Coupled System Of Wave Equations Research Articles

The Robin Problems in the Coupled System of Wave Equations on a Half-Line

This article investigates the local well-posedness of a coupled system of wave equations on a half-line, with a particular emphasis on Robin boundary conditions within Sobolev spaces. We provide estimates for the solutions to linear initial-boundary-value problems related to the coupled system of wave equations, utilizing the Unified Transform Method in conjunction with the Hadamard norm while considering the influence of external forces. Furthermore, we demonstrate that replacing the external force with a nonlinear term alters the iteration map defined by the unified transform solutions, making it a contraction map in a suitable solution space. By employing the contraction mapping theorem, we establish the existence of a unique solution. Finally, we show that the data-to-solution map is locally Lipschitz continuous, thus confirming the local well-posedness of the coupled system of wave equations under consideration.

Periodic solutions for a coupled system of wave equations with x-dependent coefficients

Abstract This paper is concerned with the periodic solutions for a coupled system of wave equations with x-dependent coefficients. Such a model arises naturally when two waves propagate simultaneously in the nonisotrpic media. In this paper, for the periods having the form T = 2 a − 1 b ( a , b are positive integers ) $T=\frac{2a-1}{b}\left(a,b \text{are\,positive\,integers}\right)$ and some types of boundary conditions, we obtain the existence of the time periodic solutions and analyze the asymptotic behaviors as the coupled parameter goes to zero, when the nonlinearities are superlinear and monotone, by using the variational method. In particular, the condition ess inf η ϱ (x) > 0 is not required.

Approximate internal controllability and synchronization of a coupled system of wave equations

Based on the uniqueness of solution to a coupled system of wave equations associated with incomplete internal observations, we establish the approximate internal synchronization by groups, the induced internal synchronization and the approximate internal synchronization in the pinning sense.

D-Observability for a Coupled System of Wave Equations on Right Angle Triangles

Based on the tiled method, we establish the equivalence between the D-observability for diagonalizable systems on the rectangular domain and the right angle triangle domain, and then get the sufficiency of Kalman’s rank condition to the D-observability for diagonalizable systems on some right angle triangles.

Stability of exactly synchronizable state by groups for a coupled system of wave equations with respect to applied controls

Stability of exactly synchronizable state by groups for a coupled system of wave equations with respect to applied controls

A note on the family of synchronizations for a coupled system of wave equations

A note on the family of synchronizations for a coupled system of wave equations

Multiplicity results of periodic solutions for a coupled system of wave equations

Multiplicity results of periodic solutions for a coupled system of wave equations

On the partially synchronizable system for a coupled system of wave equations with different wave speeds

On the partially synchronizable system for a coupled system of wave equations with different wave speeds

Approximate mixed synchronization by groups for a coupled system of wave equations

We first show that under a suitable balanced repartition of the mixed controls within the system, Kalman’s rank condition is still necessary and sufficient for the uniqueness of solution to the adjoint system associated with incomplete internal and boundary observations, therefore for the approximate controllability of the primary system by means of mixed controls. Then we study the stability of the approximately synchronizable state by groups with respect to applied controls.

Exact Internal Controllability and Synchronization for a Coupled System of Wave Equations

Exact Internal Controllability and Synchronization for a Coupled System of Wave Equations

Exactly Synchronizable State and Approximate Controllability for a Coupled System of Wave Equations with Locally Distributed Controls

For a coupled system of wave equations with locally distributed controls, we prove that the exactly synchronizable state is independent of applied controls and then can be uniquely determined by the initial data if and only if the system is not approximately controllable or, equivalently, if and only if Kalman’s rank condition (3.2) holds.

Blow-up for the coupled system of wave equations with memory terms in Schwarzschild spacetime

This paper is concerned with blow-up results of solutions to the Cauchy problem for a coupled system of wave equations in Schwarzschild spacetime, where the nonlinear terms contain power-type memory terms, derivative-type memory terms, and combined-type memory terms, respectively. Upper-bound lifespan estimates of solutions to the Cauchy problem with small initial values and certain assumptions on the exponents in the memory terms are established by taking advantage of the Regge–Wheeler coordinate and iteration approach. Our main new contribution is that lifespan estimates of solutions to the problem are associated with the well-known Strauss conjecture and Glassey conjecture in Schwarzschild spacetime. To the best of the authors’ knowledge, the results in Theorems 1.1–1.3 are new.

Algebraic Conditions to the Uniqueness Theorem for a Coupled System of Wave Equations

We propose a new rank condition on the coupling matrices, which, together with Kalman’s classic rank condition, can guarantee the uniqueness of a solution for a large class of coupled systems of wave equations with incomplete observations. We then establish the approximate controllability and the approximate synchronization of the underlying system in the general situation.

