System Of Wave Equations Research Articles

Approximate boundary synchronization by groups for a coupled system of wave equations with coupled Robin boundary conditions

In this paper, we first give an algebraic characterization of uniqueness of continuation for a coupled system of wave equations with coupled Robin boundary conditions. Then, the approximate boundary controllability and the approximate boundary synchronization by groups for a coupled system of wave equations with coupled Robin boundary controls are developed around this fundamental characterization.

Exact boundary controllability and exact boundary synchronization for a coupled system of wave equations with coupled Robin boundary controls

In this paper, we consider the exact boundary controllability and the exact boundary synchronization (by groups) for a coupled system of wave equations with coupled Robin boundary controls. Owing to the difficulty coming from the lack of regularity of the solution, we confront a bigger challenge than that in the case with Dirichlet or Neumann boundary controls. In order to overcome this difficulty, we use the regularity results of solutions to the mixed problem with Neumann boundary conditions by Lasiecka and Triggiani [J. Differ. Equ. 94 (1991) 112–164] to get the regularity of solutions to the mixed problem with coupled Robin boundary conditions. Thus we show the exact boundary controllability of the system, and by a method of compact perturbation, we obtain the non-exact boundary controllability of the system with fewer boundary controls on some special domains. Based on this, we further study the exact boundary synchronization (by groups) for the same system, the determination of the exactly synchronizable state (by groups), as well as the necessity of the compatibility conditions of the coupling matrices.

Asymptotic synchronization of a coupled system of wave equations on a rectangular domain with reflecting sides

We show the sufficiency of Kalman’s rank condition for the uniqueness of solution to a coupled system of wave equations in a rectangular domain. The approach does not need any gap condition on the spectrum of the differential operator and the usual multiplier geometrical condition. Then, the study on the asymptotic synchronization by groups can be improved for the corresponding system.

Uniform Synchronization of an Abstract Linear Second Order Evolution System

Although mathematical study on the synchronization of wave equations at the finite horizon has been well developed, there have been few results on the synchronization of wave equations for the long-time horizon. The aim of the paper is to investigate the uniform synchronization at the infinite horizon for one abstract linear second order evolution system in a Hilbert space. First, using the classical compact perturbation theory on the uniform stability of semigroups of contractions, we will establish a lower bound on the number of damping, necessary for the uniform synchronization of the considered system. Then, under the minimum number of damping, we clarify the algebraic structure of the system as well as the necessity of the conditions of compatibility on the coupling matrices. We then establish the uniform synchronization by the compact perturbation method and then give the dynamics of the asymptotic orbit. Various applications are given for the system of wave equations with boundary feedback or (and) locally distributed feedback, and for the system of Kirchhoff plate with distributed feedback. Some open questions are raised at the end of the paper for future development. The study is based on the synchronization theory and the compact perturbation of semigroups.

Generalized approximate boundary synchronization for a coupled system of wave equations

In this paper, we consider the generalized approximate boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls. We analyze the relationship between the generalized approximate boundary synchronization and the generalized exact boundary synchronization, give a sufficient condition to realize the generalized approximate boundary synchronization and a necessary condition in terms of Kalman's matrix, and show the meaning of the number of total controls. Besides, by the generalized synchronization decomposition, we define the generalized approximately synchronizable state and obtain its properties and a sufficient condition for it to be independent of applied boundary controls.

New general decay result for a system of viscoelastic wave equations with past history

<p style='text-indent:20px;'>This work is concerned with a coupled system of viscoelastic wave equations in the presence of infinite-memory terms. We show that the stability of the system holds for a much larger class of kernels. More precisely, we consider the kernels <inline-formula><tex-math id="M1">\begin{document}$ g_i : [0, +\infty) \rightarrow (0, +\infty) $\end{document}</tex-math></inline-formula> satisfying <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ g_i'(t)\leq-\xi_i(t)H_i(g_i(t)),\qquad\forall\,t\geq0 \quad\mathrm{and\ for\ }i = 1,2, $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ \xi_i $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ H_i $\end{document}</tex-math></inline-formula> are functions satisfying some specific properties. Under this very general assumption on the behavior of <inline-formula><tex-math id="M4">\begin{document}$ g_i $\end{document}</tex-math></inline-formula> at infinity, we establish a relation between the decay rate of the solutions and the growth of <inline-formula><tex-math id="M5">\begin{document}$ g_i $\end{document}</tex-math></inline-formula> at infinity. This work generalizes and improves earlier results in the literature. Moreover, we drop the boundedness assumptions on the history data, usually made in the literature.

On initial-boundary value problems for one-dimension semilinear wave equations with null conditions

We consider the initial-boundary value problem on R+×R+ for one-dimension systems of semilinear wave equations with null conditions. We show that for homogeneous Dirichlet or Neumann boundary values and sufficiently small initial data, classical solutions always globally exist. Furthermore, we also prove that the global solution is asymptotically free in the energy sense.

