Klein-Gordon Model Research Articles

Spectral stability analysis for special solutions of second order in time PDEs: The higher dimensional case

We develop a general theory to treat the linear stability of certain special solutions of second order in time evolutionary PDE. We apply these results to standing waves of the following problems: the Klein–Gordon equation, for which we consider both ground states and certain excited states, the Klein–Gordon–Zakharov system and the beam equation. We also discuss applications to excited states for the Klein–Gordon model as well as multidimensional traveling waves (not necessarily homoclinic to zero) for general second order equations of this type. In all cases, our abstract results provide a complete characterization of the linear stability of such solutions.

The effects of horizontal singular straight line in a generalized nonlinear Klein–Gordon model equation

In this paper, we investigate bounded traveling waves of the generalized nonlinear Klein–Gordon model equations by using bifurcation theory of planar dynamical systems to study the effects of horizontal singular straight lines in nonlinear wave equations. Besides the well-known smooth traveling wave solutions and the non-smooth ones, four kinds of new bounded singular traveling wave solution are found for the first time. These singular traveling wave solutions are characterized by discontinuous second-order derivatives at some points, even though their first-order derivatives are continuous. Obviously, they are different from the singular traveling wave solutions such as compactons, cuspons, peakons. Their implicit expressions are also studied in this paper. These new interesting singular solutions, which are firstly founded, enrich the results on the traveling wave solutions of nonlinear equations. It is worth mentioning that the nonlinear equations with horizontal singular straight lines may have abundant and interesting new kinds of traveling wave solution.

Collective-Coordinate Analysis of Inhomogeneous Nonlinear Klein–Gordon Field Theory

Two different sets of collective-coordinate equations for solitary solutions of Nonlinear Klein-Gordon (NKG) model is introduced. The collective-coordinate equations are derived using different approaches for adding the inhomogeneities as exrernal potentials to the soliton equation of motion. Interaction of the NKG field with a local inhomogeneity like a delta function potential wall and also delta function potential well is investigated using the presented collective-coordinate equations and the results of two different models are compared. Most of the characters of the interaction are derived analytically. Analytical results are also compared with the results of numerical simulations.

Relativistic x-ray quantum free-electron lasers: a collective Klein–Gordon model

We present a quantum-relativistic model for the nonlinear interaction between large-amplitude electromagnetic (EM) waves and a quantum plasma. The model is based on a collective Klein–Gordon equation for the relativistic electrons, which is coupled with the Maxwell equations for the EM and electrostatic fields. The model is used to derive a nonlinear dispersion relation for the EM scattering instabilities in a relativistic quantum plasma. With application to the quantum free-electron laser (FEL), a relativistic electron beam is passing through an intense EM wave used as a wiggler to produce coherent tunable radiation. The frequency upshift of the radiation is tuned by the energy of the electron beam. The nonlinear dispersion relation reveals the importance of quantum recoil effects and oblique scattering of the radiation on the gain of the quantum FEL.

Escape dynamics in the discrete repulsive [formula omitted] model

We study deterministic escape dynamics of the discrete Klein–Gordon model with a repulsive quartic on-site potential. Using a combination of analytical techniques, based on differential and algebraic inequalities and selected numerical illustrations, we first derive conditions for collapse of an initially excited single-site unit, for both the Hamiltonian and the linearly damped versions of the system and showcase different potential fates of the single-site excitation, such as the possibility to be “pulled back” from outside the well or to “drive over” the barrier some of its neighbors. Next, we study the evolution of a uniform (small) segment of the chain and, in turn, consider the conditions that support its escape and collapse of the chain. Finally, our path from one to the few and finally to the many excited sites is completed by a modulational stability analysis and the exploration of its connection to the escape process for plane wave initial data. This reveals the existence of three distinct regimes, namely modulational stability, modulational instability without escape and, finally, modulational instability accompanied by escape. These are corroborated by direct numerical simulations. In each of the above cases, the variations of the relevant model parameters enable a consideration of the interplay of discreteness and nonlinearity within the observed phenomenology.

Negative energy waves and quantum relativistic Buneman instabilities

The quantum relativistic Buneman instability is investigated theoretically using a collective Klein-Gordon model for the electrons and a cold fluid model for the ions. The growth rate and unstable wave spectrum is investigated in different parameter regimes corresponding to various degrees of relativistic and quantum effects. The results may be important for streaming instabilities involving ion dynamics in very dense plasmas.

A scale-invariant Klein–Gordon model with time-dependent potential

The goal of the present paper is to derive statements about energy estimates as well as L p −L q decay estimates for a Klein–Gordon model with a particular time-dependent mass. The study of this special case of a scale-invariant model is an important step within a systematic investigation of Klein–Gordon models with time-dependent mass.

