Nonlinear Klein-Gordon Research Articles

A symmetric low-regularity integrator for nonlinear Klein-Gordon equation

In this work, we propose a symmetric exponential-type low- regularity integrator for solving the nonlinear Klein-Gordon equation under rough data. The scheme is explicit in the physical space, and it is efficient under the Fourier pseudospectral discretization. Moreover, it achieves the second-order accuracy in time without loss of regularity of the solution, and its time-reversal symmetry ensures the good long-time behavior. Error estimates are done for both semi- and full discretizations. Numerical results confirm the theoretical results, and comparisons illustrate the improvement of the proposed scheme over the existing methods.

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.

New Path of Popularized Homogeneous Balance Method and Travelling Wave Solutions of a Nonlinear Klein-Gordon Equation

The aim of this paper is to obtain a set of traveling wave solutions for klein –Gorden equation with kerr law non-linearity. More precisely, we apply a new path of popularized homogeneous balance (HB) method in terms of using linear auxiliary equations to find the results of non-linear klein-Gorden equation, which is a fundamental approach to determine competent solutions. The solutions are achieved as the integration of exponential, hyperbolic, trigonometric and rational functions. Besides, some of the solutions are demonstrated by the3D graphics.

Adiabatic soliton management: Controlling solitary wave motion while keeping the wave envelope unchanged

We describe how to control the motion – both speed and propagation direction – of a nonlinear traveling wave in real time via soliton management with time-varying dispersion/diffusion and loss/gain terms. When carried out subject to certain parameter constraints we derive, the approach allows for real-time manipulation of wave motion without compromising the structure or stability of the wave, maintaining strict adiabaticity in time. This makes our approach a promising tool for physical systems which admit a degree of control through adjustable dispersion and loss/gain terms, including problems arising in nonlinear optics, atomic physics, Bose-Einstein condensates, and thermally driven chemical reactions, where one is interested in modifying the position or speed of a wave while still maintaining adiabaticity. We illustrate both the utility and simplicity of the method through several canonical examples, demonstrating how the motion of KdV solitons, Fisher-KPP solitary wavefronts, BBM solitons, nonlinear Klein-Gordon equations, and interacting nonlinear waves in a vector NLS system can be controlled in real time.

On asymptotic behavior of solutions to cubic nonlinear Klein-Gordon systems in one space dimension

In this paper, we consider the large time asymptotic behavior of solutions to systems of two cubic nonlinear Klein-Gordon equations in one space dimension. We classify the systems by studying the quotient set of a suitable subset of systems by the equivalence relation naturally induced by the linear transformation of the unknowns. It is revealed that the equivalence relation is well described by an identification with a matrix. In particular, we characterize some known systems in terms of the matrix and specify all systems equivalent to them. An explicit reduction procedure from a given system in the suitable subset to a model system, i.e., to a representative, is also established. The classification also draws our attention to some model systems which admit solutions with a new kind of asymptotic behavior. Especially, we find new systems which admit a solution of which decay rate is worse than that of a solution to the linear Klein-Gordon equation by logarithmic order.

Global Existence And Growth Of Solutions To Coupled Degeneratly Damped Klein-Gordon Equations

We study a system of nonlinear Klein-Gordon equations with viscoelastic and degenerate damping in a bounded domain. We prove the global existence and exponential growth of solutions with negative initial energy.

Birkhoff normal forms for Hamiltonian PDEs in their energy space

We study the long time behavior of small solutions of semi-linear dispersive Hamiltonian partial differential equations on confined domains. Provided that the system enjoys a new non-resonance condition and a strong enough energy estimate, we prove that its low super-actions are almost preserved for very long times. Roughly speaking, it means that, to exchange energy, modes have to oscillate at the same frequency. Contrary to the previous existing results, we do not require the solutions to be especially smooth. They only have to live in the energy space. We apply our result to nonlinear Klein-Gordon equations in dimension d = 1 and nonlinear Schrodinger equations in dimension d ≤ 2.

