Exact Plane Wave Solutions Research Articles

Damping in a row of locally-resonant inclusions: Dynamic homogenization and scattering of transient shear waves

The scattering of scalar waves by a periodic row of inclusions is theoretically and numerically investigated. The wavelength in the background medium is assumed to be much larger than the typical sizes of the inclusions. The latter are also much softer than the matrix, yielding localized resonances within the microstructure. Previous works in the inviscid case have concerned: (i) the derivation of effective resonant jump conditions, that are non local in time (Touboul et al. (2020) [41]); (ii) the introduction of auxiliary fields along the interface, providing a time-domain formulation of the scattering problem (Touboul et al. (2020) [40]). The present contribution extends the analysis to dissipative cases, which allows to be closer from real devices. The effective jump conditions with damping are obtained, both in the frequency domain and in the time domain. An exact plane-wave solution is proposed. A balance of energy is written, and new auxiliary fields are introduced. Practical implementation of the simulation methods is discussed. Then, numerical experiments are proposed to validate the auxiliary-field approach. The effect of dissipation is examined, and the relevance of the homogenized simulations in comparison with full-field simulations of transient waves is assessed. As an application, a numerical experiment of Coherent Perfect Absorption is finally proposed: at critical values of the attenuation parameter and close to the resonant frequencies, the waves impacting the dissipative resonant interface are fully absorbed.

Exact Plane-Wave Solution of the Equations of a Theory of Gravitation with a Massive Graviton

Exact Plane-Wave Solution of the Equations of a Theory of Gravitation with a Massive Graviton

Dispersion properties, nonlinear waves and birefringence in classical nonlinear electrodynamics

Using the very basic physics principles, we have studied the implications of quantum corrections to classical electrodynamics and the propagation of electromagnetic waves and pulses. The initial nonlinear wave equation for the electromagnetic vector potential is solved perturbatively about the known exact plane wave solution in both the case of a polarized vacuum without external field, as well as when a constant magnetic field is applied. A nonlinear wave equation with nonzero convective part for the (relatively) slowly varying amplitude of the first-order perturbation has been derived. This equation governs the propagation of electromagnetic waves with a reduced speed of light, where the reduction is roughly proportional to the intensity of the initial pumping plane wave. A system of coupled nonlinear wave equations for the two slowly varying amplitudes of the first-order perturbation, which describe the two polarization states, has been obtained for the case of constant magnetic field background. Further, the slowly varying wave amplitude behavior is shown to be similar to that of a cnoidal wave, known to describe surface gravity waves in shallow water. It has been demonstrated that the two wave modes describing the two polarization states are independent, and they propagate at different wave frequencies. This effect is usually called nonlinear birefringence.

Coupled complex Ginzburg–Landau systems with saturable nonlinearity and asymmetric cross-phase modulation

We formulate and study dynamics from a complex Ginzburg–Landau system with saturable nonlinearity, including asymmetric cross-phase modulation (XPM) parameters. Such equations can model phenomena described by complex Ginzburg–Landau systems under the added assumption of saturable media. When the saturation parameter is set to zero, we recover a general complex cubic Ginzburg–Landau system with XPM. We first derive conditions for the existence of bounded dynamics, approximating the absorbing set for solutions. We use this to then determine conditions for amplitude death of a single wavefunction. We also construct exact plane wave solutions, and determine conditions for their modulational instability. In a degenerate limit where dispersion and nonlinearity balance, we reduce our system to a saturable nonlinear Schrödinger system with XPM parameters, and we demonstrate the existence and behavior of spatially heterogeneous stationary solutions in this limit. Using numerical simulations we verify the aforementioned analytical results, while also demonstrating other interesting emergent features of the dynamics, such as spatiotemporal chaos in the presence of modulational instability. In other regimes, coherent patterns including uniform states or banded structures arise, corresponding to certain stable stationary states. For sufficiently large yet equal XPM parameters, we observe a segregation of wavefunctions into different regions of the spatial domain, while when XPM parameters are large and take different values, one wavefunction may decay to zero in finite time over the spatial domain (in agreement with the amplitude death predicted analytically). We also find a collection of transient features, including transient defects and what appear to be rogue waves, while in two spatial dimensions we observe highly localized pattern formation. While saturation will often regularize the dynamics, such transient dynamics can still be observed – and in some cases even prolonged – as the saturability of the media is increased, as the saturation may act to slow the timescale.

