System Of Nonlinear Wave Equations Research Articles

The null condition for systems of nonlinear wave equations and its related topics

In this article, we consider the Cauchy problem for systems of nonlinear wave equations, whose nonlinear terms depend mainly on derivatives of the unknown, with small, smooth, and compactly supported initial data. If the nonlinear terms have the critical power, we need some structural conditions to obtain the small data global existence. The null condition is one such condition, but recently some weaker conditions were also found. We discuss the small data global existence and the asymptotic behavior of solutions under these conditions. The corresponding result for nonlinear Schrödinger equations will be also discussed.

Existence and decay of solutions to coupled systems of nonlinear wave equations with variable exponents

In this article, we consider a coupled system of two hyperbolic equations with variable exponents in the damping and source terms, where the dampings are modilated with time-dependent coefficients \(\alpha(t), \beta(t)\). First, we state and prove an existence result of a global weak solution, using Galerkin's method with compactness arguments. Then, by a Lemma due to Martinez, we establish the decay rates of the solution energy, under suitable assumptions on the variable exponents \(m\) and \(r\) and the coefficients \( \alpha\) and \(\beta\). To illustrate our theoretical results, we give some numerical examples.
 For more information see https://ejde.math.txstate.edu/Volumes/2023/73/abstr.html

Global existence for quasilinear wave equations exterior to star-shaped regions with inhomogeneous boundary conditions

This paper establishes the global existence of classical solutions to systems of nonlinear wave equations with multiple speeds outside star-shaped regions satisfying time-independent inhomogeneous boundary conditions, provided that nonlinear terms obey the null condition. We first prove the existence and uniqueness of stationary solutions. Then we use the space-time estimates for perturbed wave equations in Metcalfe and Sogge (2007) to show that solutions of systems converge to stationary solutions as time goes to infinity.

Global stability of traveling waves for (1 + 1)-dimensional systems of quasilinear wave equations

A key feature of [Formula: see text]-dimensional nonlinear wave equations is that they admit left or right traveling waves, under appropriate algebraic conditions on the nonlinearities. In this paper, we prove global stability of such traveling wave solutions for [Formula: see text]-dimensional systems of nonlinear wave equations, given a certain asymptotic null condition and sufficient decay for the traveling wave. We first consider semilinear systems as a simpler model problem; we then proceed to treat more general quasilinear systems.

Peeling in generalized harmonic gauge

It is shown that a large class of systems of non-linear wave equations, based on the good–bad–ugly model, admit formal solutions with polyhomogeneous expansions near null infinity. A particular set of variables is introduced which allows us to write the Einstein field equations in generalized harmonic gauge (GHG) as a good–bad–ugly system and the functional form of the first few orders in such an expansion is found by applying the aforementioned result. Exploiting these formal expansions of the metric components, the peeling property of the Weyl tensor is revisited. The question addressed is whether or not the use of GHG, by itself, causes a violation of peeling. Working in harmonic gauge, it is found that log-terms that prevent the Weyl tensor from peeling do appear. The impact of gauge source functions and constraint additions on the peeling property is then considered. Finally, the special interplay between gauge and constraint addition, as well as its influence on the asymptotic system and the decay of each of the metric components, is exploited to find a particular gauge which suppresses this specific type of log-term to arbitrarily high order.

Blow up of negative initial-energy solutions of a system of nonlinear wave equations with variable-exponent nonlinearities

<p style='text-indent:20px;'>This work is concerned with a system of wave equations with variable-exponent nonlinearities acting in both equations. We, first, discuss the well-posedness then prove a blow up result for solutions with negative initial energy.</p>

Global existence and blow up for systems of nonlinear wave equations related to the weak null condition

<p style='text-indent:20px;'>We discuss how the higher-order term <inline-formula><tex-math id="M1">\begin{document}$ |u|^q $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M2">\begin{document}$ (q>1+2/(n-1)) $\end{document}</tex-math></inline-formula> has nontrivial effects in the lifespan of small solutions to the Cauchy problem for the system of nonlinear wave equations</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \partial_t^2 u-\Delta u = |v|^p, \qquad \partial_t^2 v-\Delta v = |\partial_t u|^{(n+1)/(n-1)} +|u|^q $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>in <inline-formula><tex-math id="M3">\begin{document}$ n\,(\geq 2) $\end{document}</tex-math></inline-formula> space dimensions. We show the existence of a certain "critical curve" in the <inline-formula><tex-math id="M4">\begin{document}$ pq $\end{document}</tex-math></inline-formula>-plane such that for any <inline-formula><tex-math id="M5">\begin{document}$ (p,q) $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M6">\begin{document}$ (p,q>1) $\end{document}</tex-math></inline-formula> lying below the curve, nonexistence of global solutions occurs, whereas for any <inline-formula><tex-math id="M7">\begin{document}$ (p,q) $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M8">\begin{document}$ (p>1+3/(n-1),\,q>1+2/(n-1)) $\end{document}</tex-math></inline-formula> lying exactly on it, this system admits a unique global solution for small data. When <inline-formula><tex-math id="M9">\begin{document}$ n = 3 $\end{document}</tex-math></inline-formula>, the discussion for the above system with <inline-formula><tex-math id="M10">\begin{document}$ (p,q) = (3,3) $\end{document}</tex-math></inline-formula>, which lies on the critical curve, has relevance to the study on systems satisfying the weak null condition, and we obtain a new result of global existence for such systems. Moreover, in the particular case of <inline-formula><tex-math id="M11">\begin{document}$ n = 2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M12">\begin{document}$ p = 4 $\end{document}</tex-math></inline-formula> it is observed that no matter how large <inline-formula><tex-math id="M13">\begin{document}$ q $\end{document}</tex-math></inline-formula> is, the higher-order term <inline-formula><tex-math id="M14">\begin{document}$ |u|^q $\end{document}</tex-math></inline-formula> never becomes negligible and it essentially affects the lifespan of small solutions.</p>