A uniformly accurate exponential wave integrator Fourier pseudo-spectral method with energy-preservation for long-time dynamics of the nonlinear Klein-Gordon equation

For the nonlinear Klein-Gordon (NKG) equation with weak nonlinearity characterized by a small parameter ε∈(0,1], the numerical methods for long-time dynamics have received more and more attention. For the NKG equation up to the time at O(1/ε2), the error bounds of classical numerical methods depend significantly on the small parameters ε which creates serious numerical burdens as ε→0+. Recently, an exponential wave integrator Fourier pseudo-spectral (EWIFP) method for the NKG equation has been proposed (Feng et al., 2021) which is uniformly accurate about ε and of course performs well over the classical methods. However, the EWIFP method can not preserve the discrete energy, which is an important structural feature of the true solution in the long-time dynamics from the perspective of geometric numerical integration. In addition, the EWIFP method only is conditionally stable under specific stability condition and the authors did not give a convergence analysis for the method without any CFL condition restrictions on the grid ratio. In this work, we propose an energy-preserving EWIFP (EPEWIFP) method. The proposed method is proved to be time symmetric, unconditionally stable and preserves the discrete energy. We carry out a rigorously error analysis and give uniform error bounds of the method at O(hm0+ε2−βτ2) up to the time at O(1/εβ) with β∈[0,2], mesh size h, time step τ and an integer m0 determined by the regularity conditions. In general, the NKG equation with weak nonlinearity can be converted to an oscillatory NKG equation with wavelength at O(ε2) in time. It is easy to extend the error bounds and energy-preservation properties to the oscillatory NKG equation. Numerical results confirm our error bounds and energy-preservation properties.

Wave and scattering operators for the nonlinear Klein-Gordon equation on a quarter-plane

We consider the initial-boundary value problem for the nonlinear Klein-Gordon equation posed on a quarter-plane with Dirichlet non zero boundary conditions. We construct the wave and scattering operators for this model in the scattering-supercritical regime. We study the influence of the boundary data on the asymptotic behavior of solutions. We show that if the boundary data has sufficiently slow decay rate, it affects the order of the nonlinearity for which we are able to construct the scattering operator. This suggests a modification for the critical value of the nonlinear term due to the presence of non zero boundary condition. In order to prove our results, we construct an explicit formula for the solutions to corresponding linear Klein-Gordon equation on a quarter-plane. Using this explicit formula, we obtain weighted Sobolev estimates for the solution to the complete nonlinear problem. In order to control the long-time behavior of the solution we find an operator J suitable for our settings, analogous to operator x+it∂x in the case of NLS equation.

Corrigendum to “Multi-travelling waves for the nonlinear Klein-Gordon equation”

We provide a correction to the proof of Proposition 3 by Côte and Martel [Trans. Amer. Math. Soc. 370 (2018), pp. 7461–7487]. We refer to Yuan [Nonlinearity 32 (2019), pp. 5017–5048] for a self-contained proof in the analogue context of the energy critical wave equation. We also refer to Chen and Jendrej [Trans. Amer. Math. Soc. 372 (2019), pp. 7461–7496] for an alternate proof.

Two-dimensional Gegenbauer wavelets for the numerical solution of tempered fractional model of the nonlinear Klein-Gordon equation

In the present article, we study a new version of the physical model, namely the Klein-Gordon equation involved with tempered fractional derivative. This model is numerically simulated for the tempered and the non-tempered fractional derivatives. Two-dimensional Gegenbauer Wavelets are implemented for the solution of the tempered fractional Klein-Gordon equation. The proposed method is a combination of different operational matrices of the two-dimensional Gegenbauer wavelets and the collocation nodes. The applicability and the accuracy of the suggested method has been examined by four illustrative examples and the obtained simulation results are in good agreement with the exact solution. The maximum absolute error, L2 norm error, and the root mean square error have been evaluated to analyze the precision of the proposed method. This method gives better and comparable results to the existing methods given in the literature.

