Klein-Gordon Model Research Articles

Kink-antikink collisions in hyper-massive models

We study topological kinks and their interactions in a family of scalar field models with a double well potential parametrized by the mass of small perturbations around the vacua, ranging from the mass of the ϕ4 Klein-Gordon model all the way to the limit of infinite mass, which is identified with a non-analytic potential. In particular, we look at the problem of kink-antikink collisions and analyze the windows of capture and escape of the soliton pair as a function of collision velocity and model mass. We observe a disappearance of the capture cases for intermediary masses between the ϕ4 and non-analytic cases. The main features of the kink-antikink scattering are reproduced in a collective coordinates model, including the disappearance of the capture cases.

A cortical field theory – dynamics and symmetries

We characterise cortical dynamics using partial differential equations (PDEs), analysing various connectivity patterns within the cortical sheet. This exploration yields diverse dynamics, encompassing wave equations and limit cycle activity. We presume balanced equations between excitatory and inhibitory neuronal units, reflecting the ubiquitous oscillatory patterns observed in electrophysiological measurements. Our derived dynamics comprise lowest-order wave equations (i.e., the Klein-Gordon model), limit cycle waves, higher-order PDE formulations, and transitions between limit cycles and near-zero states. Furthermore, we delve into the symmetries of the models using the Lagrangian formalism, distinguishing between continuous and discontinuous symmetries. These symmetries allow for mathematical expediency in the analysis of the model and could also be useful in studying the effect of symmetrical input from distributed cortical regions. Overall, our ability to derive multiple constraints on the fields — and predictions of the model — stems largely from the underlying assumption that the brain operates at a critical state. This assumption, in turn, drives the dynamics towards oscillatory or semi-conservative behaviour. Within this critical state, we can leverage results from the physics literature, which serve as analogues for neural fields, and implicit construct validity. Comparisons between our model predictions and electrophysiological findings from the literature — such as spectral power distribution across frequencies, wave propagation speed, epileptic seizure generation, and pattern formation over the cortical surface — demonstrate a close match. This study underscores the importance of utilizing symmetry preserving PDE formulations for further mechanistic insights into cortical activity.

Mathematical analysis of soliton solutions in space-time fractional Klein-Gordon model with generalized exponential rational function method

In this article, we investigate the space-time Klein-Gordon (KG) model, a significant framework in quantum field theory and quantum mechanics, which also describes phenomena such as wave propagation in crystal dislocations. This model is particularly important in high-energy particle physics. The novelty of this article is to examine the sufficient, useful in optical fibers, and further general soliton solutions of the nonlinear KG model using the generalized exponential rational function method (GERFM), which do not exist in the recent literature. The fractional complex wave transformation is utilized to turn the model into a nonlinear form, and the accuracy of the acquired solutions is confirmed by reintroducing them into the original models using Mathematica. The obtained solutions are expressed in hyperbolic, exponential, rational, and trigonometric forms. We elucidate the fractional effects for specific parameter values, accompanied by illustrative figures. Our results demonstrate that GERFM is effective, powerful, and versatile, providing exact traveling wave solutions for various nonlinear models in engineering and mathematical physics. Our findings reveal that the characteristics of soliton-shaped waves in both three-dimensional and two-dimensional contexts are profoundly influenced by fractional order derivative. This study advances the understanding of nonlinear wave dynamics and offers a robust method for solving complex physical models.

Investigation of Analytical Soliton Solutions to the Non-Linear Klein–Gordon Model Using Efficient Techniques

Nonlinear distinct models have wide applications in various fields of science and engineering. The present research uses the mapping and generalized Riccati equation mapping methods to address the exact solutions for the nonlinear Klein–Gordon equation. First, the travelling wave transform is used to create an ordinary differential equation form for the nonlinear partial differential equation. This work presents the construction of novel trigonometric, hyperbolic and Jacobi elliptic functions to the nonlinear Klein–Gordon equation using the mapping and generalized Riccati equation mapping methods. In the fields of fluid motion, plasma science, and classical physics the nonlinear Klein–Gordon equation is frequently used to identify of a wide range of interesting physical occurrences. It is considered that the obtained results have not been established in prior study via these methods. To fully evaluate the wave character of the solutions, a number of typical wave profiles are presented, including bell-shaped wave, anti-bell shaped wave, W-shaped wave, continuous periodic wave, while kink wave, smooth kink wave, anti-peakon wave, V-shaped wave and flat wave solitons. Several 2D, 3D and contour plots are produced by taking precise values of parameters in order to improve the physical description of solutions. It is noteworthy that the suggested techniques for solving nonlinear partial differential equations are capable, reliable, and captivating analytical instruments.

