Klein-Gordon Type Equation Research Articles

Subwavelength anti-diffracting beams propagating over more than 1,000 Rayleigh lengths

Researchers take advantage of the nonlinear response of a dipolar glass and observe non-paraxial propagation of a sub-micrometre-sized beam over 103 diffraction lengths. Propagating light beams with widths down to and below the optical wavelength require bulky large-aperture lenses and remain focused only for micrometric distances1,2. Here, we report the observation of light beams that violate this localization/depth-of-focus law by shrinking as they propagate, allowing resolution to be maintained and increased over macroscopic propagation lengths. In nanodisordered ferroelectrics3,4 we observe a non-paraxial propagation of a sub-micrometre-sized beam for over 1,000 diffraction lengths, the narrowest visible beam reported to date5,6,7,8. This unprecedented effect is caused by the nonlinear response of a dipolar glass, which transforms the leading optical wave equation into a Klein–Gordon-type equation that describes a massive particle field9. Our findings open the way to high-resolution optics over large depths of focus, and a route to merging bulk optics into nanodevices.

Existence and stability of traveling waves for a class of nonlocal nonlinear equations

In this article we are concerned with the existence and orbital stability of traveling wave solutions of a general class of nonlocal wave equations: utt−Luxx=B(±|u|p−1u)xx, p>1. The main characteristic of this class of equations is the existence of two sources of dispersion, characterized by two coercive pseudo-differential operators L and B. Members of the class arise as mathematical models for the propagation of dispersive waves in a wide variety of situations. For instance, all Boussinesq-type equations and the so-called double-dispersion equation are members of the class. We first establish the existence of traveling wave solutions to the nonlocal wave equations considered. We then obtain results on the orbital stability or instability of traveling waves. For the case L=I, corresponding to a class of Klein–Gordon-type equations, we give an almost complete characterization of the values of the wave velocity for which the traveling waves are orbitally stable or unstable by blow-up.

Control of error in the homotopy analysis of nonlinear Klein–Gordon initial value problems

In the present paper, we discuss the application of homotopy analysis to general nonlinear Klein–Gordon type equations. We first outline the method for general forms of the nonlinearity, as well as for general functional forms of the initial conditions. In particular, we discuss a method of controlling the residual error in approximate solutions which may be found via homotopy analysis, through adequate selection of the convergence control parameter. With the general problem outlined, we apply the method to various equations, including the quasilinear cubic Klein–Gordon equation, the modified Liouville equation, the sinh-Gordon equation, and the tanh-Gordon equation. For each of these equations and related initial data, we obtain residual error minimizing solutions which demonstrate the qualitative behavior of the true solutions in each case.

Nonrelativistic quantum dynamics on a cone with and without a constraining potential

In this paper we investigate the bound state problem of nonrelativistic quantum particles on a conical surface. This kind of surface appears as a topological defect in ordinary semiconductors as well as in graphene sheets. Specifically, we compare and discuss the results stemming from two different approaches. In the first one, it is assumed that the charge carriers are bound to the surface by a constraining potential, while the second one is based on the Klein-Gordon type equation on surfaces, without the constraining potential. The main difference between both theories is the presence/absence of a potential which contains the mean curvature of a given surface. This fact changes the dependence of the bound states on the angular momentum l. Moreover, there are bound states that are absent in the Klein-Gordon theory, which instead appear in the Schrödinger one.

Type N spacetimes as solutions of extended new massive gravity

We study algebraic type N spacetimes in the extended new massive gravity (NMG), considering both the Born-Infeld model (BI-NMG) and the model of NMG with any finite order curvature corrections. We show that for these spacetimes, the field equations of BI-NMG take the form of the massive (tensorial) Klein–Gordon type equation, just as it happens for ordinary NMG. This fact enables us to obtain the type N solution to BI-NMG, utilizing the general type N solution of NMG, earlier found in our work. We also obtain type N solutions to NMG with all finite order curvature corrections and show that, in contrast to BI-NMG, this model admits the critical point solutions, which are counterparts of “logarithmic” AdS pp-waves solutions of NMG.