High order asymptotic expansions of a good–bad–ugly wave equation

A heuristic method to find asymptotic solutions to a system of non-linear wave equations near null infinity is proposed. The non-linearities in this model, dubbed good–bad–ugly, are known to mimic the ones present in the Einstein field equations and we expect to be able to exploit this method to derive an asymptotic expansion for the metric in general relativity close to null infinity that goes beyond first order as performed by Lindblad and Rodnianski for the leading asymptotics. For the good–bad–ugly model, we derive formal expansions in which terms proportional to the logarithm of the radial coordinate appear at every order in the bad field, from the second order onward in the ugly field but never in the good field. The model is generalized to wave operators built from an asymptotically flat metric and it is shown that it admits polyhomogeneous asymptotic solutions. Finally we define stratified null forms, a generalization of standard null forms, which capture the behavior of different types of field, and demonstrate that the addition of such terms to the original system bears no qualitative influence on the type of asymptotic solutions found.

Corrigendum to “Existence, blow‐up, and exponential decay estimates for a system of nonlinear wave equations with nonlinear boundary conditions”

This note presents corrections to our paper. We also give remarks and additions related to other papers.

Quasi-stability and continuity of attractors for nonlinear system of wave equations

Abstract In this paper, we study the long-time behavior of a nonlinear coupled system of wave equations with damping terms and subjected to small perturbations of autonomous external forces. Using the recent approach by Chueshov and Lasiecka in [21], we prove that this dynamical system is quasi-stable by establishing a quasistability estimate, as consequence, the existence of global and exponential attractors is proved. Finally, we investigate the upper and lower semicontinuity of global attractors under autonomous perturbations.

Energy-preserving finite element methods for a class of nonlinear wave equations

In this paper for the first time, two kinds of energy-preserving finite element approximation schemes, which are based upon the standard finite element method (FEM) and the mixed FEM, respectively, are developed and analyzed for a class of nonlinear wave equations. The energy conservation and the optimal convergence properties are obtained for both finite element schemes in their respective norms, additionally, the energy-preserving mixed FEM can produce one-order higher approximation accuracy to the flux (the gradient of the primary unknown) in L2 norm in contrast with that of the standard FEM when the same degree piecewise polynomial is employed to construct their respective finite element spaces, which may likely result in a more accurate and more physical discrete energy conservation. Numerical experiments are carried out to validate all attained theoretical results. Furthermore, the developed energy-preserving finite element methods can be directly applied to the coupled system of nonlinear wave equations, whose energy conservation and optimal convergence properties are also confirmed by our numerical experiments.

Nonexistence of Global Weak Solutions of a System of Nonlinear Wave Equations with Nonlinear Fractional Damping

We consider the system of nonlinear wave equations with nonlinear time fractional dampingutt+−Δmu+CD0,tαtσuq=vp,t>0,x∈ℝN,vtt+−Δmv+CD0,tβtδvr=vs,t>0,x∈ℝN,u0,x,ut0,x=u0x,u1x,x∈ℝN,u0,x,ut0,x=u0x,u1x,x∈ℝN,whereu,v=ut,x,vt,x,mandNare positive natural numbers,p,q,r,s>1,σ,δ≥0,0<α,β<1, and CD0,tκ,0<κ<1, is the Caputo fractional derivative of orderκ. Namely, sufficient criteria are derived so that the system admits no global weak solution. To the best of our knowledge, the considered system was not previously studied in the literature.

