Gordon System Research Articles

Energy-conserving SAV-Hermite–Galerkin spectral scheme with time adaptive method for coupled nonlinear Klein–Gordon system in multi-dimensional unbounded domains

In this article, we construct an efficient numerical scheme to solve the coupled nonlinear Klein–Gordon system in a multi-dimensional unbounded domain Rd(d=1,2,3). We first use the SAV method to equivalently deform the original system, then we treat the nonlinear part of the system explicitly in temporal discretization. Thanks to the SAV method, we ensure the unconditional energy conservation of the discrete system, as a cost, we only need to solve two completely linearized decoupled systems. To avoid the errors caused by domain truncation and the construction of artificial boundary conditions, we apply the Hermite–Galerkin spectral method with scaling factor for spatial discretization. In order to save computational costs, we utilize two time adaptive strategies to adjust the time step size, through which we can improve computational efficiency without sacrificing accuracy. Finally, we presented numerical experiments to demonstrate the accuracy, efficiency, and energy conservation properties of our algorithm, and applied it to simulate the interactions of two-dimensional and three-dimensional vector solitons.

Global stability of the Dirac–Klein–Gordon system in two and three space dimensions

Global stability of the Dirac–Klein–Gordon system in two and three space dimensions

Strong Cosmic Censorship for the Spherically Symmetric Einstein–Maxwell-Charged-Klein–Gordon System with Positive $$\Lambda $$: Stability of the Cauchy Horizon and $$H^1$$ Extensions

Strong Cosmic Censorship for the Spherically Symmetric Einstein–Maxwell-Charged-Klein–Gordon System with Positive $$\Lambda $$: Stability of the Cauchy Horizon and $$H^1$$ Extensions

Global existence and completeness of classical solutions in higher dimensional Einstein–Klein–Gordon system

Global existence and completeness of classical solutions in higher dimensional Einstein–Klein–Gordon system

From Scalar Clouds to Rotating Hairy Black Holes

First, we review the solutions of a complex-valued scalar field, termed scalar clouds, with and without electric charge, when coupled to a rotating Kerr–Newman (electrically charged) or Kerr (neutral) black hole (BH), respectively. To this aim, we determine the conditions and parameters that characterize the existence of solutions that represent bound states, with an energy-momentum tensor that respect the symmetries of the underlying spacetimes, even if the backreaction of the field is not taken into account at this stage. In particular, we show that in the extremal Kerr scenario the cloud solutions exist only when the mass of the BH satisfies certain bounds, which are obtained by analyzing an effective potential associated with the radial dependency of the scalar clouds that leads to a Schrödinger-like equation. Second, when the backreaction of the field in the spacetime is taken into account, we present a family of stationary, axisymmetric and asymptotically flat solutions of the Einstein–Klein–Gordon system that represent genuine rotating hairy black holes (RHBHs) and provide different values of some global quantities associated with them, such as the Komar mass and the Komar angular momentum. We also compute RHBH solutions with nodes in the radial part of the scalar field and also for a higher azimuthal number m.

On the Fourier Analysis of the Einstein–Klein–Gordon System: Growth and Decay of the Fourier Constants

We consider the (1+3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(1+3)$$\\end{document}-dimensional Einstein equations with negative cosmological constant coupled to a spherically symmetric, massless scalar field and study perturbations around the anti-de Sitter spacetime. We derive the resonant systems, pick out vanishing secular terms and discuss issues related to small divisors. Most importantly, we rigorously establish (sharp, in most of the cases) asymptotic behaviour for all the interaction coefficients. The latter is based on uniform estimates for the eigenfunctions associated to the linearized operator as well as on some oscillatory integrals.

Large time well-posedness for a Dirac–Klein–Gordon system

In this paper, we prove well-posedness for a Dirac equation perturbed with a moving potential W that satisfies a Klein–Gordon equation. This represents a “toy model” for atoms with relativistic corrections, as the wave function of the electrons interacts with an electric field generated by a nucleus with a given charge density. One of the main ingredients we need is a new family of Strichartz estimates for time-dependent perturbations of the Dirac equation: these represent a result of independent interest.