On the Synchronizable System

In this paper, the synchronizable system is defined and studied for a coupled system of wave equations with the same wave speed or with different wave speeds.

Dynamics of a coupled system for nonlinear damped wave equations with variable exponents

AbstractWe consider a coupled system of viscoelastic wave equation with weak, strong damping and power nonlinearity. For a single viscoelastic wave equation, we have already obtained a global solution, its decay rate and finite‐time blow‐up [1]. In this paper, we extend these results to a coupled system. First, we obtain a global solution and derive its decay rate by a decreasing energy. Finally, we apply the concavity method in order to show that the solution blows up in finite time under nonclassic constraint on kernel functions.

The influence of the coefficients of a system of wave equations coupled by velocities on its stabilization

In this work, we consider a system of two wave equations coupled by velocities in a one-dimensional space, with one boundary fractional damping. First, we show that the system is strongly asymptotically stable if and only if the coupling parameter b of the two equations is outside a discrete set of exceptional real values. Next, we show that our system is not uniformly stable. Hence, we look for a polynomial decay rate for smooth initial data. Using a frequency domain approach combined with the multiplier method, we prove that the energy decay rate is greatly influenced by the nature of the coupling parameter b, the arithmetic property of the wave propagation speed a and the order of the fractional damping $$\alpha $$ . Indeed, under the equal speed propagation condition, i.e., $$a=1$$ , we establish an optimal polynomial energy decay rate of type $$t^{-\frac{2}{{1-\alpha }}}$$ if the coupling parameter $$b\notin \pi {\mathbb {Z}}$$ and of type $$t^{-\frac{2}{{5-\alpha }}}$$ if the coupling parameter $$b\in \pi {\mathbb {Z}}$$ . Furthermore, when the wave propagates with different speeds, i.e., $$a\not =1$$ , we prove that, for any rational number $$\sqrt{a}$$ and almost all irrational numbers $$\sqrt{a}$$ , the energy of our system decays polynomially to zero like as $$t^{-\frac{2}{{5-\alpha }}}$$ . This result still holds if $$a\in {\mathbb {Q}}$$ , $$\sqrt{a}\notin {\mathbb {Q}}$$ and b small enough.

Nonexistence of global solutions for a weakly coupled system of semilinear damped wave equations in the scattering case with mixed nonlinear terms

In this paper we consider the blow-up of solutions to a weakly coupled system of semilinear damped wave equations in the scattering case with nonlinearities of mixed type. The proof of the blow-up results is based on an iteration argument. We find as critical curve for the pair of exponents (p, q) in the nonlinear terms the same one found for the weakly coupled system of semilinear wave equations with the same kind of nonlinearities. In the critical and not-damped case we combine an iteration argument with the so-called slicing method to show the blow-up dynamic of a weighted version of the functionals used in the subcritical case.

Blow-up for a coupled system of semilinear wave equations with scattering dampings and combined nonlinearities

The blow-up of solutions to a coupled system of semilinear wave equations in the sub-critical and critical cases is investigated. The system contains scattering dampings and combined nonlinearities. The upper bound lifespan estimates of solutions which are related to the indexes of nonlinear terms are established by applying the functional method and iteration technique.

Time fourth-order energy-preserving AVF finite difference method for nonlinear space-fractional wave equations

In this paper, we develop and analyze a new time fourth-order energy-preserving average vector field (AVF) finite difference method for the nonlinear fractional wave equations with Riesz space-fractional derivative. To the corresponding Hamiltonian system of the nonlinear fractional wave equations, the fourth-order weighted and shifted Lubich difference operator in space and the fourth-order AVF method in time are used to develop the time fourth-order energy-preserving AVF finite difference method for the nonlinear fractional wave equations. The energy conservation in the discrete form and unique solvability of the proposed scheme are proved and error estimates of the scheme are further proved to be order of O((Δt)4+h4) in the discrete L2- norm. Numerical experiments confirm energy conservation and high-order accuracy of the proposed scheme.

A Lorentz-covariant interacting electron–photon system in one space dimension

A Lorenz-covariant system of wave equations is formulated for a quantum-mechanical two-body system in one space dimension, comprised of one electron and one photon. Manifest Lorentz covariance is achieved using Dirac's formalism of multi-time wave functions, i.e., wave functions $\Psi(\mathbf{x}_{{ph}},\mathbf{x}_{{el}})$ where $\mathbf{x}_{{el}},\mathbf{x}_{{ph}}$ are the generic spacetime events of the electron and photon, respectively. Their interaction is implemented via a Lorentz-invariant no-crossing-of-paths boundary condition at the coincidence submanifold $\{\mathbf{x}_{{el}}=\mathbf{x}_{{ph}}\}$, compatible with particle current conservation. The corresponding initial-boundary-value problem is proved to be well-posed. Electron and photon trajectories are shown to exist globally in a Hypersurface Bohm--Dirac theory, for typical particle initial conditions. Also presented are the results of some numerical experiments which illustrate Compton scattering as well as a new phenomenon: photon capture and release by the electron.

