Klein-Gordon Type Research Articles

Existence of subsonic periodic traveling waves in discrete Klein-Gordon type equations with nonlocal interaction

Existence of subsonic periodic traveling waves in discrete Klein-Gordon type equations with nonlocal interaction

Існування надзвукових періодичних біжучих хвиль в дискретних рівняннях типу Клейна-Ґордона з нелокальною взаємодією

The article is devoted to discrete Klein-Gordon type equations that describe infinite chains of nonlinear oscillators with nonlocal interactions. This implies that each oscillator interacts with several of its neighbors on both sides. The main result of the article concerns the existence of periodic traveling waves in such equations. Sufficient conditions for the existence of such waves were established using the variational method and the mountain pass theorem.

On the small noise limit in the Smoluchowski-Kramers approximation of nonlinear wave equations with variable friction

We study the validity of a large deviation principle for a class of stochastic nonlinear damped wave equations, including equations of Klein-Gordon type, in the joint small mass and small noise limit. The friction term is assumed to be state dependent. We also provide the proof of the Smolchowski-Kramers approximation for the case of variable friction, non-Lipschitz nonlinear term and unbounded diffusion.

Aplicación de las teorías de Klein-Gordon y de Bogoliubov-deGennes a Niquelatos

In the present work we show the generalities of the classical field theory (CFT), we study its extension to the quantum field theory (QFT), where as an example of numerical analysis and combination with the field theory technique, we solve a system Klein-Gordon type (KGS) in two space-time dimensions (1+1) studying its stability through the spectral parameter λ(k), principle of convergence due to the parameters of the numerical network and the solution for the field ф (x;t), obtaining novel results. Also, we briefly study the technique of creation and destruction ladder operators from the perspective of the quantum harmonic oscillator, to define some properties and extensions to the problem in canonical quantization. Finally, we apply the topics studied to a problem of unconventional superconductivity in Nickelates compounds by solving the system of Bogoliubov-deGennes (BdG) Equations in the mean expansion of the field, obtaining the superconducting energy band.

Odd periodic waves for some Klein–Gordon type equations: Existence and stability

In this paper, we establish the existence and stability properties of odd periodic waves related to the Klein–Gordon type equations, which include the well known and models. Existence of periodic waves is determined by using a general planar theory of ODE. The spectral analysis for the corresponding linearized operator is established using the monotonicity of the period map combined with an improvement of the standard Floquet theory. Orbital stability in the odd sector of the energy space is proved using exclusively the monotonicity of the period map. The orbital instability of explicit solutions for the and models is presented using the abstract approach of instability for general abstract Hamiltonian systems.

A Remark on the Essential Self-adjointness for Klein–Gordon-Type Operators

We discuss a new simplified proof of the essential self-adjointness for formally self-adjoint differential operators of real principal type with the null non-trapping condition, previously proved by Vasy (J Spectr Theory 10: 439–461, 2020) and Nakamura-Taira (Ann Henri Lebesgue 4: 1035–1059, 2021). For simplicity, here we discuss the second-order cases, i.e., Klein–Gordon-type operators only.

Spectral properties of the finite system of Klein-Gordon S-wave equations with general boundary condition

The spectral characteristics of the operator L is studied where L is defined within the Hilbert space L2(R+, CV) given by a finite system of Klein-Gordon type differential equations and boundary condition at general form. The research of the Klein-Gordon type operator continues to be an important topic for researchers due to the range of applicability of them in numerous branches of mathematics and quantum physics. Contrary to the previous works, we take the potential as complex valued and generalize the problem to the matrix Klein-Gordon operator case. The spectrum is derived by determining the Jost function and resolvent operator of the prescribed operator. Further, we provide the conditions that must be met for the certain quantitative properties of the spectrum.

Релятивистская модель межатомных взаимодействий

In this paper, we propose a method for relativistic description of the dynamics of systems of interacting particles through an auxiliary field which in the static mode is equivalent to given interatomic potentials, and in the dynamic mode is a classical relativistic field. It has been established that for static interatomic potentials of a general form, the auxiliary field is a composition of elementary fields satisfying the Klein-Gordon type equations. Each elementary field is characterized by a complex parameter which is an analogue of the real mass in the Klein-Gordon equation. The interaction between particles through an auxiliary field is nonlocal both in spatial variables and in time. The qualitative properties of solutions to equations describing the auxiliary field are studied. Relativistic mechanisms of both the thermodynamic behavior and synergetic effects in few-particle systems have been established.

