field is continuous, and that the lowest order outer equations, which are the equations for travelling waves in inviscid Hall-MHD, are exactly integrable. We show that the inner and outer solutions match, which allows us to construct a family of uniformly valid continuous composite solutions that become discontinuous when the diffusivity vanishes.

Stability of planar travelling waves for bistable reaction–diffusion equations in multiple dimensions

This paper proves the multidimensional stability with an algebraic rate as time goes to infinity of planar travelling waves for bistable reaction–diffusion equations in for all , without assuming the hypothesis on the spectral gap of the one-dimensional linearized operator in as in previous works. The convergence rate in obtained here is better than the known rate in . Compared with the stability results where the convergence rate is only given for the special initial data which are the translations of the travelling wave, the convergence rate obtained in the present paper is for much more general initial perturbations.

Linear relaxation to planar travelling waves in inertial confinement fusion

In this paper we study the linear stability of planar travelling waves for a scalar reaction–diffusion equation with nonlinear anisotropic diffusion. The mathematical model is derived from the full thermo-hydrodynamical model describing the process of inertial confinement fusion. We show that solutions of the Cauchy problem with physically relevant initial data become planar exponentially fast with the rate s(ε′, k) > 0, where ε′ = (Tmin/Tmax) ≪ 1 is a small temperature ratio and k ≫ 1 is the transversal wrinkling wavenumber of perturbations. We rigorously recover in some particular limit (ε′, k) → (0, +∞) a dispersion relation s(ε′, k) ∼ γ0kα previously computed heuristically and numerically in some physical models of inertial confinement fusion.

Dynamics of a nonlocal discrete Gross-Pitaevskii equation with defects

We study the dynamics of dipolar gas in deep lattices described by a nonlocal nonlinear discrete Gross-Pitaevskii equation. The stabilities and the propagation properties of traveling plane waves in the system with defects are discussed in detail. For a clean lattice, both energetic and dynamical stabilities of the traveling plane waves are studied. It is shown that the system with attractive local interaction can preserve the stabilities, i.e., the dipoles can stabilize the gas because of repulsive nonlocal dipole-dipole interactions. For a lattice with defects, within a two-mode approximation, the propagation properties of traveling plane waves in the system map onto a nonrigid pendulum Hamiltonian with quasimomentum-dependent nonlinearity (induced by the nonlocal interactions). Competition between defects, quasimomentum of the gas, and nonlocal interactions determines the propagation properties of the traveling plane waves. Critical conditions for crossing from a superfluid regime with propagation preserved to a normal regime with defect-induced damping are obtained analytically and confirmed numerically. In particular, the critical conditions for supporting the superfluidity strongly depend on the defect type and the quasimomentum of the plane waves. The nonlocal interaction can significantly enhance the superfluidity of the system.

Simulation of Spatially Resolved Electron Energy Loss Near-Edge Structure for Scanning Transmission Electron Microscopy

Aberration-corrected scanning transmission electron microscopy yields probe-position-dependent energy-loss near-edge structure (ELNES) measurements, potentially providing spatial mapping of the underlying electronic states. ELNES calculations, however, typically describe excitations by a plane wave traveling in vacuum, neglecting the interaction of the electron probe with the local electronic environment as it propagates through the specimen. Here, we report a methodology that combines a full electronic-structure calculation with propagation of a focused beam in a thin film. The results demonstrate that only a detailed calculation using this approach can provide quantitative agreement with observed variations in probe-position-dependent ELNES.

Multidimensional stability of planar traveling waves for an integrodifference model

This paper studies the multidimensional stability of planar traveling waves for integrodifference equations. It is proved that for a Gaussian dispersal kernel, if the traveling wave is exponentially orbitally stable in one space dimension, then the corresponding planar wave is stable in $H^m(\mathbb{R}^N)$, $N\ge 4$, $m\ge [N/2]+1$, with the perturbation decaying at algebraic rate.

