Wave-type Equations Research Articles

Blow-up for Strauss type wave equation with damping and potential

We study a kind of nonlinear wave equations with damping and potential, whose coefficients are both critical in the sense of the scaling and depend only on the spatial variables. Based on the earlier works, one may think there are two kinds of blow-up phenomenons when the exponent of the nonlinear term is small. It also means there are two kinds of law to determine the critical exponent. In this paper, we obtain a blow-up result and get the estimate of the upper bound of the lifespan in critical and sub-critical cases. All of the results support such a conjecture, although for now, the existence part is still open.

Linear Stability of Higher Dimensional Schwarzschild Spacetimes: Decay of Master Quantities

In this paper, we study solutions to the linearized vacuum Einstein equations centered at higher-dimensional Schwarzschild metrics. We employ Hodge decomposition to split solutions into scalar, co-vector, and two-tensor pieces; the first two portions respectively correspond to the closed and co-closed, or polar and axial, solutions in the case of four spacetime dimensions, while the two-tensor portion is a new feature in the higher-dimensional setting. Rephrasing earlier work of Kodama-Ishibashi-Seto in the language of our Hodge decomposition, we produce decoupled gauge-invariant master quantities satisfying Regge-Wheeler type wave equations in each of the three portions. The scalar and co-vector quantities respectively generalize the Moncrief-Zerilli and Regge-Wheeler quantities found in the setting of four spacetime dimensions; beyond these quantities, we discover a higher-dimensional analog of the Cunningham-Moncrief-Price quantity in the co-vector portion. In addition, our work provides the first verification that the scalar master quantity satisfies its putative Regge-Wheeler equation. In the analysis of the master quantities, we strengthen the mode stability result of Kodama-Ishibashi to a uniform boundedness estimate in all dimensions; further, we prove decay estimates in the case of six or fewer spacetime dimensions. In the case of more than six spacetime dimensions, we discover an obstruction to Morawetz type estimates arising from negative potential terms growing quadratically in spacetime dimension. Finally, we provide a rigorous argument that linearized solutions of low angular frequency are decomposable as a sum of pure gauge solution and linearized Myers-Perry solution, the latter solutions generalizing the linearized Kerr solutions in four spacetime dimensions.

Non-Archimedean analysis and a wave-type pseudodifferential equation on finite adèles

In this work the ring of finite adeles $${\mathbb {A}}_f$$ of the rational numbers $${\mathbb {Q}}$$ is obtained as a completion of $${\mathbb {Q}}$$ with respect to a certain non-Archimedean metric related to the second Chebyshev function, which allows us to represent any finite adele as a convergent series, generalizing m-adic analysis. This polyadic analysis allows us to introduce a novel pseudodifferential operator $$D^{\alpha }$$ on $$L^2({\mathbb {A}}_f)$$ of fractional differentiation, similar to the Vladimirov operator on the p-adic numbers. The operator $$D^{\alpha }$$ is a positive selfadjoint unbounded operator whose spectrum $$\sigma (D^{\alpha })$$ is essential and it consists of a countable number of eigenvalues, which converges to zero, and zero. Moreover, a sort of multiresolution analysis on $${\mathbb {A}}_f$$ provides us with a wavelet basis which is an orthonormal basis of eigenfunctions of $$D^{\alpha }$$ as well. The Cauchy problem of a wave-type pseudodifferential equation $$\begin{aligned} u_{tt}(x,t)+D^{\alpha }_x u(x,t) =F(x,t), \qquad (x \in {\mathbb {A}}_f), \end{aligned}$$ with appropriate initial conditions $$u(x,0)=f(x), u_t(x,0)=g(x),$$ and external force F(x, t), is solved separating variables and using the Fourier expansion of functions in $$L^2({\mathbb {A}}_f)$$ , with respect to the wavelet basis.

Optimal Error Estimate of the Extended-WKB Approximation to the High Frequency Wave-Type Equation in the Semi-classical Regime

For the Cauchy problem of the high frequency wave-type equation with Wentzel–Kramers–Brillouin (WKB) type initial data, the extended Wentzel–Kramers–Brillouin (E-WKB) ansatz is an asymptotically valid solution. This ansatz is globally defined and formulated as an integral of coherent states over the displaced Lagrangian submanifold. This paper proves the optimal first order error estimate of the proposed E-WKB ansatz in $$L^2$$ norm for the wave-type equation in the semi-classical regime. The key ingredients in the proof are the moving frame technique developed in Zheng (Commun Math Sci 11:105–140, 2013) and the deep relations between the E-WKB analysis and the classical WKB analysis. Numerical results on the linear KdV equation verify the theoretical analysis.

Long-time dynamics of Kirchhoff equations with exponential nonlinearities

Our aim in this paper is to study the initial boundary problem for the two-dimensional Kirchhoff type wave equation with an exponentially growing source term. We first prove that the Kirchhoff wave model is globally well-posed in (H01(Ω)∩L∞(Ω))×L2(Ω), which covers the case of degenerate stiffness coefficient, and then obtain that the semigroup generated by the problem has a global attractor in the corresponding phase space. We also point out that the above results are still true in the natural energy space H01(Ω)×L2(Ω).