Soliton–potential interaction in the nonlinear Klein–Gordon model

The interaction of solitons with external potentials in the nonlinear Klein–Gordon field theory is investigated using an improved model. The presented model has been constructed with a better approximation for adding the potential to the Lagrangian through the metric of background space–time. The results of the model are compared with another model and the differences are discussed.

Parameter Region for Existence of Non-classical Solitons

We study a pure nonlinear model of a chain formed by particles that are linked each other by an enharmonic type. This lattice nonlinear Klein–Gordon model is subsequently studied in its continuum version. We use the dynamical systems approach for analyzing the properties of the non-classical structures that support the model. Several non-classical structures like peakons, kink compactons and crodwon or bubble compactons are generated along the chain for the specific region of the parameter space. It is shown that the phase space trajectories are nonclassical curves and show unexpected behaviors. The first type of phase transition in the parametric space occurs when the number of centers and saddles changes while the main phase state parameter becomes critical.

Effect of damping on kink ratchets in the Klein–Gordon lattice free of the Peierls–Nabarro potential

Ratchet of kink under harmonic ac driving force in the discrete Klein–Gordon model with asymmetric potential and with a damping term is investigated numerically. A particular feature of the considered discrete model is that supported kinks are free of the Peierls–Nabarro potential (PNp). It is demonstrated that, depending on the strength of damping, the drift velocity of the kink can be positive (weak damping) or negative (strong damping). The absence of PNp results in a weak dependence of kink drift velocity on the discreteness parameter.

Emergence of Compact Structures in a Klein-Gordon Model

The Klein-Gordon model (KG) phi=P{'}(|phi|)phi/|phi| is Lorenz invariant and has a finite wave speed, yet its localized modes, whether Q balls or vortices, suffer from the same fundamental flaw as all other solitons-they extend indefinitely. Using the KG model as a case study, we demonstrate that appending the site potential, P{a}(phi|), with a subquadratic part P(|phi|)=b{2}|phi|{1+delta}+P{a}(|phi|), 0<or=delta<1, induces particlelike modes with strictly compact support. These modes are robust and shorten in the direction of motion. Their interactions, which occur only on contact, are studied in two and three dimensions and are shown to span the whole range from being nearly elastic to plastic.

General theory of oscillon dynamics

We present a comprehensive, nonperturbative analytical method to investigate the dynamics of time-dependent oscillating scalar field configurations. The method is applied to oscillons in a double well Klein-Gordon model in two and three spatial dimensions, yielding high accuracy results in the characterization of all aspects of the complex oscillon dynamics. In particular, we show how oscillons can be interpreted as long-lived perturbations about an attractor in field configuration space. By investigating their radiation rate as they approach the attractor, we obtain an accurate estimate of their lifetimes in d=3 and explain why they seem to be perturbatively stable in d=2, where d is the number of spatial dimensions.

Threshold field in nucleation process for an inhomogeneous deformable nonlinear Klein-Gordon system

We investigate the effect of local inhomogeneity on the nucleation process of kink-antikink pairs in the driven nonlinear Klein-Gordon model with the Remoissenet-Peyrard substrate potential, whose shape can be varied as a function of the shape parameter and which has the sine-Gordon shape as a particular case. From numerical integrations of the model equations under defined conditions, the configuration of the critical nucleus in the presence of a localized inhomogeneity is determined and shown to be affected by the shape parameter of the substrate potential. The expression of the depinning threshold field of kink-antikink pairs, which is the value of the applied field at which the process of the nucleation of kink-antikink pairs takes place, is also obtained. The dependence of this depinning threshold field on the shape parameter r shows that it strongly increases, for small intensity of the impurity potential when the shape of the substrate potential deviates from the sinusoidal one.

Bifurcation analysis of travelling wave solutions in the nonlinear Klein–Gordon model with anharmonic coupling

By using bifurcation method of phase portraits and theory of dynamical systems to the nonlinear Klein–Gordon model, in the range 0 ⩽ r < 1 , numbers of kink compactons, solitary wave, kink and anti-kink waves are given for each parameter condition. The parameter condition when the type of travelling wave changes is obtained.