Relativistic hydrodynamics with phase transition

Assessing the applicability of hydrodynamic expansions close to phase transition points is crucial from either theoretical or phenomenological points of view. We explore this within the gauge/gravity duality, using the Einstein–Klein–Gordon model, a bottom-up string theory construction. This model incorporates a parameter, B4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_4$$\\end{document}, that simulates different types of phase transitions in the strongly coupled field theory existing at the boundary. We thoroughly examine the thermodynamics and dynamics of time-dependent, linearized perturbations in the spin-2, spin-1, and spin-0 sectors. Our findings suggest that ‘hydrodynamic series breakdown near transition points” is valid exclusively for second-order phase transitions, not for crossovers or first-order phase transitions. Additionally, we observe that the high-temperature and low-temperature limits of the radius of convergence for the hydrodynamic series (qc2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q^2_c$$\\end{document}) are equal. We also discover that the relationship (Max|qc2|)spin-2<(Max|qc2|)spin-0<(Max|qc2|)spin-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\ ext {Max}|q^2_c|)_{\ ext {spin-2}}< (\ ext {Max}|q^2_c|)_{\ ext {spin-0}} < (\ ext {Max}|q^2_c|)_{\ ext {spin-1}}$$\\end{document} is consistent for different spin sectors, regardless of the phase transition type. At the chaos point, we observe the emergence of pole-skipping behavior for both gravity and scalar perturbations at ωn=-2πTni\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega _n = -2\\pi T n i$$\\end{document}. Lastly, comparing the chaos momentum with qc2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q^2_c$$\\end{document}, we find that qps2<qc2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q^2_{ps} < q^2_c$$\\end{document}, except for extremely high temperatures.

Spectral algorithm for two-dimensional fractional sine-Gordon and Klein–Gordon models

Solitons and waves with memory effects are two examples of nonlinear wave phenomena that can be studied mathematically using the fractional nonlinear sine-Gordon equation. It is a modification of the traditional sine-Gordon equation that takes memory effects and nonlocal interactions into account by adding fractional derivatives. A more complex explanation of particle dynamics and interactions within relativistic quantum mechanics is made possible by the fractional Klein–Gordon model, a theoretical framework that expands the standard equation to include fractional derivatives. The study uses shifted Legendre–Gauss–Lobatto and shifted Legendre–Gauss–Radau collocation techniques to solve numerically two-dimensional sine-Gordon and Klein–Gordon models. The study handles two-dimensional sine-Gordon and Klein–Gordon models by extending a collocation approach using basis functions. It suggests a collocation technique that uses a suggested basis to automatically satisfy the conditions. The suggested methods’ spectral accuracy and efficiency are validated by numerical results.

Analytical and numerical solutions to the Klein–Gordon model with cubic nonlinearity

In this paper, the nonlinear Klein–Gordon equation’s exact solutions are obtained through the application of an appropriate transformation based on He’s semi-inverse approach. This equation considered a generalization of other famous models in applied science, such as Phi-4 equation, Duffing equation, Fisher–Kolmogorov model through population dynamics and Hodgkin–Huxley equation that characterizes the propagation of electrical signals via nervous system. The suggested approach is simple, robust, and efficient, and its application in other partial differential equations in applied science seems promising. Numerical solution of the nonlinear Klein–Gordon equation is presented using finite difference method. The method’s accuracy is demonstrated by contrasting it with the exact solution that we obtained. The Von Neumann stability technique is applied to obtain the stability condition and a convergent test is presented Some 2D and 3D graphs matching to chosen solutions are simulated by taking into account appropriate values for the parameters.

