Klein-Gordon Type Research Articles

Time domain normal mode analysis of underwater acoustic wave propagation for a single layered acoustic channel in two dimensional Cartesian coordinates

A new analytical time domain Normal Mode solution in a single layer acoustic waveguide in Cartesian coordinates is presented in the thesis. The method is based on the separation of variables technique for the two dimensional time domain wave equation. After separation, Sturm-Liouville type ordinary differential equation for depth dependency and inhomogeneous Klein-Gordon type partial differential equation for time and range dependency are obtained. Whenever the ordinary differential equation is solved in a classical manner with the application of Dirichlet boundary condition on the upper and lower boundaries of the waveguide, Inhomogeneous Klein-Gordon type partial differential equation is solved by using the Green function technique in time domain. Then the exact solution satisfying the causality principle is constructed for the acoustic pressure. The main advantage of the proposed causal solution is not necessary to use the Fourier Transformation to obtain time domain response of arbitrary acoustical source signals having the monochromatic and Gaussian pulse type time dependency in the waveguide. The excellent agreement for the comparisons of the range dependent transmission losses is observed between the proposed solution and the KRAKEN program.

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.

NONEXPANSIVE RETRACTIONS ONTO CLOSED CONVEX CONES IN BANACH SPACES

We investigate Lorentzian minimal surfaces in Lorentzian complex space forms. First, we prove that for such surfaces the equation of Ricci is a consequence of the equations of Gauss and Codazzi. Next, we classify Lorentzian minimal surfaces in the Lorentzian complex plane ${\bf C}^2_1$. Finally, we classify minimal slant surfaces in the Lorentzian complex projective plane $CP^2_1(4)$ and in the Lorentzian complex hyperbolic plane $CH^2_1(-4)$. In particular, our latter results show that if a minimal slant surface in $CP^2_1(4)$ or in $CH^2_1(-4)$ contains no open subset of constant curvature, then it is of Klein-Gordon type which arises from the solutions of certain nonlinear Klein-Gordon equations.

NONEXPANSIVE RETRACTIONS ONTO CLOSED CONVEX CONES IN BANACH SPACES

We investigate Lorentzian minimal surfaces in Lorentzian complex space forms. First, we prove that for such surfaces the equation of Ricci is a consequence of the equations of Gauss and Codazzi. Next, we classify Lorentzian minimal surfaces in the Lorentzian complex plane ${\bf C}^2_1$. Finally, we classify minimal slant surfaces in the Lorentzian complex projective plane $CP^2_1(4)$ and in the Lorentzian complex hyperbolic plane $CH^2_1(-4)$. In particular, our latter results show that if a minimal slant surface in $CP^2_1(4)$ or in $CH^2_1(-4)$ contains no open subset of constant curvature, then it is of Klein-Gordon type which arises from the solutions of certain nonlinear Klein-Gordon equations.

The invariant factor of the chiral determinant

The coupling of spin 0 and spin 1 external fields to Dirac fermions defines a theory which displays gauge chiral symmetry. Quantum-mechanically, functional integration of the fermions yields the determinant of the Dirac operator, known as the chiral determinant. Its modulus is chiral invariant but not so its phase, which carries the chiral anomaly through the Wess–Zumino–Witten term. Here we find the remarkable result that, upon removal from the chiral determinant of this known anomalous part, the remaining chiral-invariant factor is just the square root of the determinant of a local covariant operator of the Klein–Gordon type. This procedure bypasses the integrability obstruction allowing one to write down a functional that correctly reproduces both the modulus and the phase of the chiral determinant. The technique is illustrated by computing the effective action in two dimensions at leading order (LO) in the derivative expansion. The results previously obtained by indirect methods are indeed reproduced.

Global attractors for nonlinear wave equations with some nonlinear dissipative terms in exterior domains

We show the existence, size and some absorbing properties of global attractor of a Klein-Gordon type nonlinear wave equation with some nonlinear dissipation in an exterior domain.

A nonlinear elliptic equation with singular potential and applications to nonlinear field equations

We study existence and asymptotic properties of solutions to a semilinear elliptic equation in the whole space. The equation has a cylindrical symmetry and we find cylindrical solutions. The main features of the problem are that the potential has a large set of singularities (i.e. a subspace), and that the nonlinearity has a double power-like behaviour, subcritical at infinity and supercritical near the origin. We also show that our results imply the existence of solitary waves with nonvanishing angular momentum for nonlinear evolution equations of Schrodinger and Klein-Gordon type.

Equations for relativistic particles, binaries, and ternaries

Relativistic differential equations for bound states of n Dirac particles (n=1,2,3) are put into forms that depend only on the square of the total center-of-mass system energy. Instead of coupling 4{sup n} Dirac components, they decompose into two equivalent equations for 4{sup n}/2 components {psi} and {chi} of total chiralities +1 and -1, respectively. Their time-dependent versions are of the Klein-Gordon type; they reproduce the relativistic kinematics for the emission of photons or pions. Although the equations are presently extended to mesons and baryons within perturbative QCD only, the necessity of free Klein-Gordon equations for closed systems implies E{sup 2} spectra not only for mesons, but also for baryons. For binaries, there exists an intricate transformation which turns the equation for {psi} into an effective one-body Dirac equation. The corresponding transformation for {chi} is derived here.

