Klein Gordon Zakharov Research Articles

On the decay problem for the Zakharov and Klein–Gordon–Zakharov systems in one dimension

We are interested in the long time asymptotic behaviour of solutions to the scalar Zakharov system $$\begin{aligned} \begin{array}{ll} i u_{t} + \Delta u = nu,\\ n_{tt} - \Delta n= \Delta |u|^2 \end{array} \end{aligned}$$and the Klein–Gordon–Zakharov system $$\begin{aligned} \begin{array}{ll} u_{tt} - \Delta u + u = - nu,\\ n_{tt} - \Delta n= \Delta |u|^2 \end{array} \end{aligned}$$in one dimension of space. For these two systems, we give two results proving decay of solutions for initial data in the energy space. The first result deals with decay over compact intervals assuming smallness and parity conditions (u odd). The second result proves decay in far field regions along curves for solutions whose growth can be dominated by an increasing \(C^1\) function. No smallness condition is needed to prove this last result for the Zakharov system. We argue relying on the use of suitable virial identities appropriate for the equations and follow the technics of [22, 24] and [33].

Abundant novel wave solutions of nonlinear Klein–Gordon–Zakharov (KGZ) model

Abundant novel wave solutions of nonlinear Klein–Gordon–Zakharov (KGZ) model

Fast evaluation for the two-dimensional nonlinear coupled time–space fractional Klein–Gordon–Zakharov equations

In this paper, we develop a fast algorithm to solve the two-dimensional nonlinear coupled time–space fractional Klein–Gordon–Zakharov (KGZ) equations. The L2−1σ method based on an efficient sum-of-exponentials (SOE) approximation and a Fourier spectral method are used to approximate the time and space direction, respectively. And we use the previous time levels to deal with the nonlinear terms to obtain a linearized numerical scheme. Finally, a numerical example is given to show that our numerical method is of second order accuracy in time, spectral accuracy in space and the fast algorithm is effective.

Asymptotic Behavior of the Solution to the Klein–Gordon–Zakharov Model in Dimension Two

Consider the Klein-Gordon-Zakharov equations in $\mathbb{R}^{1+2}$, and we are interested in establishing the small global solution to the equations and in investigating the pointwise asymptotic behavior of the solution. The Klein-Gordon-Zakharov equations can be regarded as a coupled semilinear wave and Klein-Gordon system with quadratic nonlinearities which do not satisfy the null conditions, and the fact that wave components and Klein-Gordon components do not decay sufficiently fast makes it harder to conduct the analysis. In order to conquer the difficulties, we will rely on the hyperboloidal foliation method and a minor variance of the ghost weight method. As a side result of the analysis, we are also able to show the small data global existence result for a class of quasilinear wave and Klein-Gordon system violating the null conditions.

A uniformly first-order accurate method for Klein-Gordon-Zakharov system in simultaneous high-plasma-frequency and subsonic limit regime

We present a uniformly first order accurate numerical method for solving the Klein-Gordon-Zakharov (KGZ) system with two dimensionless parameters 0<ε≤1 and 0<γ≤1, which are inversely proportional to the plasma frequency and the acoustic speed, respectively. In the simultaneous high-plasma-frequency and subsonic limit regime, i.e. ε<γ→0+, the KGZ system collapses to a cubic Schrödinger equation, and the solution propagates waves with O(ε2)-wavelength in time and meanwhile contains rapid outgoing initial layers with speed O(1/γ) in space due to the incompatibility of the initial data. By presenting a multiscale decomposition of the KGZ system, we propose a multiscale time integrator Fourier pseudospectral method which is explicit, efficient and uniformly accurate for solving the KGZ system for all 0<ε<γ≤1. Numerical results are reported to show the efficiency and accuracy of scheme. Finally, the method is applied to investigate the convergence rates of the KGZ system to its limiting models when ε<γ→0+.

Analysis of fractional Klein–Gordon–Zakharov equations using efficient method

Analysis of fractional Klein–Gordon–Zakharov equations using efficient method

An energy-preserving and efficient scheme for a double-fractional conservative Klein–Gordon–Zakharov system

In this work, we consider a fractional extension of the Klein–Gordon–Zakharov system which describes the propagation of strong turbulences on the Langmuir wave in a high-frequency plasma. Both components consider space-fractional derivatives of the Riesz type, and initial-boundary conditions are imposed on a closed and bounded interval of the real numbers. In a first stage, we show that the total energy of the system is conserved, and that the global solutions of the system are bounded. Motivated by these results, we propose a finite-difference scheme to approximate the solutions, and a discrete form of the energy functional. The advantage of the discretization proposed in this work lies in that the difference equations to solve the component equations are decoupled. This implies that the numerical schemes can be solved separately at each temporal step. We establish rigorously the existence of solutions, as well as the capability of the scheme to conserve the discrete energy. The method has a second-order consistency in both space and time. Moreover, using a discrete form of the energy method, we establish mathematically that the finite-difference scheme is stable and quadratically convergent. We provide some simulations to show that the proposed methodology is quadratically convergent and that it preserves the total energy of the system.