Interplay effects on blow-up of weakly coupled systems for semilinear wave equations with general nonlinear memory terms

In this paper, we study weakly coupled systems for semilinear wave equations with distinct nonlinear memory terms in general forms, and the corresponding single semilinear equation with general nonlinear memory terms. Thanks to fixed point theory, we prove local (in time) existence of solutions with the L1 assumption on the memory kernels. Then, blow-up results for energy solutions are derived by applying iteration methods associated with slicing procedure. We investigate interactions on the blow-up conditions under different decreasing assumptions on the memory term. Particularly, a new threshold for the kernels on interplay effect is found. Additionally, we give some applications of our results on semilinear wave equations and acoustic wave equations.

Influence of the hidden regularity on the stability of partially damped systems of wave equations

We consider the stability of a system of two wave equations with only one boundary feedback and we show that the stability of the partially damped system depends on the transmission of energy between the two equations. The study confirms that the hidden regularity is an essential ingredient for the stability property. In particular, using a sharp regularity for Neumann problem of wave equation, we improve the usual results on the energy decay rate. This new approach can certainly be applied to other situations of partially damped systems.

On Agemi-type structural conditions for a system of semilinear wave equations

We consider a two-component system of cubic semilinear wave equations in two space dimensions satisfying the Agemi-type structural condition (Ag) but violating (Ag0) and (Ag[Formula: see text]). For this system, we show that small amplitude solutions are asymptotically free as [Formula: see text].

Permittivity Gradient Induced Depolarization: Electromagnetic Propagation With the Maxwell Vector Wave Equation

Recent interest in 3-D vectorial sensors requires the development of vectorial propagation methods, rather than scalar wave equation approaches. We derive a vector wave equation from Maxwell’s equations for a medium which has an inhomogeneous dielectric permittivity dominated by variation along one dimension. It is well known that the electric field components decouple for homogeneous media. However, 1-D permittivity variations yield an upper triangular system of scalar wave equations with the wave polarization component parallel to the inhomogeneous direction/axis acting as a forcing term for the orthogonal components. The main implication is that waves with polarization oriented parallel to the permittivity gradient will act as a forcing term and excite other polarization components and, thus, induce depolarization. Contemporary studies treat the permittivity as a constant when deriving a wave equation or paraxial approximation, and then re-introduce via inhomogeneous wave speed, variable permittivity, thus missing important terms and physical mechanisms in their resulting equations. Contemporary studies neglect the term in the Maxwell vector wave equation responsible for this effect. Application of the electromagnetic propagation depolarization effect is demonstrated numerically for an air–sea interface evaporation duct with a 500 MHz source.

Dispersive soliton solutions for shallow water wave system and modified Benjamin-Bona-Mahony equations via applications of mathematical methods

We have utilized three novel methods, called generalized direct algebraic, modified F-expansion and improved simple equation methods to construct traveling wave solutions of the system of shallow water wave equations and modified Benjamin-Bona-Mahony equation. After substituting particular values of the parameters, different solitary wave solutions are derived from the exact traveling wave solutions. It is shown that these employed methods are more powerful tools for nonlinear wave equations.

3D elastic waveform modeling with an optimized equivalent staggered-grid finite-difference method

Equivalent staggered-grid (ESG) as a new family of schemes has been utilized in seismic modeling, imaging, and inversion. Traditionally, the Taylor series expansion is often applied to calculate finite-difference (FD) coefficients on spatial derivatives, but the simulation results suffer serious numerical dispersion on a large frequency zone. We develop an optimized equivalent staggered-grid (OESG) FD method that can simultaneously suppress temporal and spatial dispersion for solving the second-order system of the 3D elastic wave equation. On the one hand, we consider the coupling relations between wave speeds and spatial derivatives in the elastic wave equation and give three sets of FD coefficients with respect to the P-wave, S-wave, and converted-wave (C-wave) terms. On the other hand, a novel plane wave solution for the 3D elastic wave equation is derived from the matrix decomposition method to construct the time–space dispersion relations. FD coefficients of the OESG method can be acquired by solving the new dispersion equations based on the Newton iteration method. Finally, we construct a new objective function to analyze P-wave, S-wave, and C-wave dispersion concerning frequencies. The dispersion analyses show that the presented method produces less modeling errors than the traditional ESG method. The synthetic examples demonstrate the effectiveness and superiority of the presented method.