Essential Self-Adjointness of Klein-Gordon Type Operators on Asymptotically Static, Cauchy-Compact Spacetimes

Let $X=\mathbb{R}\times M$ be the spacetime, where $M$ is a closed manifold equipped with a Riemannian metric $g$, and we consider a symmetric Klein-Gordon type operator $P$ on $X$, which is asymptotically converges to $\partial_t^2-\triangle_g$ as $|t|\to\infty$, where $\triangle_g$ is the Laplace-Beltrami operator on $M$. We prove the essential self-adjointness of $P$ on $C_0^\infty(X)$. The idea of the proof is closely related to a recent paper by the authors on the essential self-adjointness for Klein-Gordon operators on asymptotically flat spaces.

Third quantization for scalar and spinor wave functions of the Universe in an extended minisuperspace

We consider the third quantization in quantum cosmology of a minisuperspace extended by the Eisenhart–Duval lift. We study the third quantization based on both Klein–Gordon type and Dirac-type equations in the extended minisuperspace. Spontaneous creation of ‘Universes’ is investigated upon the quantization of a simple model. We find that the quantization of the Dirac-type wave function reveals that the number density of universes is expressed by the Fermi–Dirac distribution. We also calculate the entanglement entropy of the multi-universe system.

Nonlocal elasticity of Klein–Gordon type: Fundamentals and wave propagation

A nonlocal elasticity theory with nonlocality in space and time is developed by considering nonlocal constitutive equations with a dynamical scalar nonlocal kernel function. The proposed theory is specified to isotropic nonlocal elasticity of Klein–Gordon type, which is an extension of nonlocal elasticity of Helmholtz type, and it possesses one characteristic internal length scale parameter as well as one characteristic internal time scale parameter in addition to the elastic material parameters. Dynamical nonlocal kernels are analytically derived as retarded Green functions of the Klein–Gordon operator. The dispersion relations of the considered isotropic nonlocal elasticity model of Klein–Gordon type are analytically determined. The obtained results reveal the advantage of the proposed nonlocal elasticity model, that is, its ability to predict for the first time in the framework of nonlocal elasticity in addition to the acoustic modes (low-frequency modes), optic modes (high-frequency modes) as well as frequency band-gaps between the acoustic and optic modes. A qualitative examination of the phase and group velocities for all four modes (acoustic and optic branches of longitudinal and transverse waves) shows that the proposed model allows for physically realistic dispersive wave propagation.

Uniform stabilization for a strongly coupled semilinear/linear system

Abstract In this manuscript, we analyze the exponential stability of a strongly coupled semilinear system of Klein-Gordon type, posed in an inhomogeneous medium Ω \Omega , subject to local dampings of different natures distributed around a neighborhood of the boundary according to the geometric control condition (GCC). The first one is of the type viscoelastic and is distributed around a neighborhood ω \omega of the boundary ∂ Ω \partial \Omega of Ω \Omega , according to the GCC. The second one is a frictional damping and we consider it hurting the GCC. The third dissipation, which acts only in the second equation (according to the GCC), is of the frictional type. We show that the energy of the system goes uniformly and exponentially to zero for all initial data of finite energy taken in bounded sets of finite energy phase space. We also prove the exponential decay for the linear problem associated with this same system, and in this case, no restrictions are made with respect to dimension of the space nor with respect to the limitation of the initial data in the phase space.

Nonlinear topological edge states: From dynamic delocalization to thermalization

We consider a mechanical lattice inspired by the Su-Schrieffer-Heeger model along with cubic Klein-Gordon type nonlinearity. We investigate the long-time dynamics of the nonlinear edge states, which are obtained by nonlinear continuation of topological edge states of the linearized model. Linearly unstable edge states delocalize and lead to chaos and thermalization of the lattice. Linearly stable edge states also reach the same fate, but after a critical strength of perturbation is added to the initial edge state. We show that the thermalized lattice in all these cases shows an effective renormalization of the dispersion relation. Intriguingly, this renormalized dispersion relation displays a unique symmetry, i.e., its square is symmetric about a finite squared frequency, akin to the chiral symmetry of the linearized model.

Time domain analysis of impulse electromagnetic field at the interface of two media