Dynamical behavior of soliton solutions to the fractional phi-four model via two analytical techniques

This study explores solutions for a mathematical equation called the time-space fractional phi-four equation using two methods: the Sardar-subequation method and the modified extended auxiliary equation method. The phi-four equation is connected to the Klein–Gordon model and is important in different scientific areas like biology and nuclear physics. Understanding its solutions is crucial. By using a specific wave transformation, the equation is changed into a simpler form for analysis. The methods proposed give a variety of solutions, such as Kink, bright singular, dark, combo dark bright, periodic, and singular periodic solutions. Each solution we find using these methods has specific rules that determine when it’s correct. We carefully choose specific values for the parameters to help us understand more about the solutions. This helps us see the detailed features of the solutions and improves our understanding of how the model behaves in the real world. These methods create a strong framework for studying solitons, which are specific types of mathematical solutions. The study compares the outcomes of these methods with earlier ones to get a complete understanding. Graphical illustrations are used to visually represent some of these solutions, helping us grasp their characteristics. Visual representations in two- and three-dimensional figures add originality to the findings. Importantly, these methods can be applied to solve similar problems with fractional derivatives in various scientific contexts. In summary, this research not only deepens our understanding of the phi-four equation but also introduces powerful methods with broad applications in fractional differential equations.

Retracted: Novel Analysis of Fuzzy Fractional Klein-Gordon Model via Semianalytical Method

Retracted: Novel Analysis of Fuzzy Fractional Klein-Gordon Model via Semianalytical Method

Exploration of solitons and analytical solutions by sub-ODE and variational integrators to Klein-Gordon model

In this paper, we use the sub-ODE method to analyze soliton solutions for the renowned nonlinear Klein-Gordon model (NLKGM). This method provides a variety of soliton solutions, including three positive solitons, three Jacobian elliptic function solutions, bright solitons, dark solitons, periodic solitons, rational solitons and hyperbolic function solutions. Applications for these solitons can be found in optical communication, fiber optic sensors, plasma physics, Bose-Einstein condensation and other areas. We also study some numerical solutions by using forward, backward, and central difference techniques. Moreover, we discuss variational integrators (VIs) using the projection technique for NLKGM. We develop a numerical solution for NLKGM using the discrete Euler lagrange equation, the Lagrangian and the Euler lagrange equation. At the end, in various dimensions, covering 3D, 2D, and contour, we will also plot several graphs for the obtained NLKGM solutions. A contour plot is a type of graphic representation that displays a three-dimensional surface on a two-dimensional plane by using contour lines. Each contour line in the plotted function represents one of the function's constant values, mapping the function's value across the plane. This model has been studied across multiple soliton solutions using various methods in the open literature, but this model for VIs and finite deference scheme (FDS) is the first time it has been studied. Within the various numerical techniques accessible for solving Hamiltonian systems, variational integrators distinguish themselves because of their symplectic quality. Here are some of the symplectic properties: symplectic orthogonality, energy conservation, area preservation, and structure preservation.

New analytical method for cubic Klein-Gordon equation

In this paper, the (2+1)-dimensional cubic Klein-Gordon model is investigated, which is used to described the propagation of dislocation in crystals. A simple and efficient analytical technology is successfully employed to seek some new periodic and solitary wave solutions, which is called sine-cosine method. The physics properties of these obtained periodic and solitary wave solutions are illustrated by corresponding graphs.

A robust way to justify the derivative NLS approximation

The derivative nonlinear Schrödinger (DNLS) equation can be derived as an amplitude equation via multiple scaling perturbation analysis for the description of the slowly varying envelope of an underlying oscillating and traveling wave packet in dispersive wave systems. It appears in the degenerated situation when the cubic coefficient of the similarly derived NLS equation vanishes. It is the purpose of this paper to prove that the DNLS approximation makes correct predictions about the dynamics of the original system under rather weak assumptions on the original dispersive wave system if we assume that the initial conditions of the DNLS equation are analytic in a strip of the complex plane. The method is presented for a Klein–Gordon model with a cubic nonlinearity.