TypeDsolutions of 3D new massive gravity

In a recent reformulation of three-dimensional new massive gravity (NMG), the field equations of the theory consist of a massive (tensorial) Klein-Gordon type equation with a curvature-squared source term and a constraint equation. Using this framework, we present all algebraic type D solutions of NMG with constant and nonconstant scalar curvatures. For constant scalar curvature, they include homogeneous anisotropic solutions which encompass both solutions originating from topologically massive gravity (TMG), Bianchi types II, VIII, IX, and those of non-TMG origin, Bianchi types VI_{0} and VII_{0} . For a special relation between the cosmological and mass parameters, \lambda=m^2, they also include conformally flat solutions, and in particular those being locally isometric to the previously-known Kaluza-Klein type AdS_2xS^1 or dS_2x S^1 solutions. For nonconstant scalar curvature, all the solutions are conformally flat and exist only for \lambda=m^2 . We find two general metrics which possess at least one Killing vector and comprise all such solutions. We also discuss some properties of these solutions, delineating among them black hole type solutions.

Classification of integrable discrete Klein–Gordon models

The Lie algebraic integrability test is applied to the problem of classification of integrable Klein–Gordon-type equations on quad graphs. The list of equations passing the test is presented, containing several well-known integrable models. A new integrable example is found; its higher symmetry is presented.

Coordinate representation of particle dynamics in AdS and generic static spacetimes

We discuss the quantum dynamics of a particle in static curved spacetimes in a coordinate representation. The scheme is based on the analysis of the squared energy operator E2, which is quadratic in momenta and contains a scalar curvature term. Our main emphasis is on AdS spaces, where this term is fixed by the isometry group. As a byproduct the isometry generators are constructed and the energy spectrum is reproduced. In the massless case the conformal symmetry is realized as well. We show the equivalence between this quantization and the covariant quantization, based on the Klein–Gordon-type equation in AdS. We further demonstrate that the two quantization methods in an arbitrary (N + 1)-dimensional static spacetime are equivalent to each other if the scalar curvature terms both in the operator E2 and in the Klein–Gordon-type equation have the same coefficient equal to .

Exact Solutions in 3D New Massive Gravity

We show that the field equations of new massive gravity (NMG) consist of a massive (tensorial) Klein-Gordon-type equation with a curvature-squared source term and a constraint equation. We also show that, for algebraic type D and N spacetimes, the field equations of topologically massive gravity (TMG) can be thought of as the "square root" of the massive Klein-Gordon-type equation. Using this fact, we establish a simple framework for mapping all types D and N solutions of TMG into NMG. Finally, we present new examples of types D and N solutions to NMG.

On precise center stable manifold theorems for certain reaction–diffusion and Klein–Gordon equations

We consider positive, radial and exponentially decaying steady state solutions of the general reaction–diffusion and Klein–Gordon type equations and present an explicit construction of infinite-dimensional invariant manifolds in the vicinity of these solutions. The result is a precise stable manifold theorem for the reaction–diffusion equation and a precise center-stable manifold theorem for the Klein–Gordon equation, which include the co-dimension of the manifolds and the decay rates for even perturbations.

Uniqueness of the Fock quantization of the GowdyT3model

After its reduction by a gauge-fixing procedure, the family of linearly polarized Gowdy ${T}^{3}$ cosmologies admits a scalar field description whose evolution is governed by a Klein-Gordon type equation in a flat background in $1+1$ dimensions with the spatial topology of ${S}^{1}$, though in the presence of a time-dependent potential. The model is still subject to a homogeneous constraint, which generates ${S}^{1}$-translations. Recently, a Fock quantization of this scalar field was introduced and shown to be unique under the requirements of unitarity of the dynamics and invariance under the gauge group of ${S}^{1}$-translations. In this work, we extend and complete this uniqueness result by considering other possible scalar field descriptions, resulting from reasonable field reparametrizations of the induced metric of the reduced model. In the reduced phase space, these alternate descriptions can be obtained by means of a time-dependent scaling of the field, the inverse scaling of its canonical momentum, and the possible addition of a time-dependent, linear contribution of the field to this momentum. Demanding again unitarity of the field dynamics and invariance under the gauge group, we prove that the alternate canonical pairs of fieldlike variables admit a Fock representation if and only if the scaling of the field is constant in time. In this case, there exists essentially a unique Fock representation, provided by the quantization constructed by Corichi, Cortez, and Mena Marug\'an. In particular, our analysis shows that the scalar field description proposed by Pierri does not admit a Fock quantization with the above unitarity and invariance properties.