The memory effect for plane gravitational waves

We give an account of the gravitational memory effect in the presence of the exact plane wave solution of Einstein's vacuum equations. This allows an elementary but exact description of the soft gravitons and how their presence may be detected by observing the motion of freely falling particles. The theorem of Bondi and Pirani on caustics (for which we present a new proof) implies that the asymptotic relative velocity is constant but not zero, in contradiction with the permanent displacement claimed by Zel'dovich and Polnarev. A non-vanishing asymptotic relative velocity might be used to detect gravitational waves through the “velocity memory effect”, considered by Braginsky, Thorne, Grishchuk, and Polnarev.

Numerical computations for N-coupled nonlinear Schrödinger equations by split step spectral methods

In this paper, various split step spectral (SSSP) schemes are proposed for N-coupled nonlinear Schrödinger (N-CNLS) equations, especially for the systems with N⩾3 for which numerical studies are few. These schemes are spectrally accurate in space, and sth (s=1,2,4,6,8) order in time. They are proved to be conservative and to admit the exact plane wave solution. Extensive numerical experiments are carried out for the 3- and 4-CNLS systems to confirm the theoretical analysis. Half of the schemes are shown to possess wonderful ability of capturing high-frequency waves. However, the eighth-order schemes which seems to be optimal fail in this test. Accuracy and efficiency of the schemes are compared with each other, and the high-order schemes exhibit better. Since the eighth-order schemes are not better than the sixth-order ones, it is believed that constructing too much high-order schemes are unnecessary. Moreover, interactions of two-soliton solutions are also well simulated by the fourth-order and the sixth-order schemes, so the two methods are sufficient for use.

Plane waves in the generalized Galileon theory

We present an exact plane wave solution of the most general shift-symmetric Horndeski (generalized Galileon) theory. The solution consists of the scalar part, and the gravitational part with two polarization modes. The former is due to the presence of the non-trivial Galileon scalar field, and it is parametrized by an arbitrary function of the light-cone coordinate. For a trivial scalar field configuration the solution is equivalent to the plane gravitational wave in General Relativity. When the metric is Minkowski, we reproduce known results for the plane waves of k-essence and a soliton-like solutions of a non-covariant Galileon model in a flat space-time.

CONFORMAL GRAVITY AND GRAVITATIONAL WAVES

We consider monochromatic, plane gravitational waves in a conformally invariant theory of general relativity. We show that the simple, standard ansatz for the metric, usually that which is taken for the linearized theory of these waves, is reducible to the metric of Minkowski spacetime via a sequence of conformal and coordinate transformations. This implies that we have in fact, exact plane wave solutions. However they are simply coordinate/conformal artifacts. As a consequence, they carry no energy.

Monochromatic plane-fronted waves in conformal gravity are pure gauge

We consider plane-fronted, monochromatic gravitational waves on a Minkowski background, in a conformally invariant theory of general relativity. By this we mean waves of the form: $g_{\mu\nu}=\eta_{\mu\nu}+\epsilon_{\mu\nu}F(k\cdotx)$, where $\epsilon_{\mu\nu}$ is a constant polarization tensor, and $k_\mu$ is a lightlike vector. We also assume the coordinate gauge condition $|g|-1/4\partial_\tau(|g|1/4g^{\sigma\tau})=0$ which is the conformal analog of the harmonic gauge condition $g^{\mu\nu}\Gamma_{\mu\nu}^\sigma=-|g|-1/2\partial_\tau(|g|1/2g^{\sigma\tau})=0, where $\det[g_{\mu\nu}]\equivg$. Requiring additionally the conformal gauge condition $g=-1$ surprisingly implies that the waves are both transverse and traceless. Although the ansatz for the metric is eminently reasonable when considering perturbative gravitational waves, we show that the metric is reducible to the metric of Minkowski space-time via a sequence of coordinate transformations which respect the gauge conditions, without any perturbative approximation that \in{\mu}{\nu} be small. This implies that we have, in fact, exact plane-wave solutions; however, they are simply coordinate/conformal artifacts. As a consequence, they carry no energy. Our result does not imply that conformal gravity does not have gravitational wave phenomena. A different, more generalized ansatz for the deviation, taking into account the fourth-order nature of the field equation, which has the form $g_{\mu\nu}=\eta_{\mu\nu}+B_{\mu\nu}(n\cdotx)G(k\cdotx)$, indeed yields waves which carry energy and momentum [P.D. Mannheim, Gen. Relativ. Gravit. 43, 703 (2010)]. It is just surprising that transverse, traceless, plane-fronted gravitational waves, those that would be used in any standard, perturbative, quantum analysis of the theory, simply do not exist.