New solitary waves for the Klein–Gordon–Zakharov equations

This paper studies new solitary waves for the Klein–Gordon–Zakharov equations. The obtained results are diverse and some specific ones emerge as dark, bright and bell-shape. The two integration schemes lead to obtaining waves which propagate without deformation as illustrated in graphical representations. Our obtained results are more specific compared to those obtained by Refs. 8 and 31 [C. H. Zhao and Z. M. Sheng, Acta Phys. Sin. 53 (2004) 29; S. Yakada, B. Depelair, G. Betchewe and S. Y. Doka, Optik 197 (2019) 163108].

Existence of solutions of an explicit energy-conserving scheme for a fractional Klein–Gordon–Zakharov system

In the article Martínez et al. (2019) [7], the authors assumed the triviality of the existence of solutions of a finite-difference model for a fractional Klein–Gordon–Zakharov equation. It turns out that property is not trivial at all. In this short communication, we provide a proof of that result and, in the way, we mention some wrong proofs of similar results available in the literature. We believe that the report on the demonstration of this result will be a useful tool for researchers working on the numerical analysis of partial differential equations.

The Zakharov limit of Klein–Gordon–Zakharov like systems in case of analytic solutions

ABSTRACT We give a new proof for the validity of the Zakharov approximation of the Klein–Gordon–Zakharov system under the assumption that the solutions of the Zakharov system are analytic in a strip in the complex plane. The approach is conceptually more simple than for solutions which are only finitely many times differentiable and so it can simply be applied to more general systems, too.

Corrigendum to “A numerically efficient and conservative model for a Riesz space-fractional Klein–Gordon–Zakharov system”

In the article [Commun Nonlinear Sci Numer Simul 2019;71:22–37], the authors provided an erroneous proof for the existence of solutions of a finite-difference model for a fractional Klein–Gordon–Zakharov equation. This letter is intended to provide a correct proof of that theorem.

Numerical simulation for solitary wave of Klein–Gordon–Zakharov equation based on the lattice Boltzmann model

This work reports a study of Klein–Gordon–Zakharov equation by using the lattice Boltzmann method. The Klein–Gordon–Zakharov equation describes the interaction of the Langmuir wave and the ion acoustic wave in plasma. Numerical simulations of the constructed lattice Boltzmann model for the Klein–Gordon–Zakharov equation show the propagation and the interaction of the (1+1) and (2+1) dimensional solitary waves. These simulation results are examined and compared by using different parameters and different schemes. This work indicates that the lattice Boltzmann method can be a very effective tool for the simulation of Klein–Gordon–Zakharov equation.

Effective Slow Dynamics Models for a Class of Dispersive Systems

We consider dispersive systems of the form $$\begin{aligned} \partial _t U = \Lambda _U U + B_U(U,V) , \qquad \varepsilon \partial _t V = \Lambda _V V + B_V(U,U) \end{aligned}$$ ∂ t U = Λ U U + B U ( U , V ) , ε ∂ t V = Λ V V + B V ( U , U ) in the singular limit $$ \varepsilon \rightarrow 0 $$ ε → 0 , where $$ \Lambda _U,\Lambda _V $$ Λ U , Λ V are linear and $$ B_U,B_V $$ B U , B V bilinear mappings. We are interested in deriving error estimates for the approximation obtained through the regular limit system $$\begin{aligned} \partial _t \psi _U = \Lambda _U \psi _U - B_U(\psi _U, \Lambda _V^{-1} B_V(\psi _U,\psi _U) ) \end{aligned}$$ ∂ t ψ U = Λ U ψ U - B U ( ψ U , Λ V - 1 B V ( ψ U , ψ U ) ) from a more general point of view. Our abstract approximation theorem applies to a number of semilinear systems, such as the Dirac–Klein–Gordon system, the Klein–Gordon–Zakharov system, and a mean field polaron model. It extracts the common features of scattered results in the literature, but also gains an approximation result for the Dirac–Klein–Gordon system which has not been documented in the literature before. We explain that our abstract approximation theorem is sharp in the sense that there exists a quasilinear system of the same structure where the regular limit system makes wrong predictions.

Theoretical analysis of an explicit energy-conserving scheme for a fractional Klein–Gordon–Zakharov system

Departing from an initial-boundary-value problem governed by a Klein–Gordon–Zakarov system with fractional derivatives in the spatial variable, we provide an explicit finite-difference scheme to approximate its solutions. In agreement with the continuous system, the method proposed in this work is also capable of preserving the energy of the system, and the energy quantities are nonnegative under flexible parameter conditions. The boundedness, the consistency, the stability and the convergence of the technique are also established rigorously. The method is easy to implement, and the computer simulations confirm the main analytical and numerical properties of the new model.