The energy-preserving finite difference methods and their analyses for system of nonlinear wave equations in two dimensions

The coupled sine-Gordon (SG) equations and the coupled Klein-Gordon (KG) equations play an important role in scientific fields, such as nonlinear optics, solid state physics, quantum mechanics. As their energies are conservative, it is of importance to develop energy preserving finite difference method (EP-FDM) for these systems of nonlinear wave equations. However, the energy preserving finite difference methods (EP-FDMs) for one-dimensional single sine-Gordon equation and one-dimensional single Klein-Gordon equation, can not directly be generalized to solve the systems of coupled SG or coupled KG equations, and the theoretical analysis technique used for 1D single SG equation or for 1D single KG equation is not suitable for the analysis of high dimensional problems. In this paper, we develop and analyze two kinds of energy preserving FDMs for the systems of coupled SG equations or coupled KG equations in two dimensions. One proposed scheme is a two-level scheme and the other is a three-level scheme. We prove the schemes to satisfy the energy conservations in the discrete forms. By using the fixed point theorem, it is shown that they are solvable. Also, it is further proved that they have the second order convergence rate in both time and space steps. Numerical tests show the performance of the methods and confirm the theoretical findings.

Two-Dimensional Dynamics of Solitons under the Conditions of Zakharov–Benney Resonance

The generation of terahertz radiation by optical pulses under quasi-resonance conditions in a medium of asymmetric quantum particles is considered. A system of nonlinear wave equations describing this process is derived for a noncollinear geometry when the direction of the phase velocity of optical pulses does not coincide with that of the terahertz pulse radiation. Different types of solutions of the resulting system of equation are studied.

Exact equations of state for nematics

We propose a novel approach to the solution of nematic Liquid Crystal models based on the derivation of a system of nonlinear wave equations for order parameters such that the occurrence of uniaxial and biaxial phase transitions can be interpreted as the propagation of a two-dimensional shock wave in the space of thermodynamic parameters. We obtain the exact equations of state for an integrable model of biaxial nematic liquid crystals and show that the classical transition from isotropic to uniaxial phase in absence of external fields is the result of a van der Waals type phase transition, where the jump in the order parameters is a classical shock generated from a gradient catastrophe at a non-zero isotropic field. The study of the equations of state provides the first analytical description of the rich structure of nematics phase diagrams in presence of external fields.

The weak null condition and Kaluza–Klein spacetimes

We prove the non-linear stability of a system of non-linear wave equations satisfying the weak null condition. In particular, this includes the case of the non-linear stability of Minkowski spacetime times a [Formula: see text]-torus subject to perturbations depending only on the non-compact coordinates. Our argument closely follows the proof of the non-linear stability of Minkowski spacetime by Lindblad and Rodnianski.

Numerical Technique for a System of Nonlinear Partial Differential Equations of the Time-Fractional Order

In this article, generalized differential transform method (GDTM) has been employed to obtain traveling wave solutions for some Systems of nonlinear wave equations of time-fractional order. This demonstrates the validity and great potential of the generalized differential transform method (GDTM) for solving system of nonlinear fractional partial differential equations NFDEs. Thus NFDEs can be easily solved with less computational work for any analytic nonlinearity due to the properties and available algorithms of the generalized differential transform method (GDTM). The fractional derivative is described in the Caputo fractional derivative sense. The solutions are obtained in the form of rapidly convergent infinite series with easily computable terms.

New Analytical Technique for Solving a System of Nonlinear Fractional Partial Differential Equations

This paper introduces a new analytical technique (NAT) for solving a system of nonlinear fractional partial differential equations (NFPDEs) in full general set. Moreover, the convergence and error analysis of the proposed technique is shown. The approximate solutions for a system of NFPDEs are easily obtained by means of Caputo fractional partial derivatives based on the properties of fractional calculus. However, analytical and numerical traveling wave solutions for some systems of nonlinear wave equations are successfully obtained to confirm the accuracy and efficiency of the proposed technique. Several numerical results are presented in the format of tables and graphs to make a comparison with results previously obtained by other well-known methods.

A general decay result of a coupled system of nonlinear wave equations

This work is concerned with a nonlinear system of two coupled viscoelastic equations. Under some restrictions on the coupled functions, we establish, for a wider class of relaxation functions, a generalized stability result for this system. This result improves many results in the literature, such as the ones in Messaoudi and Tatar (Appl Anal 87(3): 247–263, 2008) and Liu (Non-linear Anal 71, 2257–2267, 2009).

Blow up of solutions for 3D quasi-linear wave equations with positive initial energy

In this paper we investigate blow up property of solutions for a system of nonlinear wave equations with nonlinear dissipations and positive initial energy in a bounded domain in R3. Our result improves and extends earlier results in the literature such as the ones in [Zhou, J. and Mu, C., The lifespan for 3D quasilinear wave equations with nonlinear damping terms, Nonlinear Anal., 74 (2011), 5455–5466] and [Pis¸kin, E., Uniform decay and blow-up of solutions for coupled nonlinear Klein-Gordon equations with nonlinear damping terms, Math. Meth. Appl. Scie., 37 (2014), No. 18, 3036–3047] in which the nonexistence results obtained only for negative initial energy or the one in [Ye, Y., Global existence and nonexistence of solutions for coupled nonlinear wave equations with damping and source terms, Bull. Korean Math. Soc., 51 (2014), No. 6, 1697–1710] where blow up results have been not addressed. Estimate for the lower bound of the blow up time is also given.