System Of Nonlinear Wave Equations Research Articles

A self-consistent regime of generation of terahertz radiation by an optical pulse with a tilted intensity front

We derived a self-consistent system of nonlinear wave equations describing the terahertz generation in dielectric uniaxial crystals by optical pulsed radiation with a tilted wavefront. The numerical analysis of the system of equations showed that the generation of a broadband one-period terahertz signal is accompanied by a red shift of the carrier frequency of the optical pulse, the magnitude of the shift being proportional to the pulse intensity. The generation efficiency with respect to energy reached a maximum at a certain distance of propagation in the crystal, after which the efficiency decreased. A satisfactory agreement was obtained between theoretical calculations and experimental data of other investigations.

Global solutions and finite time blow up for some system of nonlinear wave equations

An initial boundary value problem for systems of semi-linear wave equations in a bounded domain is considered. The global existence, uniqueness and blow-up of solutions by energy methods and give some estimates for the lifespan of solutions are proved.

On a nonlinear scattering model

We obtain conditions for the existence and blow-up of global solutions of systems of nonlinear wave equations with compactly supported initial data and critical nonlinearities arising from the scattering theory of electromagnetic waves.

Global existence and decay of energy to systems of wave equations with damping and supercritical sources

This paper is concerned with a system of nonlinear wave equations with supercritical interior and boundary sources and subject to interior and boundary damping terms. It is well-known that the presence of a nonlinear boundary source causes significant difficulties since the linear Neumann problem for the single wave equation is not, in general, well-posed in the finite-energy space H1(Ω) × L2(∂Ω) with boundary data from L2(∂Ω) (due to the failure of the uniform Lopatinskii condition). Additional challenges stem from the fact that the sources considered in this article are non-dissipative and are not locally Lipschitz from H1(Ω) into L2(Ω) or L2(∂Ω). With some restrictions on the parameters in the system and with careful analysis involving the Nehari Manifold, we obtain global existence of a unique weak solution and establish (depending on the behavior of the dissipation in the system) exponential and algebraic uniform decay rates of energy. Moreover, we prove a blow-up result for weak solutions with nonnegative initial energy.

Periodic and solitary wave solutions of coupled nonlinear wave equations using the first integral method

In this paper, the periodic and solitary wave solutions of two selected systems of nonlinear wave equations are obtained using the first integral method. The long–short-wave interaction system and Bogoyavlenskii equations are considered as examples, and this approach can also be applied to other coupled nonlinear differential equations.

On decay and blow-up of solutions for a system of nonlinear wave equations

The initial boundary value problem for a system of viscoelastic wave equations of Kirchhoff type with the nonlinear damping and the source terms in a bounded domain is considered. We prove that, under suitable conditions on the nonlinearity of the damping and the source terms and certain initial data in the stable set and for a wider class of relaxation functions, the decay estimates of the energy function is exponential or polynomial depending on the exponents of the damping terms in both equations by using Nakao’s method. Conversely, for certain initial data in the unstable set, we obtain the blow-up of solutions in finite time when the initial energy is nonnegative. This improves earlier results in the literature.

Global existence and decay of solutions of a nonlinear system of wave equations

This work is concerned with a system of two wave equations with nonlinear damping and source terms acting in both equations. Under some restrictions on the nonlinearity of the damping and the source terms, we show that our problem has a unique local solution. Also, we prove that, for some restrictions on the initial data, the rate of decay of the total energy is exponential or polynomial depending on the exponents of the damping terms in both equations.

On exact controllability of networks of nonlinear elastic strings in 3-dimensional space

This paper concerns a system of nonlinear wave equations describing the vibrations of a 3-dimensional network of elastic strings. The authors derive the equations and appropriate nodal conditions, determine equilibrium solutions, and, by using the methods of quasilinear hyperbolic systems, prove that for tree networks the natural initial, boundary value problem has classical solutions existing in neighborhoods of the “stretched” equilibrium solutions. Then the local controllability of such networks near such equilibrium configurations in a certain specified time interval is proved. Finally, it is proved that, given two different equilibrium states satisfying certain conditions, it is possible to control the network from states in a small enough neighborhood of one equilibrium to any state in a suitable neighborhood of the second equilibrium over a sufficiently large time interval.

EXACT CONTROLLABILITY FOR A SYSTEM OF SEMILINEAR WAVE EQUATIONS

The exact controllability for a system of interacting nonlinear wave equations, with controls in the interior and on a portion of the boundary, is established. Closed and open loop problems are considered and feedback laws for the closed loop problem are established.