On a partially synchronizable system for a coupled system of wave equations in one dimension

<abstract><p>In this paper, we study a partially synchronizable system for a coupled system of wave equations with different wave speeds in the framework of classical solutions in one dimensional. A partially synchronizable system is defined as a system with at least one partial synchronized solutions. In fact, we cannot consider partial synchronization as the case that the system has the same wave speeds, because the influence of different wave speeds cause only some of the function in a given space being in a partially synchronized state, rather than all functions. Therefore, we can only consider under what conditions the coupled system can have partially synchronized solutions. We will consider it in two ways. On the one hand, under the necessary conditions, we obtain an unclosed characteristic equation associated with the partially synchronizable state. We add conditions to the wave speed matrix and coupling matrix to make the equation closed. From this, the characteristic function can be obtained, and all partially synchronized solutions are obtained; then we obtain the conditions under which the initial value should be satisfied. On the other hand, we consider a system of three variables first, where there are only two synchronized variables. By subtracting them to obtain a new variable, the problem can be transformed into the problem wherein the system that satisfies the new variable should have only zero solutions. Then solving this problem can lead to obtaining the conditions required for a partially synchronized solution. After extending it to the case of $ N $ variables, similar conclusions can be obtained.</p></abstract>

Uniqueness theorem for a coupled system of wave equations with incomplete internal observation and application to approximate controllability

Uniqueness theorem for a coupled system of wave equations with incomplete internal observation and application to approximate controllability

A Note on the Indirect Controls for a Coupled System of Wave Equations

A Note on the Indirect Controls for a Coupled System of Wave Equations

Lifespan Estimates of Solutions for a Coupled System of Wave Equations with Damping terms and Negative Mass terms∗

The main purpose of this paper is to study the formation of singularity for a coupled system of wave equations with damping terms, negative mass terms, and divergence form nonlinearities. Upper bound lifespan estimates of solutions to the system are obtained by using the iteration method. The results are the same as the corresponding coupled system of the wave equation with power nonlinearities |v|pand |u|q. To the best of our knowledge, the results in Theorems 1–5 are new. In addition, the variation trend of the wave is analyzed by using numerical simulation.

Exact boundary synchronization for a kind of first order hyperbolic system

In recent years there have been many in-depth researches on the boundary controllability and boundary synchronization for coupled systems of wave equations with various types of boundary conditions. In order to extend the study of synchronization from wave equations to a much larger range of hyperbolic systems, in this paper we will define and establish the exact boundary synchronization for the first order linear hyperbolic system based on previous work on its exact boundary controllability. The determination and estimate of exactly synchronizable states and some related problems are also discussed. This work can be applied to a great deal of diverse systems, and a new perspective to study the synchronization problem for the coupled system of wave equations can be also provided.

Eikonal quasinormal modes of black holes beyond general relativity. III. Scalar Gauss-Bonnet gravity

In a recent series of papers, we have shown how the eikonal/geometrical optics approximation can be used to calculate analytically the fundamental quasinormal mode frequencies associated with coupled systems of wave equations, which arise, for instance, in the study of perturbations of black holes in gravity theories beyond General Relativity. As a continuation to this series, we focus here on the quasinormal modes of nonrotating black holes in scalar Gauss-Bonnet gravity assuming a small-coupling expansion. We show that the axial perturbations are purely tensorial and are described by a modified Regge-Wheeler equation, while the polar perturbations are of mixed scalar-tensor character and are described by a system of two coupled wave equations. When applied to these equations, the eikonal machinery leads to axial quasinormal modes that deviate from the general relativistic results at quadratic order in the Gauss-Bonnet coupling constant. We show that this result is in agreement with an analysis of unstable circular null orbits around black holes in this theory, allowing us to establish the geometrical optics--null geodesic correspondence for the axial quasinormal modes. For the polar quasinormal modes, the small-coupling approximation forces us to consider the ordering between eikonal and small-coupling perturbative parameters, one of which we show, by explicit comparison against numerical data, yields the correct identification of the quasinormal modes of the scalar-tensor coupled system of wave equations. These corrections lift the general relativistic degeneracy between scalar and tensorial eikonal quasinormal modes at quadratic order in Gauss-Bonnet coupling in a way reminiscent of the Zeeman effect. In general, our analytic, eikonal quasinormal mode frequencies (normalized to the General Relativity ones) agree with numerical results with an error of $\mathcal{O}(10%)$ in the regime of small coupling constant. Finally, we find that the analytical expressions for the quasinormal modes are common to a broad class of scalar-Gauss-Bonnet theories to leading eikonal order, showing a degeneracy between the quasinormal modes of nonrotating black holes in particular scalar-Gauss-Bonnet theories in the geometrical optics limit.

Fractional boundary stabilization for a coupled system of wave equations

In this paper,we consider a system of a coupled wave equations in the presence of a boundary control of fractional derivative type. We prove well-posedness by using the semigroup theory. Also we establish an optimal decay result by frequency domain method and Borichev-Tomilov theorem.