Background. Ultrashort pulses of electromagnetic field are widely used in construction, archeology and demining, etc., by constructing effective georadars of the underlying surface, but theoretical study of physical processes of irradiation of medium is had a limit range of simplified model problems, usually in frequency domain. Therefore, the solutions of the problems of penetration of a pulsed wave with arbitrary time dependence into material medium are of special importance for understanding the possibilities and limitations of georadar’s study. Objectives. To obtain the analytical solution in time domain of the problems of reflection and propagation of a impulse electromagnetic wave through the interface of two media, which is the first model approximation to the description of physical processes that occur during operation of pulsed ultrawideband radar. Materials and methods. The problem of irradiation by nonstationary electric field of a lossless medium with a given permittivity is solved analytically by application of evolutionary approach. It consists in solving of Cauchy’s problem for the second-order partial differential equation Klein–Gordon type with respect to evolutionary coefficients. The components of the electromagnetic field in free space are found by integration by spectral parameters and summation by angular modes with appropriate combinations of basis functions. Results. Cauchy’s problems for differential equations that describe the behavior of reflected and refracted waves are solved. The electrical transverse components of the reflected and refracted waves as a function of time on the longitudinal axis were found for the case of irradiation with the step-like time dependence. Graphs of dependence of electric components on time and coordinates are plotted and analyzed. Conclusion. The phenomenon of an electromagnetic missiles in the medium that was irradiated by a pulsed electromagnetic wave of ultrashort duration was demonstrated for the first time. The obtained results can be generalized for the case of an arbitrary impulse by the Duhamel’s integral method. In addition, the electric field for observation point that do not lie on the longitudinal axis can be considered. An even more interesting continuation of the researches in terms of energy analysis is the study of the behavior of longitudinal electric and transverse magnetic components.

Well-posedness of the Cauchy problem for system of oscillators on 2D–lattice in weighted $l^2$-spaces

We consider an infinite system of ordinary differential equations that describes the dynamics of an infinite system of 
 linearly coupled nonlinear oscillators on a two dimensional integer-valued lattice. It is assumed that each oscillator
 interacts linearly with its four nearest neighbors and the oscillators are at the rest at infinity. We study the initial value problem (the Cauchy problem) for such system. This system naturally can be considered as an operator-differential equation
 in the Hilbert, or even Banach, spaces of sequences. We note that $l^2$ is the simplest choice of such spaces. With this choice of the configuration space, the phase space is $l^2\times l^2$, and the equation can be written in the Hamiltonian form with the Hamiltonian $H$. Recall that from a physical point of view the Hamiltonian represents the full energy of the system, i.e., the sum of kinetic and potential energy. Note that the Hamiltonian $H$ is a conserved quantity, i.e., for any solution of equation the Hamiltonian is constant. For this space, there are some results on the global solvability of the corresponding Cauchy problem. In the present paper, results on the $l^2$-well-posedness are extended to weighted $l^2$-spaces $l^2_\Theta$. We suppose that the weight $\Theta$ satisfies some regularity assumption.
 Under some assumptions for nonlinearity and coefficients of the equation, we prove that every solution of the Cauchy problem from $C^2\left((-T, T); l^2)$ belongs to $C^2\left((-T, T); l^2_\Theta\right)$. 
 And we obtain the results on existence of a unique global solutions of the Cauchy problem for system of oscillators on a two-dimensional lattice in a wide class of weighted $l^2$-spaces. These results can be applied to discrete sine-Gordon type equations and discrete Klein-Gordon type equations on a two-dimensional lattice. In particular, the Cauchy problems for these equations are globally well-posed in every weighted $l^2$-space with a regular weight.

Duality family of scalar field

We show that there exists a duality family of self-interacting massive scalar fields. The scalar fields in a duality family are related by a duality transformation. Such a duality of scalar fields is a field version of the Newton-Hooke duality in classical mechanics. The duality transformation preserves the type of the field equation: transforming a Klein-Gordon type equation to another Klein-Gordon type equation with a different self-interacting potential. Once a field in a duality family is solved, all other family members are solved by the transformation. That is, a series of exactly solvable models can be constructed from one exactly solvable model. The dual field of the power-interaction field, the sine-Gordon field, etc., are considered. Moreover, as a comparison, we show an analogue of the duality in classical and quantum mechanics.

Methods for Constructing Complex Solutions of Nonlinear PDEs Using Simpler Solutions

This paper describes a number of simple but quite effective methods for constructing exact solutions of nonlinear partial differential equations that involve a relatively small amount of intermediate calculations. The methods employ two main ideas: (i) simple exact solutions can serve to construct more complex solutions of the equations under consideration and (ii) exact solutions of some equations can serve to construct solutions of other, more complex equations. In particular, we propose a method for constructing complex solutions from simple solutions using translation and scaling. We show that in some cases, rather complex solutions can be obtained by adding one or more terms to simpler solutions. There are situations where nonlinear superposition allows us to construct a complex composite solution using similar simple solutions. We also propose a few methods for constructing complex exact solutions to linear and nonlinear PDEs by introducing complex-valued parameters into simpler solutions. The effectiveness of the methods is illustrated by a large number of specific examples (over 30 in total). These include nonlinear heat equations, reaction–diffusion equations, wave type equations, Klein–Gordon type equations, equations of motion through porous media, hydrodynamic boundary layer equations, equations of motion of a liquid film, equations of gas dynamics, Navier–Stokes equations, and some other PDEs. Apart from exact solutions to ‘ordinary’ partial differential equations, we also describe some exact solutions to more complex nonlinear delay PDEs. Along with the unknown function at the current time, u=u(x,t), these equations contain the same function at a past time, w=u(x,t−τ), where τ>0 is the delay time. Furthermore, we look at nonlinear partial functional-differential equations of the pantograph type, which, in addition to the unknown u=u(x,t), also contain the same functions with dilated or contracted arguments, w=u(px,qt), where p and q are scaling parameters. We propose an efficient approach to construct exact solutions to such functional-differential equations. Some new exact solutions of nonlinear pantograph-type PDEs are presented. The methods and examples in this paper are presented according to the principle “from simple to complex”.