Comparison between lattice Boltzmann method and Navier–Stokes high order schemes for computational aeroacoustics

Computational aeroacoustic (CAA) simulation requires accurate schemes to capture the dynamics of acoustic fluctuations, which are weak compared with aerodynamic ones. In this paper, two kinds of schemes are studied and compared: the classical approach based on high order schemes for Navier–Stokes-like equations and the lattice Boltzmann method. The reference macroscopic equations are the 3D isothermal and compressible Navier–Stokes equations. A Von Neumann analysis of these linearized equations is carried out to obtain exact plane wave solutions. Three physical modes are recovered and the corresponding theoretical dispersion relations are obtained. Then the same analysis is made on the space and time discretization of the Navier–Stokes equations with the classical high order schemes to quantify the influence of both space and time discretization on the exact solutions. Different orders of discretization are considered, with and without a uniform mean flow. Three different lattice Boltzmann models are then presented and studied with the Von Neumann analysis. The theoretical dispersion relations of these models are obtained and the error terms of the model are identified and studied. It is shown that the dispersion error in the lattice Boltzmann models is only due to the space and time discretization and that the continuous discrete velocity Boltzmann equation yield the same exact dispersion as the Navier–Stokes equations. Finally, dispersion and dissipation errors of the different kind of schemes are quantitatively compared. It is found that the lattice Boltzmann method is less dissipative than high order schemes and less dispersive than a second order scheme in space with a 3-step Runge–Kutta scheme in time. The number of floating point operations at a given error level associated with these two kinds of schemes are then compared.

Mode decomposition scheme for ideal magnetohydrodynamic plane waves in space‐time coordinates

Exact nonsinusoidal plane wave solutions of the linearized equations of ideal magnetohydrodynamics are used to develop a decomposition scheme for extracting the forward and backward propagating components of the fast, slow, and Alfvén modes from measured data. The decomposition technique is formulated in the space‐time domain for waves propagating in one direction. The different wave modes are extracted by means of projection operators that are expressed in matrix form. Because the elements of these matrices are constants, the same projection operators can be used to obtain the mode decomposition in the frequency (Fourier) domain. The projection operators are identical to those obtained by Glassmeier et al. (1995) although they are derived here by different means. In the case of wave propagation parallel or perpendicular to the background magnetic field the wave modes are degenerate and require separate treatment. These special cases are included in the present analysis so the resulting decomposition scheme encompasses all possible propagation directions.

Gravitational Field of Ultrarelativistic Objects with Angular Momentum

A brief review of recently found gyraton metrics which describe the gravitational field of objects having an angular momentum and moving with the velocity of light is given. The gyraton metrics belong to a class of exact plane wave solutions of four and higher dimensional Einstein equations in vacuum or in the presence of a negative cosmological constant.

Explicit and exact travelling plane wave solutions of the (2+1)-dimensional Boussinesq equation

The deformation mapping method is applied to solve a system of (2+1)-dimensional Boussinesq equations. Many types of explicit and exact travelling plane wave solutions, which contain solitary wave solutions, periodic wave solutions, Jacobian elliptic function solutions and others exact solutions, are obtained by a simple algebraic transformation relation between the (2+1)-dimensional Boussinesq equation and the cubic nonlinear Klein-Gordon equation.