복제 방지용 PUF 모델링을 위한 전자계 해석

본 논문에서는 Debye 분산 특성을 가지는 복제 방지용 PUF를 설계하기 위한 전자계 해석 방안을 고려하였다. 공기층과 유전체 기판 위에 형성된 분산매질(Si)로 구성된 1차원 모델 내에 전파하는 펄스를 모형하기 위해 FDTD법을 이용하였다. 불연속 경계면에 도달한 펄스는 일부 반사되고 일부는 투과되어 빠르게 감쇠되는 것으로 나타났다. 그 결과 FDTD법에 의한 유전체 기판을 고려한 Debye 분산특성을 가지는 1차원 복제방지용 PUF 모델링에 적용 가능한 것을 확인하였다. Electromagnetic analysis to design unclonable PUFs with frequency-dependant materials with Debye dispersion was considered. To simulate FDTD calculations consider that 1-D problem of pulsed plane wave traveling in free space normally incident on air-silicon material interface on dielectric substrate. The pulse traveling wave at a vacuum-medium interface was reflected, and transmitted wave was dissipated. As a result, 1-D PUF modeling with Debye dispersion on dielectric substrate structure can be applied and FDTD calculation for PUF modeling is a good approximation.

Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity

In this paper, we study a class of nonlocal dispersionequation with monostable nonlinearity in $n$-dimensional space\[\begin{cases} u_t - J\ast u +u+d(u(t,x))=\displaystyle \int_{\mathbb{R}^n} f_\beta(y) b(u(t-\tau,x-y)) dy, \\u(s,x)=u_0(s,x), \ \ s\in[-\tau,0], \ x\in \mathbb{R}^n,\end{cases}\]where the nonlinear functions $d(u)$ and $b(u)$ possess the monostable characters like Fisher-KPP type, $f_\beta(x)$ is the heat kernel, and the kernel $J(x)$ satisfies ${\hat J}(\xi)=1-\mathcal{K}|\xi|^\alpha+o(|\xi|^\alpha)$ for $00$. After establishing the existence for both the planar traveling waves $\phi(x\cdot{\bf e}+ct)$ for $c\ge c_*$ ($c_*$ is the critical wave speed) and the solution $u(t,x)$ for the Cauchy problem, as well as the comparison principles, we prove that, all noncritical planar wavefronts $\phi(x\cdot{\bf e}+ct)$ are globally stable withthe exponential convergence rate $t^{-n/\alpha}e^{-\mu_\tau t}$ for $\mu_\tau>0$, and the criticalwavefronts $\phi(x\cdot{\bf e}+c_*t)$ are globally stable in the algebraic form $t^{-n/\alpha}$, and these rates are optimal. As application,we also automatically obtain the stability of traveling wavefronts to the classical Fisher-KPP dispersion equations. The adopted approach is Fourier transformand the weighted energymethod with a suitably selected weight function.

Planar traveling waves in capillary fluids

By capillary fluids we mean compressible, inviscid fluids whose energy depends not only on their density but also on their density gradient. Their motion is thus governed by systems of conservation laws, either in Eulerian coordinates or in Lagrangian coordinates, that are a higher-order modification of the usual equations of gas dynamics. In both settings, we receive models that also arise in other fields, in particular in water-waves theory and quantum hydrodynamics. Those Hamiltonian systems typically admit three types of planar traveling waves, namely, heteroclinic, homoclinic, and periodic ones. The purpose here is to review the main tools and results regarding the stability of those waves, under most general assumptions on the energy law. Special attention is devoted to the correspondence between traveling waves in Eulerian coordinates and those in Lagrangian coordinates.

Theoretical Investigation of Striped and Non-Striped Broad Area Lasers With Off-Axis Feedback

We report a method to improve the beam quality of broad area lasers by using a V-shaped external cavity formed by two off-axis feedback mirrors that allow to select a single transverse mode with transversally modulated intensity distribution. In the case when one of the two feedback mirrors is absent a spontaneous formation of self-induced transverse population grating leading to a reduction of the lasing threshold is observed. Most favorable conditions for stabilization of a single transverse supermode and creation of a high power and high brightness plane wave traveling in the extended cavity are obtained for equal reflectivities of the two external reflectors.