Stabilization of a Background Magnetic Field on a 2 Dimensional Magnetohydrodynamic Flow

This paper rigorously establishes the stabilization effect of a background magnetic field on electrically conducting fluids, a phenomenon that has been widely observed in physical experiments and numerical simulations. This study is based on a 2 dimensional (2D) magnetohydrodynamic (MHD) system in which the velocity equation involves no dissipation and there is only damping in the vertical component equation. Without the magnetic field, the corresponding vorticity equation is a 2D Euler-like equation with an extra Riesz transform type term. The global in time regularity and the stability near the trivial solution are well known open problems. When coupled with the magnetic field through the MHD system, the background magnetic field stabilizes the fluid, and the velocity as well as the vorticity remain small if they are initially so and decay algebraically in time. To overcome the difficulties due to the lack of full dissipation or damping, we construct suitable Lyapunov functionals and reduce the system to wave type equations.

Attractors for the degenerate Kirchhoff wave model with strong damping: Existence and the fractal dimension

In this paper, we study the long time behavior of the Kirchhoff type wave equation in the space H01(Ω)×L2(Ω). We prove the existence of the global attractor for the equation which covers the case of possible generation of the stiffness coefficient. We also consider the geometrical property of the global attractor. By means of the Z2 index, we provide, under suitable assumptions, the fractal dimension of the global attractor is infinite.

Dispersive and propagation of shallow water waves as a higher order nonlinear Boussinesq-like dynamical wave equations

The propagation of small amplitude long capillary–gravity waves on the surface of shallow water described by the higher order nonlinear Boussinesq type wave equation. This model is arising in mathematical physics, shallow water waves, fluid dynamics and fluid flow. We applied the modified Jacobi elliptic function expansion method to obtained exact solutions in the type of solitary wave and elliptic functions solutions. These solutions helps the mathematicians and physicians to understand the physical phenomena of this model. This technique can be utilized on other models to launch further exclusively novel solutions for other categories of nonlinear PDEs occurring in mathematical Physics.

Torsional Waves in a Fiber Composite Medium at a Loosely Bonded Interface Constrained Between Dry Sandy Layer and Gravitating Poroelastic Substrate

The objective of this paper is to study the effect of loosely bonded interface on torsional surface wave propagation in a fiber reinforced composite medium constrained between dry sandy layer and an anisotropic gravitating poroelastic substrate. All the media are assumed to be under initial stress. The dispersion relation on this proposed multilayer ground structure has been derived in closed form under certain boundary conditions, which contain Whittaker function and its derivative, which is further expanded asymptotically, retaining up to only the linear terms. The numerical solution for the limiting case of torsional surface waves is also discussed. As a special case of the problem, when the entire medium is isotropic and one of the upper layer vanishes and removing the initial stress and gravity, the dispersion relation obtained is in agreement with the classical Love type wave equation. The influence of various technical constants, such as sandy parameter, reinforcement parameter, porosity parameter, Biot’s gravity parameter, loosely bonded parameters, initial stress of both the layers and half spaces on the phase velocity of torsional surface wave has been pointed out by means of graphs.

Stationary Phase Methods and the Splitting of Separatrices

Using stationary phase methods, we provide an explicit formula for the Melnikov function of the one and a half degrees of freedom system given by a Hamiltonian system subject to a rapidly oscillating perturbation. Remarkably, the Melnikov function turns out to be computable without an explicit knowledge of the separatrix and in the case of non-analytic systems. This is related to a priori stable systems coupled with low regularity perturbations. It also applies to perturbations controlled by wave-type equations, so in particular we also illustrate this result with the motion of charged particles in a rapidly oscillating electromagnetic field. Quasiperiodic perturbations are discussed too.

Admissible equivalence transformations for linearization of nonlinear wave type equations

In the present paper, we consider a general family of two‐dimensional wave equations, which represents a great variety of linear and nonlinear equations within the framework of the transformations of equivalence groups. We have investigated the existence problem of point transformations that lead mappings between linear and nonlinear members of particular families and determined the structure of the nonlinear terms of linearizable equations. We have also given examples about some equivalence transformations between linear and nonlinear equations and obtained exact solutions of nonlinear equations via the linear ones.

Wave equations & energy

The focus of this work is apply Fourier analytic methods based on Parseval's equality to the computation of kinetic and potential energy of solutions of initial boundary value problems for general wave type equations on a finite interval. As a consequence, an energy equipartion principle for the solution is obtained. Within our methods are some new results regarding eigenfunction expansions arising from regular Sturm-Liouville problems in Sobolev spaces.