Thermal nucleation of kink-antikink pairs in the presence of impurities: The case of a Remoissenet-Peyrard substrate potential

Thermal nucleation of kink-antikink pairs in a nonlinear Klein-Gordon (NKG) model with a Remoissenet-Peyrard (RP) substrate potential in the presence of impurities and coupled to an applied field is analyzed in the limits of moderate temperature and strong damping. Using the Kolmogorov method, the average velocity of particles of the lattice is calculated and its dependence on the intensity of impurities is discussed in connection with the deformability parameter or the shape of the RP substrate potential. Numerical values are carried out by making use of parameters of the hydrogen atom adsorbed in the tungsten and ruthenium substrates. We show that, for large values of the applied field, the presence of impurities in the system makes the nucleation process of kink-antikink pairs more favorable in the high-temperature regime while they contribute to make it less favorable in the low-temperature regime.

Discrete gap breathers in a diatomic Klein-Gordon chain: Stability and mobility

A one-dimensional diatomic chain with harmonic intersite potential and nonlinear external potential is considered (the Klein-Gordon model). Localized solutions of the corresponding nonlinear differential equations with frequencies inside the gap of the linear wave spectrum--"gap breathers"--are studied numerically. The linear stability analysis for these solutions is performed while changing the system parameters from the anticontinuous to the continuous limit. Two different types of solutions are considered: symmetric centered at a heavy atom and antisymmetric centered at a light atom, respectively. Different mechanisms of instability, oscillatory as well as nonoscillatory, of the gap breathers are studied, and the influence of the instabilities on the breather solutions is investigated in the dynamics simulations. In particular, the presence of an "inversion of stability" regime, with simultaneous nonoscillatory instabilities of symmetric and antisymmetric solutions with respect to antisymmetric perturbations, is found, yielding practically radiationless mobility.

Thermodynamical observables in a finite temperature window from the Monte Carlo Hamiltonian

The Monte Carlo (MC) Hamiltonian is a new stochastic method to solve many-body problems. The MC Hamiltonian represents an effective Hamiltonian in a finite energy window. We construct it from the classical action via Monte Carlo with importance sampling. The MC Hamiltonian yields the energy spectrum and corresponding wave functions in a low energy window. This allows to compute thermodynamical observables in a low temperature window. We show the working of the MC Hamiltonian by an example from lattice field theory (Klein–Gordon model).

Fine structure of oscillons in the spherically symmetricφ4Klein-Gordon model

We present results from a study of the fine structure of oscillon dynamics in the 3+1 spherically symmetric Klein-Gordon model with a symmetric double-well potential. We show that in addition to the previously understood longevity of oscillons, there exists a resonant (and critical) behavior which exhibits a time-scaling law. The mode structure of the critical solutions is examined, and we also show that the upper-bound to oscillon formation (in $r_0$ space) is either non-existent or higher than previously believed. Our results are generated using a novel technique for implementing non-reflecting boundary conditions in the finite difference solution of wave equations. The method uses a coordinate transformation which blue-shifts and ``freezes'' outgoing radiation. The frozen radiation is then annihilated via dissipation explicitly added to the finite-difference scheme, with very little reflection into the interior of the computational domain.

Interplay of nonlinearity and geometry in a DNA-related, Klein-Gordon model with long-range dipole-dipole interaction.

Most of the studies on mathematical models of DNA are limited to next neighbor interaction. However, the coupling between base pairs is thought to be caused by dipole interaction, and, when the DNA strand is bent, the distances between base pairs become shorter, therefore the interactions with distant base pairs have to be taken into account. In this paper we analyze the existence and stability of breathers, i.e., localized oscillations in a simple model of bent DNA with long-range dipole interaction. Breathers have been suggested as precursors of the denaturation bubble.

Topological discrete kinks

A spatially discrete version of the general kink-bearing nonlinear Klein-Gordon model in (1 + 1) dimensions is constructed which preserves the topological lower bound on kink energy. It is proved that, provided the lattice spacing h is sufficiently small, there exist static kink solutions attaining this lower bound centred anywhere relative to the spatial lattice. Hence there is no Peierls-Nabarro (PN) barrier impeding the propagation of kinks in this discrete system. An upper bound on h is derived and given a physical interpretation in terms of the radiation of the system. The construction, which works most naturally when the nonlinear Klein-Gordon model has a squared polynomial interaction potential, is applied to a recently proposed continuum model of polymer twistons. Numerical simulations are presented which demonstrate that kink pinning is eliminated, and radiative kink deceleration is greatly reduced in comparison with the conventional discrete system. So even on a very coarse lattice, kinks behave much as they do in the continuum. It is argued, therefore, that the construction provides a natural means of numerically simulating kink dynamics in nonlinear Klein-Gordon models of this type. The construction is compared with the inverse method of Flach, Zolotaryuk and Kladko. Using the latter method, alternative spatial discretizations of the twiston and sine-Gordon models are obtained which are also free of the PN barrier.