Quantum GowdyT3model: A unitary description

The quantization of the family of linearly polarized Gowdy $T^3$ spacetimes is discussed in detail, starting with a canonical analysis in which the true degrees of freedom are described by a scalar field that satisfies a Klein-Gordon type equation in a fiducial time dependent background. A time dependent canonical transformation, which amounts to a change of the basic (scalar) field of the model, brings the system to a description in terms of a Klein-Gordon equation on a background that is now static, although subject to a time dependent potential. The system is quantized by means of a natural choice of annihilation and creation operators. The quantum time evolution is considered and shown to be unitary, allowing both the Schr\"odinger and Heisenberg pictures to be consistently constructed. This has to be contrasted with previous treatments for which time evolution failed to be implementable as a unitary transformation. Possible implications for both canonical quantum gravity and quantum field theory in curved spacetime are commented.

On a system of Klein–Gordon type equations with acoustic boundary conditions

We prove the existence, uniqueness and uniform stabilization of global solutions for a generalized system of Klein–Gordon type equations with acoustic boundary conditions on a portion of the boundary and the Dirichlet boundary condition on the rest.

Solutions of compact and noncompact structures for nonlinear Klein–Gordon-type equation

The focusing and the defocusing branches of nonlinear Klein–Gordon-type equation KG( n, n) are considered. A framework is implemented to show that the first model exhibits compactons: solitons that do not have exponential tails, whereas the second model demonstrates solitary patterns solutions. The two variants of KG( n, n) equation will be examined in one and higher dimensions. General formulas will be developed to present a fairly complete understanding of the solutions of compact and noncompact structures.

Global small solutions to Klein–Gordon-type equations with non-local non-linearities

Global small solutions to Klein–Gordon-type equations with non-local non-linearities

Axial Shear Instability in a “Tachion” Region

Plasma axial-shear flow instability arises due to a variation in an equilibrium E × B rotation along the axial direction in which the magnetic field is aligned. The two fluid MHD equations for incompressible perturbation (taking into account the FLR effects) being treated in WKB approximation in transversal direction yield one scalar Klein-Gordon type equation with one-dimensional effective potential U(s) and effective mass m(s). Only axisymmetric, paraxial geometry is analyzed in order to separate the desired effects from the effects related to a variation in cross-sectional shape of the magnetic flux tube. In this work the effective potential was considered for a semi-infinite bounded plasma, first in the form of a square well for analytical study and then in a linear nature to study in the so called “tachion” region. Growth rates as a function of the potential well depth and other parameters were calculated. The cases where effective mass is real and imaginary “tachion” regime were considered. The results obtained are interesting for the stability problem of such open devices as GDT, GAMMA-10, AMBAL-M and the scrape-off layer in tokamak divertors.

THE TWO-DIMENSIONAL STRINGY BLACK HOLE: A NEW APPROACH AND A NEW EFFECT

The string propagation in the two-dimensional stringy black hole is investigated from a new approach. We completely solve the classical and quantum string dynamics in the Lorentzian and Euclidean regimes. In the Lorentzian case all the physics reduces to a massless scalar particle described by a Klein-Gordon type equation with a singular effective potential. The scattering matrix is found and it reproduces the results obtained by coset CFT techniques. It factorizes into two pieces: an elastic Coulombian amplitude and an absorption part. In both parts, an infinite sequence of imaginary poles in the energy appears. The generic features of string propagation in curved D-dimensional backgrounds (string stretching, fall into space-time singularities) are analyzed in the present case. A new physical phenomenon specific to the present black hole is found: the quantum renormalization of the speed of light. We find that [Formula: see text] where k is the integer in front of the WZW action. Only for k→∞ does this new effect disappear (although the conformal anomaly is present). We analyze all the classical Euclidean string solutions and exactly compute the quantum partition function. No critical Hagedorn temperature appears here.

Finite dimensional exponential attractors for semilinear wave equations with damping

We consider the initial value problem for a class of second order evolution equations that includes, among others, the 3D sine-Gordon equation with damping and the 3D Klein-Gordon type equations with damping. We show the existence of a set with finite fractal dimension that contains the global attractor and attracts all smooth solutions at an exponential rate.

The use of quantum mechanics in the treatment of waveguides

In this paper we give a quantum description of guided waves. A Klein-Gordon type equation is derived for wave propagation in an ideal uniform waveguide, and the appropriate quantum-mechanical interpretation is given. Making use of such methods, we give an analysis of guided waves, of inhomogeneously filled waveguides, of waveguide discontinuities, and generally of passive microwave circuits. A spinorial formalism is employed to derive the equivalent Dirac-type equation. The general applicability of the quantum-mechanical concepts to waveguides is also discussed.

Spectral and scattering theory for the J-selfadjoint operators associated with the perturbed Klein-Gordon type equations

Spectral and scattering theory for the J-selfadjoint operators associated with the perturbed Klein-Gordon type equations