Klein‐Gordon‐Zakharov system in energy space: Blow‐up profile and subsonic limit

In this paper, we prove finite‐time blowup in energy space for the three‐dimensional Klein‐Gordon‐Zakharov (KGZ) system by modified concavity method. We obtain the blow‐up rates of solutions in local and global space, respectively. In addition, by using the energy convergence, we study the subsonic limit of the Cauchy problem for KGZ system and prove that any finite energy solution converges to the corresponding solution of Klein‐Gordon equation in energy space.

An analysis of implicit conservative difference solver for fractional Klein–Gordon–Zakharov system

In this paper, we propose an efficient linearly implicit conservative difference solver for the fractional Klein–Gordon–Zakharov system. First of all, we present a detailed derivation of the energy conservation property of the system in the discrete setting. Then, by using the mathematical induction, it is proved that the proposed scheme is uniquely solvable. Subsequently, by virtue of the discrete energy method and a ‘cut-off’ function technique, it is shown that the proposed solver possesses the convergence rates of O(Δt2+h2) in the sense of L∞- and L2- norms, respectively, and is unconditionally stable. Finally, numerical results testify the effectiveness of the proposed scheme and exhibit the correctness of theoretical results.

A numerically efficient and conservative model for a Riesz space-fractional Klein–Gordon–Zakharov system

Departing from a fractional extension of the well-known one-dimensional Klein–Gordon–Zakharov system, we propose a numerically efficient model to approximate its solutions. The continuous model under investigation considers fractional derivatives of the Riesz type in space, with orders of differentiation in (1, 2]. In analogy with the non-fractional regime, the existence of a positive conserved energy quantity is established in this work. Motivated by this fact, the design of the numerical model focuses on the preservation of the energy. Using fractional-order centered differences to approximate the fractional partial derivatives, we propose a numerical model that preserves a positive discrete form of the energy. The existence and uniqueness of solutions are thoroughly established using fixed-point arguments, and the usual argument with Taylor polynomials is employed to prove the consistency of the numerical model. Some suitable bounds in terms of the energy invariants are found for the solutions of the numerical model. Moreover, using an extension of the energy method for fractional systems, we establish the stability and the convergence properties of the methodology. Finally, we provide some examples to illustrate the accuracy of our numerical implementation, including a numerical study of the convergence rate of the scheme that confirms the validity of the analytical results derived in this work.

Group classification, conservation laws and Painlevé analysis for Klein–Gordon–Zakharov equations in (3 $$+$$ + 1)-dimension

In this paper, we study Klein–Gordon–Zakharov equations which describe the propagation of strong turbulence of the Langmuir wave in a high-frequency plasma. Using the symbolic manipulation tool Maple, the classifications of symmetry algebra are carried out, and the construction of several local non-trivial conservation laws based on a direct method of Anco and Bluman is illustrated. Starting with determination of symmetry algebra, the one- and two-dimensional optimal systems are constructed, and optimality is also established using various invariant functions of full adjoint action. Apart from classification and construction of several conservation laws, the Painleve analysis is also performed in a symbolic manner which describes the non-integrability of equations.

On the new wave behavior to the Klein–Gordon–Zakharov equations in plasma physics

In this study, the extended sinh-Gordon equation expansion method is used in constructing various exact solitary wave solutions to the Klein–Gordon–Zakharov equations. The Klein–Gordon–Zakharov equations is a nonlinear model describing the interaction between the Langmuir wave and the ion acoustic wave in a high-frequency plasma. We successfully construct some topological, non-topological, compound topological and non-topological, singular, compound singular solitons and singular periodic wave solutions. Under the choice of suitable values of the parameters, the 2D, 3D and contour graphs to some of the reported solutions are also plotted.

A uniformly and optimally accurate multiscale time integrator method for the Klein–Gordon–Zakharov system in the subsonic limit regime

We present a uniformly and optimally accurate numerical method for discretizing the Klein–Gordon–Zakharov system (KGZ) with a dimensionless parameter 0<ε≤1, which is inversely proportional to the acoustic speed. In the subsonic limit regime, i.e., 0<ε≪1, the solution of KGZ system propagates waves with O(ε)- and O(1)-wavelength in time and space, respectively, and rapid outspreading initial layers with speed O(1∕ε) in space due to the singular perturbation of the wave operator in KGZ and/or the incompatibility of the initial data. Based on a multiscale decomposition by frequency and amplitude, we propose a multiscale time integrator Fourier pseudospectral method by applying the Fourier spectral discretization for spatial derivatives followed by using the exponential wave integrator in phase space for integrating the decomposed system at each time step. The method is explicit and easy to be implemented. Extensive numerical results show that the MTI-FP method converges optimally in both space and time, with exponential and quadratic convergence rate, respectively, which is uniformly for ε∈(0,1]. Finally, the method is applied to study the convergence rates of the KGZ system to its limiting models in the subsonic limit and wave dynamics and interactions of the KGZ system in 2D.