Asymptotic stability for the Dirac–Klein–Gordon system in two space dimensions

We study the Dirac–Klein–Gordon system in 1 + 2 spacetime dimensions. We show global existence of solutions, as well as sharp time decay and linear scattering. One key advance is that we provide the first asymptotic stability result for the Dirac–Klein–Gordon system in 1 + 2 spacetime dimensions in the case of a massive Klein–Gordon field and a massless Dirac field. The nonlinearities are below critical in two spatial dimensions, and so our method requires the identification of special structures within the system and novel weighted energy estimates. Another key advance is that our proof allows us to weaken certain conditions on the nonlinear structures that have been assumed in the literature.

The Dirac–Klein–Gordon system in the strong coupling limit

The Dirac–Klein–Gordon system in the strong coupling limit

Linear superposition for a novel nonlocal PT -symmetric mKdV-sG system

A novel nonlocal -symmetric modified Korteweg–de Vries-sine Gordon (mKdV-sG) system is established via the consistent correlated bang (CCB) method from the normal mKdV-sG equation. The obtained nonlocal mKdV-sG system is Lax integrable and can be reduced to the normal mKdV-sG equation with suitable function selections. Using symmetry-antisymmetry separation method, a linear superposition solution theorem for the nonlocal mKdV-sG system is concluded. Abundant nonlinear wave structures of the obtained nonlocal system, such as the periodic waves, the kinks, the breathers and periodic-periodic interaction waves, periodic-kink interaction waves, periodic-breathers interaction waves solutions are obtained from the linear superposition solution theorem.

Numerical solutions for the f(R)-Klein–Gordon system

We construct a numerical relativity code based on the Baumgarte–Shapiro–Shibata–Nakamura (BSSN) formulation for the gravitational quadratic f(R) Starobinsky model. By removing the assumption that the determinant of the conformal 3-metric is unity, we first generalize the BSSN formulation for general f(R) gravity theories in the metric formalism to accommodate arbitrary coordinates for the first time. We then describe the implementation of this formalism to the paradigmatic Starobinsky model. We apply the implementation to three scenarios: the Schwarzschild black hole solution, flat space with non-trivial gauge dynamics, and a massless Klein–Gordon (KG) scalar field. In each case, long-term stability and second-order convergence is demonstrated. The case of the massless KG scalar field is used to exercise the additional terms and variables resulting from the f(R) contributions. For this model, we show for the first time that additional damped oscillations arise in the subcritical regime as the system approaches a stable configuration.

Global Stability of Minkowski Space for the Einstein–Maxwell–Klein–Gordon System in Generalized Wave Coordinates

Global Stability of Minkowski Space for the Einstein–Maxwell–Klein–Gordon System in Generalized Wave Coordinates

Charged boson stars revisited

We consider again stationary solutions to the spherically symmetric Einstein–Maxwell–Klein–Gordon system, commonly known as “charged boson stars”, originally studied by Jetzer and Van Der Bij (Phys Lett B 227:341, 1989). We construct families of charged boson stars in the ground state, for different values of the charge parameter q, and different values of the central scalar field. Following Jetzer and Van Der Bij, one can define a critical value for the charge q=q_c that corresponds to the value for which the Coulomb repulsion of the bosonic particles exactly cancels their newtonian gravitational attraction. We confirm the claim made by Pugliese et al. (Phys Rev D, 2013. https://doi.org/10.1103/physrevd.88.024053) that super-critical solutions exist for a limited range of charges above the critical value q>q_c (though we find an even smaller range of q for which this is possible). Our analysis indicates, however, that all such super-critical solutions are gravitationally unbound, and are therefore expected to be unstable. One of the main results of our analysis is the fact that, even though we do find a family of slightly super-critical solutions in the sense that q>q_c, there are no super-critical solutions in the sense that the total charge Q is larger than the total mass M of the system.