Fifth-order nonlinear Schrödinger equation as Routhian reduction of the nonlinear Klein–Gordon model

We consider the nonlinear Klein–Gordon model with polynomial nonlinearity involving odd and even powers. We elaborate a hybrid Hamiltonian and Lagrangian approach, which is generally referred to as Routh’s procedure, to describe the nonlinear modulation of wave packets assuming the wave field envelope to be a slow function of time and coordinate. An extended fifth-order nonlinear Schrödinger equation is obtained from the Routhian equation for a complex canonical variable that couples the wave field and its momentum. This canonical variable is expressed in terms of the amplitude of fundamental harmonic and its derivatives with respect to coordinate. The resulting fifth-order nonlinear Schrödinger equation is demonstrated to be a Hamiltonian system under a certain constraint imposed on its model parameters. When compared with other techniques used to derive high-order nonlinear Schrödinger models, our approach preserves the Hamiltonian structure of the governing equations for the fundamental harmonic, while the input of higher harmonics is taken into account by means of variational principle applied to the averaged Routhian. In this way, it represents the most general tool for the reduction of nonlinear wave equations to high-order envelope models for slowly modulated wave packets.

Analysis of the response of damped and parametrically driven, strongly anharmonic Klein-Gordon chain - Part 2: Beat-waves

This paper continues a series of publications devoted to the analysis of strongly nonlinear wave phenomena, emerging in genuinely anharmonic, Klein-Gordon cyclic chains, subjected to a traveling-wave parametric excitation. In this study, we analyze a special type of nonlinear waves, which will be further referred to as beat-waves. These are spatially extended, and strongly modulated waves manifested by the extreme amplitude modulation, which can be of stationary or nonstationary type. Stationary beat-waves of Klein-Gordon chains propagate with constant group velocity, that depends on the beat-wave amplitude. In the vicinity of 2:1 parametric resonance, we derive the parametrically driven, discrete p-Schrödinger (PD-DpS) model, which depicts the slow modulation of the response of a parametric oscillator chain. Special coordinates' transformation applied on the PD-DpS model allows the exact reduction of extremely modulated beat-waves to an invariant manifold of this slow-flow system. The system dynamics reduced to the beat-wave invariant manifold is depicted by three collective coordinates. In the undamped case, the analysis of the slow-flow system dynamics on the invariant manifold reveals a special family of nonlinear beat-waves characterized by an extreme amplitude modulation. Among these beat-wave solutions lying on the manifold, our analysis identifies some special solutions manifested by the solitary-pulse like, temporal evolution of the beat-wave amplitude which can either decay to zero or to a constant nonzero value. We derive the exact analytical solutions for these special types of waves as well as obtain an explicit analytical approximation for the beat-waves of other types. In the damped case, analysis of the slow-flow system dynamics on the invariant manifold fully recovers the mechanism of system synchronization on the special, stationary beat-wave solutions characterized by the harmonically modulated envelope. Importantly, the derived reduced order model accurately captures the transient evolution stage as well as the steady-state stage of the collective beat-wave response of PD-DpS model, initialized on the beat-wave invariant manifold. A detailed bifurcation analysis of the stationary undamped as well as the stationary damped beat-waves recovers their entire bifurcation structure and enables to fully characterize the zones of system parameters guaranteeing the existence of this special type of non-linear wave solutions. The results of analytical study of various types of beat wave solutions are in excellent agreement with the results of numerical simulation of both PD-DpS as well as the original, parametrically driven, genuinely an-harmonic, discrete Klein-Gordon model.

Experimental observation of curved light-cones in a quantum field simulator

We investigate signal propagation in a quantum field simulator of the Klein-Gordon model realized by two strongly coupled parallel one-dimensional quasi-condensates. By measuring local phononic fields after a quench, we observe the propagation of correlations along sharp light-cone fronts. If the local atomic density is inhomogeneous, these propagation fronts are curved. For sharp edges, the propagation fronts are reflected at the system's boundaries. By extracting the space-dependent variation of the front velocity from the data, we find agreement with theoretical predictions based on curved geodesics of an inhomogeneous metric. This work extends the range of quantum simulations of nonequilibrium field dynamics in general space-time metrics.