On the Steady Streaming Induced by Vibrating Walls

We consider incompressible viscous flows between two transversely vibrating solid walls and construct an asymptotic expansion of solutions of the Navier--Stokes equations in the limit when both the amplitude of the vibration and the thickness of the oscillatory boundary layers (the Stokes layers) are small and have the same order of magnitude. Our asymptotic expansion is valid up to the flow boundary. In particular, we derive equations and boundary conditions for the averaged flow. At leading order, the averaged flow is described by the stationary Navier--Stokes equations with an additional term which contains the leading-order Stokes drift velocity. The same equations had been derived by Craik and Leibovich in 1976 when they proposed their model of Langmuir circulations in the ocean [A. D. D. Craik and S. Leibovich, J. Fluid Mech., 73 (1976), pp. 401--426]. In the context of flows induced by an oscillating conservative body force, these equations had also been derived by Riley [Ann. Rev. Fluid Mech., 33 (2001), pp. 43--65]. The general theory is applied to two particular examples of steady streaming induced by transverse vibrations of the walls in the form of standing and traveling plane waves. In particular, in the case of waves traveling in the same direction, the induced flow is plane-parallel and the Lagrangian velocity profile can be computed analytically. This example may be viewed as an extension of the theory of peristaltic pumping to the case of high Reynolds numbers.

Effects of reduced discrete coupling on filament tension in excitable media

Wave propagation in the heart has a discrete nature, because it is mediated by discrete intercellular connections via gap junctions. Although effects of discreteness on wave propagation have been studied for planar traveling waves and vortexes (spiral waves) in two dimensions, its possible effects on vortexes (scroll waves) in three dimensions are not yet explored. In this article, we study the effect of discrete cell coupling on the filament dynamics in a generic model of an excitable medium. We find that reduced cell coupling decreases the line tension of scroll wave filaments and may induce negative filament tension instability in three-dimensional excitable lattices.

Impact of Depth of Penetration on Mutual Inductance and Electrical Resistance of Individual Toroidal Coils Using Analytical and Finite Element Methods Applicable to Tokamak Reactors

A toroidal field coil is composed of several individual toroidal coils (ITCs) which are connected in a series and distributed in a toroidal and symmetrical form. Cross section of ITCs is rectangular or negligible. This paper presents analytical equations of mutual inductance of two ITCs applicable to tokamak reactors using the filament method. These equations are based on those formulated by Neumann. The numerical analysis of the integrations resulting from these equations is solved using the extended three-point Gaussian algorithm. The finite element method (FEM) is employed to verify the mutual inductance equations of ITCs. The results obtained using FEM, when dimensional parameters of ITCs are changed, confirm the analytical and empirical results showing an error of less than 0.2043% in the worst case. This indicates the reliability of the presented equations. This paper also employs FEM to obtain depth of penetration of electromagnetic waves in high-conductivity conductors (HCCs). The results show that the ratio of real depth of penetration, obtained by FEM, to ideal depth of penetration in HCCs is independent of frequency. This ratio shows that the magnitude of traveling plane waves in HCC decreases with <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$e^{-1.7916}$</tex></formula> rather than <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX"> $e^{-1}$</tex></formula> .

Patterns of Sources and Sinks in the Complex Ginzburg–Landau Equation with Zero Linear Dispersion

The complex Ginzburg–Landau equation with zero linear dispersion occurs in a wide variety of contexts as the modulation equation near the supercritical onset of a homogeneous oscillation. The analysis of its coherent structures is therefore of great interest. Its fundamental spatiotemporal pattern is wavetrains, which are spatially periodic solutions moving with constant speed (also known as periodic travelling waves and plane waves). In the past decade interfaces separating regions with different wavetrains have been studied in detail, as they occur both in simulations and in real experiments. The basic interface types are sources and sinks, distinguished by the signs of the opposing group velocities of the adjacent wavetrains. In this paper we study existence conditions for propagating patterns composed of sources and sinks. Our analysis is based on a formal asymptotic expansion in the limit of large source-sink separation and small speed of propagation. The main results concern the possible relative locations of sources and sinks in such a pattern. We show that sources and sinks are to leading order coupled only to their nearest neighbors, and that the separations of a source from its neighboring sinks, $L_+$ and $L_-$ say, satisfy a phase locking condition, whose leading order form is derived explicitly. Significantly this leading order phase locking condition is independent of the propagation speed. The solutions of the condition form a discrete infinite sequence of curves in the $L_+$–$L_-$ plane.