Exponential growth of positive initial energy solutions for coupled nonlinear Klein-Gordon equations with degenerate damping and source terms

In this paper we will prove that the positive initial-energy solution for coupled nonlinear Klein-Gordon equations with degenerate damping and source terms grows exponentially.

Analysis of nonlinear time-fractional Klein-Gordon equation with power law kernel

<abstract><p>We investigate the nonlinear Klein-Gordon equation with Caputo fractional derivative. The general series solution of the system is derived by using the composition of the double Laplace transform with the decomposition method. It is noted that the obtained solution converges to the exact solution of the model. The existence of the model in the presence of Caputo fractional derivative is performed. The validity and precision of the presented method are exhibited with particular examples with suitable subsidiary conditions, where good agreements are obtained. The error analysis and its corresponding surface plots are presented for each example. From the numerical solutions, we observe that the proposed system admits soliton solutions. It is noticed that the amplitude of the wave solution increases with deviations in time, that concludes the factor $ \omega $ considerably increases the amplitude and disrupts the dispersion/nonlinearity properties, as a result, may admit the excitation in the dynamical system. We have also depicted the physical behavior that states the advancement of localized mode excitations in the system.</p></abstract>

Quadratic Klein-Gordon equations with a potential in one dimension

Abstract This paper proposes a fairly general new point of view on the question of asymptotic stability of (topological) solitons. Our approach is based on the use of the distorted Fourier transform at the nonlinear level; it does not rely only on Strichartz or virial estimates and is therefore able to treat low-power nonlinearities (hence also nonlocalised solitons) and capture the global (in space and time) behaviour of solutions. More specifically, we consider quadratic nonlinear Klein-Gordon equations with a regular and decaying potential in one space dimension. Additional assumptions are made so that the distorted Fourier transform of the solution vanishes at zero frequency. Assuming also that the associated Schrödinger operator has no negative eigenvalues, we obtain global-in-time bounds, including sharp pointwise decay and modified asymptotics, for small solutions. These results have some direct applications to the asymptotic stability of (topological) solitons, as well as several other potential applications to a variety of related problems. For instance, we obtain full asymptotic stability of kinks with respect to odd perturbations for the double sine-Gordon problem (in an appropriate range of the deformation parameter). For the $\phi ^4$ problem, we obtain asymptotic stability for small odd solutions, provided the nonlinearity is projected on the continuous spectrum. Our results also go beyond these examples since our framework allows for the presence of a fully coherent phenomenon (a space-time resonance) at the level of quadratic interactions, which creates a degeneracy in distorted Fourier space. We devise a suitable framework that incorporates this and use multilinear harmonic analysis in the distorted setting to control all nonlinear interactions.

Einstein-Klein-Gordon System in Higher Dimensional

In this present work, we study the Einstein equation coupled with the nonlinear Klein-Gordon equation. We obtain Ricci tensor, scalar curvature, and Einstein equation of the Einstein-Klein-Gordon system in higher dimensional. If we put D=4, our formulations reduce to the four dimensional Einstein-Klein-Gordon system.

Numerical solutions of nonlinear time fractional Klein-Gordon equation via natural transform decomposition method and iterative Shehu transform method

The time-fractional Klein-Gordon equation was analyzed in this article using the natural transform decomposition method (NTDM) and the iterative Shehu transform method (ISTM) with a singular kernel derivative. We employed natural transform (NT) and Shehu transform (ST), followed by inverse natural transform and inverse Shehu transform respectively, to obtain the solution. The numerical simulations are provided to ensure that the methods under consideration are efficient. The current framework captures the behaviour of the obtained findings for various fractional orders. The findings of this study indicate that the proposed methods are effective and reliable in analyzing fractional differential equations.