Time-Changed Dirac–Fokker–Planck Equations on the Lattice

A time-changed discretization for the Dirac equation is proposed. More precisely, we consider a Dirac equation with discrete space and continuous time perturbed by a time-dependent diffusion term $\sigma^2Ht^{2H-1}$ that seamlessly describes a latticizing version of the time-changed Fokker-Planck equation carrying the Hurst parameter $0<H<1$. Our model problem formulated on the space-time lattice $\mathbb{R}_{h,\alpha}^n\times [0,\infty)$ ($h>0$ and $0<\alpha<\frac{1}{2}$) preserves the main features of the Dirac-K\"ahler type discretization over the space-time lattice $h\mathbb{Z}^n\times [0,\infty)$ in case of $\alpha,H \rightarrow 0$, and encompasses a regularization of Wilson's approach [Physical review D, 10(8), 2445, 1974] for values of $H$ in the range $0<H\leq \frac{1}{2}$ (limit condition $\alpha \rightarrow \frac{1}{2}$). The main focus here is the representation of the solutions by means of discrete convolution formulae involving a kernel function encoded by (unnormalized) Hartman-Watson distributions -- ubiquitous on stochastic processes of Bessel type -- and the solutions of a semi-discrete equation of Klein-Gordon type. Namely, on our main construction the ansatz function $\widehat{\varPsi}_H(y)$ appearing on the discrete convolution representation may be rewritten as a Mellin convolution type integral involving the solutions $\varPsi(x,t|p)$ of a semi-discrete equation of Klein-Gordon type and a L\'evy one-sided distribution $L_H(u)$ in disguise. Interesting enough, by employing Mellin-Barnes integral representations it turns out that the underlying solutions of Klein-Gordon type may be represented through generalized Wright functions of type ${~}_1\Psi_1$, that converge uniformly in case that the quantity $\alpha+\frac{1}{2}$ may be regarded as an lower estimate for the Hurst parameter in the superdiffusive case (that is, if $\alpha+\frac{1}{2}\leq H<1$).

Dynamic Input Reconstruction Algorithm for a Nonlinear Equation with Distributed Parameters

We consider a differential equation of the Klein–Gordon type with an unknown input. The solution of this equation is assumed to be measured with some errors. We indicate a reconstruction algorithm whose input is formed by the feedback principle and serves as approximations to the unknown input of the original equation.

The Wave-Front Equation of Gravitational Signals in Classical General Relativity

In this paper the dynamical equation for propagating wave-fronts of gravitational signals in classical general relativity (GR) is determined. The work relies on the manifestly-covariant Hamilton and Hamilton–Jacobi theories underlying the Einstein field equations recently discovered (Cremaschini and Tessarotto, 2015–2019). The Hamilton–Jacobi equation obtained in this way yields a wave-front description of gravitational field dynamics. It is shown that on a suitable subset of configuration space the latter equation reduces to a Klein–Gordon type equation associated with a 4-scalar field which identifies the wave-front surface of a gravitational signal. Its physical role and mathematical interpretation are discussed. Radiation-field wave-front solutions are pointed out, proving that according to this description, gravitational wave-fronts propagate in a given background space-time as waves characterized by the invariant speed-of-light c. The outcome is independent of the actual shape of the same wave-fronts and includes the case of gravitational waves which are characterized by an eikonal representation and propagate in a generic curved space-time along a null geodetics. The same waves are shown: (a) to correspond to the geometric-optics limit of the same curved space-time solutions; (b) to propagate in a flat space-time as plane waves with constant amplitude; (c) to display also the corresponding form of the wave-front in curved space-time. The result is consistent with the theory of the linearized Einstein field equations and the existence of gravitational waves achieved in such an asymptotic regime. Consistency with the non-linear Trautman boundary-value theory is also displayed.