Numerical Investigation of Time-Fractional Phi-Four Equation via Novel Transform

This paper examines two methods for solving the nonlinear fractional Phi-four problem with variable coefficients. One of the distinct states of the Klein–Gordon model yields the Phi-four equation. It is also used to simulate the kink and anti-kink solitary wave connections that have recently emerged in biological systems and nuclear particle physics. The approaches that are being suggested consist of the Yang transform, the homotopy perturbation approach, the decomposition approach, and the fractional derivative as stated by Caputo. The advantages of the proposed techniques are their capability of combining two dominant approaches for attaining precise and approximate solutions of nonlinear equations. It is important to keep in mind that the suggested methods can perform better in general as they need less computational effort than the alternative methods, while keeping a high level of numerical precision. The actual and estimated outcomes are demonstrated in graphs and tables to be quite similar, demonstrating the usefulness of the proposed approaches. Additionally, several simulations are used to show the physical behaviors of the found solutions with regard to fractional order. The article’s results possess complimentary properties that relate to the symmetry of partial differential equations.

Dynamical structure of truncated M−fractional Klein–Gordon model via two integral schemes

In this work, the Truncated M−fractional Klein–Gordon (tM-fKG) equation is integrated by implementing the exp(-ϕ(ξ))-expansion and new form of modified Kudryashov’s schemes. The novel exponential, hyperbolic, trigonometric, rational solutions and multi-soliton solutions with free parameters are achieved with the help of the schemes. Taking special values of the parameters, the multi-bell wave, periodic rogue, linked rogue, soliton solution through the exp(-ϕ(ξ))-expansion and bright and dark bell shape, periodic wave and linked rogue wave solutions through the new form of modified Kudryashov’s schemes are successfully constract, which are significantly stimulating phenomena in mathematical physics. Furthermore, the effects of Truncated M−fractional time on the obtained solutions of the tM-fK-G equation are shown with proper graphical representations.

Hamiltonian form of extended cubic-quintic nonlinear Schrödinger equation in a nonlinear Klein-Gordon model.

We derive an extended cubic-quintic nonlinear Schrödinger equationwith Hamiltonian structure in a nonlinear Klein-Gordon model with cubic-quintic nonlinearity. We use the nonlinear dispersion relation to properly take into account the input of high-order nonlinear effects in the Hamiltonian perturbation approach to nonlinear modulation. We demonstrate that changing the balance between the cubic and quintic nonlinearities has a significant effect on the stability of unmodulated wave packets to long-wave modulations.

Effect of RGUP on the nonlinear Klein-Gordon model with spontaneous symmetry breaking

In the framework of the relativistic generalized uncertainty principle (RGUP), we study the nonlinear Klein-Gordon field with ϕ4 self-interaction. This generalization comes from the quantum gravity theories that predict the existence of a minimal measurable length scale. The quantum gravity effects slightly modify the momentum and change the Lagrangian density of the field. This model, specifically in the context of spontaneous symmetry breaking, is so fruitful and leads us to many interesting phenomena. Due to the complexity of the consequent equations, the model is normally studied via perturbation mechanisms. However, here, we apply a generalized tanh-method which supposed that the field solutions should be sech-functions of segments of space and time. By this extended method, we find some more general solutions with respect to our previous work [1]. Several solutions can be found but we select only the normalizable and bounded solitary fields. The energy spectrum of each field solution is obtained in an analytical form and discussed. The modification parameter of the corresponding RGUP is estimated by using the rest energy of the mass of the Higgs boson.

Bright, Dark, and Rogue Wave Soliton Solutions of the Quadratic Nonlinear Klein–Gordon Equation

This article reflects on the Klein–Gordon model, which frequently arises in the fields of solid-state physics and quantum field theories. We analytically delve into solitons and composite rogue-type wave propagation solutions of the model via the generalized Kudryashov and the extended Sinh Gordon expansion approaches. We obtain a class of analytically exact solutions in the forms of exponential and hyperbolic functions involving some arbitrary parameters with the help of Maple, which included comparing symmetric and non-symmetric solutions with other methods. After analyzing the dynamical behaviors, we caught distinct conditions on the accessible parameters of the solutions for the model. By applying conditions to the existing parameters, we obtained various types of rogue waves, bright and dark bells, combing bright–dark, combined dark–bright bells, kink and anti-kink solitons, and multi-soliton solutions. The nature of the solitons is geometrically explained for particular choices of the arbitrary parameters. It is indicated that the nonlinear rogue-type wave packets are restricted in two dimensions that characterized the rogue-type wave envelopes.