Construction of excited multi-solitons for the focusing 4D cubic wave equation

Consider the focusing 4D cubic wave equation∂ttu−Δu−u3=0,on(t,x)∈[0,∞)×R4. The main result states the existence in energy space H˙1×L2 of multi-solitary waves where each traveling wave is generated by Lorentz transform from a specific excited state, with different but collinear Lorentz speeds. The specific excited state is deduced from the non-degenerate sign-changing state constructed in Musso-Wei [34]. The proof is inspired by the techniques developed for the 5D energy-critical wave equation and the nonlinear Klein-Gordon equation in a similar context by Martel-Merle [30] and Côte-Martel [6].The main difficulty originates from the strong interactions between solutions in the 4D case compared to other dispersive and wave-type models. To overcome the difficulty, a sharp understanding of the asymptotic behavior of the excited states involved and of the kernel of their linearized operator is needed.

Pseudospectral methods with PML for nonlinear Klein-Gordon equations in classical and non-relativistic regimes

Two different Perfectly Matched Layer (PML) formulations with efficient pseudo-spectral numerical schemes are derived for the standard and non-relativistic nonlinear Klein-Gordon equations (NKGE). A pseudo-spectral explicit exponential integrator scheme for a first-order formulation and a linearly implicit preconditioned finite-difference scheme for a second-order formulation is proposed and analyzed. To obtain a high spatial accuracy, new regularized Bermúdez type absorption profiles are introduced for the PML. It is shown that the two schemes are efficient, but the linearly implicit scheme should be preferred for accuracy purpose when used within the framework of pseudo-spectral methods combined with the regularized Bermúdez type functions. In addition, in the non-relativistic regime, numerical examples lead to the conclusion that the error related to regularized Bermúdez type absorption functions is insensitive to the small parameter ε involved in the NKGE. The paper ends by two extensions of the proposed strategy: one is for the long-term dynamics of NKGE with weak nonlinearity and the other is for a two-dimensional rotating NKGE where the vortex dynamics is very well-reproduced.

Crank-Nicolson/finite element approximation for the Schrödinger equation in the de Sitter spacetime

Central to much of science, engineering, and society today is the building of mathematical models to represent complex processes. Recently, the non-relativistic limit of nonlinear Klein-Gordon equations in de Sitter spacetime has been used to derive Schrödinger equations with weighted nonlinear terms In this paper, numerical simulations are constructed to clarify the behavior of the solution in both one- and two-dimensions. These simulations are constructed based on the Crank-Nicolson scheme in the time direction and the Galerkin finite element in the spatial direction. The nonlinear system of algebraic equations resulting from the constructed scheme is solved using Newton’s method. It is also demonstrated that the numerical approximation converges to the exact one.

The energy-preserving time high-order AVF compact finite difference scheme for nonlinear wave equations in two dimensions

In this paper, energy-preserving time high-order average vector field (AVF) compact finite difference scheme is proposed and analyzed for solving two-dimensional nonlinear wave equations including the nonlinear sine-Gordon equation and the nonlinear Klein-Gordon equation. We first present the corresponding Hamiltonian system to the two-dimensional nonlinear wave equations, and further apply the compact finite difference (CFD) operator and AVF method to develop an energy conservative high-order scheme in two dimensions. The Lp-norm boundedness of two-dimensional numerical solution is obtained from the energy conservation property, which plays an important role in the analysis of the scheme for the two-dimensional nonlinear wave equations in which the nonlinear term satisfies local Lipschitz continuity condition. We prove that the proposed scheme is energy conservative and uniquely solvable. Furthermore, optimal error estimate for the developed scheme is derived for the nonlinear sine-Gordon equation and the nonlinear Klein-Gordon equation in two dimensions. Numerical experiments are carried out to confirm the theoretical findings and to show the performance of the proposed method for simulating the propagation of nonlinear waves in layered media.