Traveling plane wave solutions of delayed lattice differential systems in competitive Lotka-Volterra type

In this work we consider the existence of traveling plane wave solutions of systems of delayed lattice differential equations in competitive Lotka-Volterra type. Employing iterative method coupled with the explicit construction of upper and lower solutions in the theory of weak quasi-monotone dynamical systems, we obtain a speed, c *, and show the existence of traveling plane wave solutions connecting two different equilibria when the wave speeds are large than c *.

Small planar travelling waves in two-dimensional networks of coupled oscillators

The existence of several small planar travelling waves with arbitrary direction of propagation is shown for two-dimensional cubic networks of oscillators with linear nearest-neighbour coupling. The analysis is based on a spatial dynamics reformulation of the relevant advance–delay equation. Whereas important aspects of the analysis are discontinuous in the direction of travel, and continuity of the travelling wave is shown with respect to the direction.

Shock structure in continuum models of gas dynamics: stability and bifurcation analysis

The problem of shock structure in gas dynamics is analysed through a comparative study of two continuum models: the parabolic Navier–Stokes–Fourier model and the hyperbolic system of 13 moments equations modeling viscous, heat-conducting monatomic gases within the context of extended thermodynamics. When dissipative phenomena are neglected these models both reduce to classical Euler's equations of gas dynamics. The shock profile solution, assumed in the form of a planar travelling wave, reduces the problem to a system of ordinary differential equations, and equilibrium states appear to be stationary points of the system. It is shown that in both models an upstream equilibrium state suffers an exchange of stability when the shock speed crosses the critical value which coincides with the highest characteristic speed of the Euler's system. At the same time a downstream equilibrium state could be seen as a steady bifurcating solution, while the shock profile represents a heteroclinic orbit connecting the two stationary points. Using centre manifold reduction it is demonstrated that both models, although mathematically different, obey the same transcritical bifurcation pattern in the neighbourhood of the bifurcation point corresponding to the critical value of shock speed, the speed of sound.

Pointwise nonlinear scaling for reaction–diffusion equations

Parabolic reaction–diffusion systems may develop sharp moving reaction fronts which pose a challenge even for adaptive finite element methods. We propose a method to transform the equation into an equivalent form that usually exhibits solutions which are easier to discretize, giving higher accuracy for a given number of degrees of freedom. The transformation is realized as an efficiently computable pointwise nonlinear scaling that is optimized for prototypical planar travelling wave solutions of the underlying reaction–diffusion equation. The gain in either performance or accuracy is demonstrated on different numerical examples.

Modelling He I–He II phase transformation in long channels containing superconductors

The problem of heat transport during sub-cooling below λ point in long, narrow channels containing superconductors immersed in liquid helium is discussed. In order to describe the propagation of phase transformation front the travelling plane wave description with suitable change of variables has been adopted. Nonlinear temperature profile corresponding to the conductivity function in He II has been assumed. The heat diffusion equation is solved in the volume of He I with the assumption that the main contribution to the specific heat is due to helium and the main contribution to longitudinal conduction is due to superconductor. For one dimensional problem with the temperatures clamped at both extremities of the channel the closed form analytical solutions for temperature profile are obtained. Channels with discontinuity in diameter are also analysed. Experimental validation of the model is based on 54 m long superconducting lines applied in the Large Hadron Collider (LHC).

Robust feed-back control of travelling waves in a class of reaction–diffusion distributed biological systems

Reaction–Diffusion (RD) mechanisms can describe many biological phenomena such as neuron firing in the brain, the heartbeat, cellular organization activities or even biological disorders such as fibrillation. The FitzHugh–Nagumo (FHN) model is a particular case of RD systems. It is able to capture the key features of many biological processes and since it is relatively simple it has been widely employed during recent years. Some examples of its predictive capabilities include the representation of the normal behavior of some physiological phenomena, related to a travelling plane wave, as well as biological disorders associated with spiral or irregular fronts. The objective of this work is to design a control law that is able to stabilize complex behaviors (travelling plane wave) in biological systems using the FHN model as a case study. Since, in biological systems there usually exists a lack of detailed information on the system structure, our control law will be designed to be robust, i.e., it must be able to reach the predefined reference regardless the presence of structural uncertainties. To this purpose, we will extend some classical results on the finite-dimensional robust control theory to RD systems by means of order reduction techniques, in particular the Proper Orthogonal Decomposition method.