NONLINEAR PLANE WAVES IN A LONG-WAVELENGTH CONVECTION MODEL

The partial differential equation (PDE) associated with the long-wavelength regime that describes convection between rigid and poorly conducting boundaries in an infinite Prandtl number fluid is studied in this document. This PDE is reduced to a set of ordinary differential equations (ODE) by looking for the exact nonlinear plane wave solutions. Solving this ODE several kinds of oscillatory attractors are obtained. These attractors are compared to oscillations observed in experiments on fluids and some features are also recovered.

Atomic interferometry as two-level particle scattering by a periodic potential

The beam splitting process of an atomic beam is described by an exactly solvable theo- ry. We use exact plane wave solutions of an atomic two-level system interacting with laser beams and employ appropriate jump conditions between regions with and without laser beam. The trans- mission matrix as well as the reflection matrix are given explicitely. Also, the conditions for atom optical analogues of the Borrmann effect and of the pendellosung effect are treated exactly. The high-energy limit of these phenomena is discussed which is sufficient for many practical purposes.

Atomic interferometry as two‐level particle scattering by a periodic potential

AbstractThe beam splitting process of an atomic beam is described by an exactly solvable theory. We use exact plane wave solutions of an atomic two‐level system interacting with laser beams and employ appropriate jump conditions between regions with and without laser beam. The transmission matrix as well as the reflection matrix are given explicitely. Also, the conditions for atom optical analogues of the Borrmann effect and of the pendellösung effect are treated exactly. The high‐energy limit of these phenomena is discussed which is sufficient for many practical purposes.

KALUZA–KLEIN–GORDON FIELD THEORY

We extend the induced-matter interpretation of Kaluza–Klein field theory to obtain the Klein–Gordon equation for a scalar particle. The motion of the particle is correctly recovered, and we give an exact plane-wave solution whose induced energy-momentum tensor depends only on first derivatives of the scalar field, as in flat-space quantum field theory. However, in general the energy and momenta of the particle also include terms in second derivatives, which should repay investigation.

Plane-wave solution of systems of einstein-weyl equations

This article presents an exact plane-wave solution for systems of Einstein-Dirac equations; it is of type N in the Petrov classification, and contains four arbitrary functions of the wave variable.

A variety of stable persistent waves in intrinsically bistable reaction-diffusion systems. From one-dimensional periodic waves to one-armed and two-armed rotating spiral waves

Abstract Various types of non-trivial stable persistent waves in intrinsically bistable reaction-diffusion equations are studied with an emphasis on initiation problems of these waves. The reaction-diffusion equation of Rinzel and Keller's type allows us to obtain exact plane wave solutions and their dispersion relation curves with three branches in most cases. Coexistence of six plane wave solutions is also discussed. On the basis of the knowledge of the stable plane wave solutions, the initiation problem for multi-armed rotating spiral waves is discussed. Numerical simulations show that a one-armed spiral wave maintains its wave pattern, and that a two-armed spiral wave survives for a long time before it eventually dies out.

Four-dimensional plane wave string solutions with coset CFT description

We present a number of D = 4 bosonic and heterotic string solutions with a covariantly constant null Killing vector which, like the solution of Nappi and Witten (NW), correspond to (gauged) WZW models and thus have a direct conformal field theory interpretation. A class of exact plane wave solutions (which includes the NW solution) is constructed by ‘boosting’ the twisted products of two D = 2 ‘cosmological’ or ‘black-hole’ cosets related to (G ⊗ G′)/(H ⊗ H′) (G, G′ = SL(2, R) or SU(2); H, H ′ = SO(1, 1) or SO(2)) gauged WZW models. We describe a general limiting procedure by which one can construct new solutions with a covariantly constant null Killing vector starting with known string backgrounds. By applying a non-abelian duality transformation to the NW model we find a D = 4 solution which admits a covariantly constant null Killing vector but is not a plane wave. Higher dimensional bosonic backgrounds with isometries can be interpreted as lower dimensional ones with extra gauge fields. Some of them are at the same time solutions of the heterotic string theory. In particular, the NW model represents also a D = 3 gravi-electromagnetic heterotic string plane wave. In addition to the (1, 1) supersymmetric embeddings of bosonic solutions we construct a number of non-trivial (1, 0) supersymmetric exact D = 4 heterotic string plane wave solutions some of which are related (by a boost and analytic continuation) to limiting cases of D = 4 heterotic black hole solutions.