A hierarchy of long wave-short wave type equations: quasi-periodic behavior of solutions and their representation

Based on the Lenard recursion relation and the zero-curvature equation, we derive a hierarchy of long wave-short wave type equations associated with the 3 × 3 matrix spectral problem with three potentials. Resorting to the characteristic polynomial of the Lax matrix, a trigonal curve is defined, on which the Baker-Akhiezer function and two meromorphic functions are introduced. Analyzing some properties of the meromorphic functions, including asymptotic expansions at infinite points, we obtain the essential singularities and divisor of the Baker-Akhiezer function. Utilizing the theory of algebraic curves, quasi-periodic solutions for the entire hierarchy are finally derived in terms of the Riemann theta function.

Asymptotic spreading speed for the weak competition system with a free boundary

This paper is concerned with a diffusive Lotka-Volterra type competition system with a free boundary in one space dimension. Such a system may be used to describe the invasion of a new species into the habitat of a native competitor, and its long-time dynamical behavior can be described by a spreading-vanishing dichotomy. The main purpose of this paper is to determine the asymptotic spreading speed of the invading species when its spreading is successful, which involves two systems of traveling wave type equations.

Approximation of controls for linear wave equations: A first order mixed formulation

This paper deals with the numerical approximation of null controls for the wave equation posed in a bounded domain of \begin{document}$ \mathbb{R}^n $\end{document} . The goal is to compute approximations of controls that drive the solution from a prescribed initial state to zero at a large enough controllability time. In [Cindea & Munch, A mixed formulation for the direct approximation of the control of minimal \begin{document}$ L^2 $\end{document} -norm for linear type wave equations], we have introduced a space-time variational approach ensuring strong convergent approximations with respect to the discretization parameter. The method, which relies on generalized observability inequality, requires \begin{document}$ H^2 $\end{document} -finite element approximation both in time and space. Following a similar approach, we present and analyze a variational method still leading to strong convergent results but using simpler \begin{document}$ H^1 $\end{document} -approximation. The main point is to preliminary restate the second order wave equation into a first order system and then prove an appropriate observability inequality.

Reconstruction of the initial state from the data measured on a sphere for plasma-acoustic wave equations

We establish a reconstruction formula for the initial information of a density (pressure) from the time records of data measured on a sphere for a certain class of wave-type equations. We first derive a reconstruction formula of integral form, and by applying the theory of Fourier Bessel series for this, we also obtain a formula of discrete version, which is more robust and numerically advantageous in the presence of noises. The reconstruction formulas we propose in this work are explicit and applicable to a wider class of wave-type equations including the plasma-acoustic wave equations.

Homogenization Method in the Problem of Long Wave Propagation from a Localized Source in a Basin over an Uneven Bottom

In the framework of the linearized shallow water equations, the homogenization method for wave type equations with rapidly oscillating coefficients that generally cannot be represented as periodic functions of the fast variables is applied to the Cauchy problem for the wave equation describing the evolution of the free surface elevation for long waves propagating in a basin over an uneven bottom. Under certain conditions on the function describing the basin depth, we prove that the solution of the homogenized equation asymptotically approximates the solution of the original equation. Model homogenized wave equations are constructed for several examples of one-dimensional sections of the real ocean bottom profile, and their numerical and asymptotic solutions are compared with numerical solutions of the original equations.

Unified error analysis for nonconforming space discretizations of wave-type equations

AbstractThis paper provides a unified error analysis for nonconforming space discretizations of linear wave equations in the time domain. We propose a framework that studies wave equations as first-order evolution equations in Hilbert spaces and their space discretizations as differential equations in finite-dimensional Hilbert spaces. A lift operator maps the semidiscrete solution from the approximation space to the continuous space. Our main results are a priori error bounds in terms of interpolation, data and conformity errors of the method. Such error bounds are the key to the systematic derivation of convergence rates for a large class of problems. To show that this approach significantly eases the proof of new convergence rates, we apply it to an isoparametric finite element discretization of the wave equation with acoustic boundary conditions in a smooth domain. Moreover, our results reproduce known convergence rates for already investigated conforming and nonconforming space discretizations in a concise and unified way. The examples discussed in this paper comprise discontinuous Galerkin discretizations of Maxwell’s equations and finite elements with mass lumping for the acoustic wave equation.

Energy decay rates for solutions of the Kirchhoff type wave equation with boundary damping and source terms

In this work, we are concerned with uniform stabilization for an initial-boundary value problem associated with the Kirchhoff type wave equation with feedback terms and memory condition at the boundary. We prove that the energy decays exponentially when the boundary damping term has a linear growth near zero and polynomially when the boundary damping term has a polynomial growth near zero. Furthermore, we study the decay rate of the energy without imposing any restrictive growth assumption on the damping term near zero.

Addendum to foundations of multidimensional wave field signal theory: Gaussian source function

Many important physical phenomena are described by wave or diffusion-wave type equations. Recent work has shown that a transform domain signal description from linear system theory can give meaningful insight to multi-dimensional wave fields. In N. Baddour [AIP Adv. 1, 022120 (2011)], certain results were derived that are mathematically useful for the inversion of multi-dimensional Fourier transforms, but more importantly provide useful insight into how source functions are related to the resulting wave field. In this short addendum to that work, it is shown that these results can be applied with a Gaussian source function, which is often useful for modelling various physical phenomena.