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.

Solvable Nonlinear Evolution PDEs in Multidimensional Space

A class of solvable (systems of) nonlinear evolution PDEs in multidimensional space is discussed. We focus on a rotation-invariant system of PDEs of Schrodinger type and on a relativistically-invariant system of PDEs of Klein-Gordon type. Isochronous variants of these evolution PDEs are also considered.

POSITIVE SOLUTIONS TO A NON-RADIAL SUPERCRITICAL KLEIN–GORDON-TYPE EQUATION

AbstractWe find positive solutions to a nonlinear equation of Klein–Gordon type. Our analysis is carried out by truncating the related functional and estimating mountain pass solutions by Moser’s iterative scheme.

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.

Exact traveling breather solutions in a discrete Klein-Gordon ring

Following an earlier work relating to traveling kinks and pulses in a discrete reaction-diffusion system, we construct exact traveling breather solutions on a closed Klein-Gordon type lattice with a double-well potential at each site. The dynamics consists of a succession of linear regimes and the problem reduces to an appropriate matching for successive regimes. The analysis shows that the so-called marginal modes are not essential for the existence of traveling breathers. Results are shown for a ring with N = 3 sites for the sake of illustration of the principles involved, while a number of basic results are also presented for the open lattice N →, see below.

ON A SYSTEM OF DIRAC-KLEIN-GORDON TYPE IN $1+1$ DIMENSIONS

We establish a local and a global existence results for a Dirac- Klein-Gordon type system in 1+1 dimensions with a pseudoscalar bilinear form.

Discrete Klein–Gordon models with static kinks free of the Peierls–Nabarro potential

For the nonlinear Klein-Gordon type models, we describe a general method of discretization in which the static kink can be placed anywhere with respect to the lattice. These discrete models are therefore free of the {\it static} Peierls-Nabarro potential. Previously reported models of this type are shown to belong to a wider class of models derived by means of the proposed method. A relevant physical consequence of our findings is the existence of a wide class of discrete Klein-Gordon models where slow kinks {\it practically} do not experience the action of the Peierls-Nabarro potential. Such kinks are not trapped by the lattice and they can be accelerated by even weak external fields.

Discrete peakons

We demonstrate the possibility for explicit construction in a discrete Hamiltonian model of an exact solution of the form exp ⁡ ( − | n | ) , i.e., a discrete peakon. These discrete analogs of the well-known, continuum peakons of the Camassa–Holm equation [R. Camassa, D.D. Holm, Phys. Rev. Lett. 71 (1993) 1661] are found in a model different from the one for their continuum siblings and from that of earlier studies in the discrete setting [A.A. Ovchinnikov, S. Flach, Phys. Rev. Lett. 83 (1999) 248]. Namely, we observe discrete peakons in Klein–Gordon-type and nonlinear Schrödinger-type chains with long-range interactions. The interesting linear stability differences between these two chains are examined numerically and illustrated analytically. Additionally, inter-site centered peakons are also obtained in explicit form and their stability is studied. We also prove the global well-posedness for the discrete Klein–Gordon equation, show the instability of the peakon solution, and the possibility of a formation of a breathing peakon.

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.

Bifurcations of steady states in a class of lattices of nonlinear discrete Klein–Gordon type with double-quadratic on-site potential

By using the method of symbolic dynamics, we study the bifurcations of steady states in a class of lattices of nonlinear discrete Klein–Gordon type with double-quadratic on-site potential. We derive by virtue of the admissible condition the critical value ε0 of the coupling strength, below which the steady states persist without bifurcations. If the coupling coefficient ε passes through the critical value, some of the steady states disappear. Meanwhile there are no new steady states created as ε varies. We obtain bifurcation values of some lower-order spatially periodic steady states by introducing the concept 'characteristic polynomial' of periodic sequences.

The Stochastic Nonlinear Damped Wave Equation

Abstract. We prove the existence of an invariant measure for the transition semigroup associated with a nonlinear damped stochastic wave equation in R n of the Klein—Gordon type. The uniqueness of the invariant measure and the structure of the corresponding Kolmogorov operator are also studied.

Hilbert space structures on the solution space of Klein–Gordon-type evolution equations

We use the theory of pseudo-Hermitian operators to address the problem of the construction and classification of positive-definite invariant inner-products on the space of solutions of a Klein–Gordon-type evolution equation. This involves dealing with the peculiarities of formulating a unitary quantum dynamics in a Hilbert space with a time-dependent inner product. We apply our general results to obtain possible Hilbert space structures on the solution space of the equation of motion for a classical simple harmonic oscillator, a free Klein–Gordon equation and the Wheeler–DeWitt equation for the FRW-massive-real-scalar-field models.