The lifespan for 3D quasilinear wave equations with nonlinear damping terms

In this paper, we study the initial-boundary value problem for a system of nonlinear wave equations, involving nonlinear damping terms, in a bounded domain Ω . The nonexistence of global solutions is discussed under some conditions on the given parameters. Estimates on the lifespan of solutions are also given. Our results extend and generalize the recent results in [K. Agre, M.A. Rammaha, System of nonlinear wave equations with damping and source terms, Differential Integral Equations 19 (2006) 1235–1270], especially, the blow-up of weak solutions in the case of non-negative energy.

On a system of nonlinear wave equations associated with the helical flows of Maxwell fluid

The paper is devoted to the study a system of nonlinear wave equations associated with the mixed nonhomogeneous conditions. Existence of a weak solution is proved by using the Faedo–Galerkin method. Uniqueness, regularity and decay properties of solutions are also discussed.

Blow-up for coupled nonlinear wave equations with damping and source

We consider the system of nonlinear wave equations { u t t + u t + | u t | m − 1 u t = div ( ρ 1 ( | ∇ u | 2 ) ∇ u ) + f 1 ( u , v ) , ( x , t ) ∈ Ω × ( 0 , T ) , v t t + v t + | v t | r − 1 v t = div ( ρ 2 ( | ∇ v | 2 ) ∇ v ) + f 2 ( u , v ) , ( x , t ) ∈ Ω × ( 0 , T ) , with initial and Dirichlet boundary conditions. Under some suitable assumptions on the functions f 1 , f 2 , ρ 1 , ρ 2 , parameters r , m and the initial data, the result on blow-up of solutions and upper bound of blow-up time are given.

Global solutions of the evolutionary Faddeev model with small initial data

We consider the Cauchy problem for evolutionary Faddeev model corresponding to maps from the Minkowski space ℝ1+n to the unit sphere $$ \mathbb{S} $$ 2, which obey a system of non-linear wave equations. The nonlinearity enjoys the null structure and contains semi-linear terms, quasi-linear terms and unknowns themselves. We prove that the Cauchy problem is globally well-posed for sufficiently small initial data in Sobolev space.

Global existence for nonlinear system of wave equations in 3-D domains

We study the initial-boundary problem for a nonlinear system of wave equations with Hamilton structure under Dirichlet's condition. We use the local-in-time Strichartz estimates from [Burq et al., J. Amer. Math. Soc. 21 (2008), 831–845], Morawetz&#8

Global Nonexistence of Positive Initial‐Energy Solutions for Coupled Nonlinear Wave Equations with Damping and Source Terms

This work is concerned with a system of nonlinear wave equations with nonlinear damping and source terms acting on both equations. We prove a global nonexistence theorem for certain solutions with positive initial energy.

Critically and degenerately damped systems of nonlinear wave equations with source terms

This article is concerned with the global well-posedness of the critically and degenerately damped system of nonlinear wave equations in a bounded domain Ω ⊂ ℝ n , n = 1, 2, 3, with Dirichlét boundary conditions. The nonlinearities f 1(u, v) and f 2(u, v) act as a strong source in the system. Under some restriction on the parameters in the system we obtain several results concerning the existence of local solutions, global solutions, uniqueness and the blow up in finite time.

The extended tanh method for solving systems of nonlinear wave equations

The extended tanh method with a computerized symbolic computation is used for constructing the traveling wave solutions of coupled nonlinear equations arising in physics. The obtained solutions include solitons, kinks and plane periodic solutions. The applied method will be used to solve the generalized coupled Hirota Satsuma KdV equation.

The exponential decay of a system of nonlinear wave equations

The exponential decay of a system of nonlinear wave equations with initial boundary values is considered. We have some sufficient conditions that ensure that the energy admits exponential decay by a compactness uniqueness argument and the energy estimates.

Finite-Energy Global Well-Posedness of the Maxwell–Klein–Gordon System in Lorenz Gauge

It is known that the Maxwell–Klein–Gordon system (M–K–G), when written relative to the Coulomb gauge, is globally well-posed for finite-energy initial data. This result, due to Klainerman and Machedon, relies crucially on the null structure of the main bilinear terms of M–K–G in Coulomb gauge. It appears to have been believed that such a structure is not present in Lorenz gauge, but we prove here that it is, and we use this fact to prove finite-energy global well-posedness in Lorenz gauge. The latter has the advantage, compared to Coulomb gauge, of being Lorentz invariant, hence M–K–G in Lorenz gauge is a system of nonlinear wave equations, whereas in Coulomb gauge the system has a less symmetric form, as it contains also an elliptic equation.

ON COUPLED NONLINEAR WAVE EQUATIONS OF KIRCHHOFF TYPE WITH DAMPING AND SOURCE TERMS

The initial boundary value problem for a system of nonlinear wave equations of Kirchhoff type with strong damping in a bounded domain is considered. The existence, asymptotic behavior and blow-up of solutions are discussed under some conditions. The decay estimates of the energy function and the estimates for the lifespan of solutions are given.