Global solution to the 3D Dirac–Klein–Gordon system with uniform energy bounds

On the (1+3) dimensional Minkowski spacetime, for small, regular initial data, it is well-known that the Dirac–Klein–Gordon system admits a global solution. In the present paper, we aim to establish the uniform boundedness of the total energy of the solution for this system. The proof relies on Klainerman’s vector field and Alinhac’s ghost weight methods. The main difficulty originates from the slow decay nature of the Dirac and wave components in three space dimensions. To overcome the difficulty, a sharp understanding of the structure for this system, and a new weighted conformal energy estimate are implemented. In addition, we also provide a few scattering results for the system.

Regularity of weak solutions for the relativistic Vlasov–Klein–Gordon system

This article is devoted to the study of the relativistic Vlasov–Klein–Gordon system in space dimension three. The dynamical model describes a collisionless ensemble of classical particles coupled with a Klein–Gordon field. By combining the Fourier method and smoothing effect of low velocity particles, we prove a higher regularity of the field, which is widely used for the study of Vlasov equation.

Global existence of classical static solutions of four dimensional Einstein–Klein–Gordon system

In this paper, we prove the global existence of classical static solutions of Einstein’s gravitational theory coupled to a real scalar field where spacetime admits spherically symmetry. The equations of motions can then be reduced into a single first-order integro-differential equation. First, we obtain the decay estimates of the solutions. Then, to prove the global existence, we use the contraction mapping theorem in the appropriate function spaces.

Stability of solitary waves in nonlinear Klein–Gordon equations

The stability of topological solitary waves and pulses in one-dimensional nonlinear Klein–Gordon systems is revisited. The linearized equation describing small deviations around the static solution leads to a Sturm–Liouville problem, which is solved in a systematic way for the -potential, showing the orthogonality and completeness relations fulfilled by the set of its solutions for all values . This approach enables the linear stability of kinks and pulses of certain nonlinear Klein–Gordon equations to be determined. The inverse problem, which starts from Sturm–Liouville problem and obtains nonlinear Klein–Gordon potentials, is also revisited and solved in a direct way. The exact solutions (kinks and pulses) for these potentials are calculated, even when the nonlinear potential is not explicitly known. The kinks are found to be stable, whereas the pulses are unstable. The stability of the pulses is achieved by introducing certain spatial inhomogeneities.

Global in retarded time solutions to the Einstein–Maxwell–Klein–Gordon system with positive cosmological constant

Global in retarded time solutions to the Einstein–Maxwell–Klein–Gordon system with positive cosmological constant

On robustness of the speed-gradient sampled-data energy control for the sine–Gordon equation: The simpler the better

The paper studies robustness with respect to time-sampling of the energy regulation for one-dimensional sine–Gordon system. Such a problem is a new to control of invariants for hyperbolic partial differential equations (PDEs). In the absence of analytic results, the problem is studied numerically. The properties of four sampled-data algorithms are computationally studied with respect to three performance criteria. The four speed gradient algorithms are the “proportional”, “relay”, “adaptive-relay” and combined ones, by using state feedback with in-domain actuators. The three performance criteria are limit error, transient time, and threshold of stability for the sampling interval. An unexpected result is that the best performance for all three criteria was exhibited by the simplest, speed-gradient-proportional, algorithm. Simulation results are also presented for other energy tracking controllers to add insight into the parameter choice for improving the closed-loop robustness in the PDE setting over sampled-data algorithms.

Einstein–Klein–Gordon spacetimes in the harmonic near-Minkowski regime

We study the initial value problem for the Einstein–Klein–Gordon system and establish the global nonlinear stability of massive matter in the near-Minkowski regime when the initial geometry is a perturbation of an asymptotically flat, spacelike hypersurface in Minkowski spacetime and the metric enjoys the harmonic decay 1/r (in terms of a suitable distance function r at spatial infinity). Our analysis encompasses matter fields that have small energy norm and solely enjoys a slow decay at spacelike infinity. Our proof is based on the Euclidean-hyperboloidal foliation method recently introduced by the authors, and distinguishes between the decay along asymptotically hyperbolic slices and the decay along asymptotically Euclidean slices. We carefully analyze the decay of metric components at the harmonic level 1/r , especially the metric component in the direction of the light cone. In presence of such a slow-decaying matter field, we establish a global existence theory for the Einstein equations expressed as a coupled system of nonlinear wave and Klein–Gordon equations.