Narrow Results

In this paper, the Cauchy problem for linear and nonlinear wave equations is studied. The equation involves abstract operator $A$ in a Banach space $E$ and convolution terms. Here, assuming enough smoothness on the initial data and on coefficients, the existence, uniqueness and regularity properties of local and global solutions are established in terms of fractional powers of a given sectorial operator $A$. We obtain the regularity properties of a wide class of wave equations by choosing the space $E$ and the operator $A$ which appear in the field of physics.

New equations describing particles with spin-3/2 are derived. The nonlocal equation with the unique mass can be considered as "square root" of the Proca equation in the same sense as the Dirac equation is related to the Klein–Gordon–Fock equation. The local equation describes spin-3/2 particles with three mass states. The equations considered involve fields with spin-3/2 and spin-1/2, i.e. multispin 1/2, 3/2. The projection operators extracting states with definite energy, spin, and spin projections are obtained. All independent solutions of the local equation are expressed through projection matrices. The first order relativistic wave equation in the 20-dimensional matrix form, the relativistically invariant bilinear form and the corresponding Lagrangian are given. Two parameters characterizing nonminimal electromagnetic interactions of fermions are introduced, and the quantum-mechanical Hamiltonian is found. It is proved that there is only causal propagation of waves in the approach considered.

We try to stabilize steady solutions of a physical model described by wave equations with nonlinear boundary conditions. This system is a distributed parameter system in which ideal turbulence, introduced by Sharkovsky et al., occurs. Although the behavior of the system is quite intricate both in time and space, by using d'Alembert's solution, the analysis of the dynamic characteristics can be reduced to that of a finite-dimensional difference equation. In this report, based on this analytical method using d'Alembert's solution, we design control laws to stabilize steady solutions (equilibrium solutions and periodic solutions) and synchronize a pair of identical systems.

We examine the dynamics of materials characterized by the presence of a deformation threshold beyond which no deformation is possible. The class of bodies that we are interested in studying are described by an implicit constitutive relationship between the Cauchy stress and the deformation gradient.

A specific one-dimensional dynamical problem is studied, showing that the mathematical model takes the form of a hyperbolic free boundary problem in which the free boundary conditions can be of two different types, selected according to whether the stress is continuous at the interface (separating the deformable region from the fully strained region), or whether it is discontinuous. Both situations have been analyzed. Sample numerical computations are carried out using data that are relevant to biological materials.

A comparison with the problem obtained from a limiting procedure for a constitutive model with a piecewise linear elastic response is performed, showing a very interesting feature, namely that the limit does not lead to the solution of the model with a threshold. This is however not surprising as the latter exhibits dissipative behavior.

The variational method in a reformulated Hamiltonian formalism of quantum electrodynamics (QED) is used to derive relativistic wave equations for systems consisting of n fermions and antifermions of various masses. The derived interaction kernels of these equations include one-photon exchange interactions. The equations have the expected Schrödinger non-relativistic limit. Application to some exotic few lepton systems is discussed briefly.

In this paper we define Maxwell's equations in the setting of the intrinsic complex of differential forms in Carnot groups introduced by M. Rumin. It turns out that these equations are higher-order equations in the horizontal derivatives. In addition, when looking for a vector potential, we have to deal with a new class of higher-order evolution equations that replace usual wave equations of the Euclidean setting and that are no more hyperbolic. We prove equivalence of these equations with the "geometric equations" defined in the intrinsic complex, as well as existence and properties of solutions.

A superconvergent isogeometric formulation is presented for the transient analysis of wave equations with particular reference to quadratic splines. This formulation is developed in the context of Newmark time integration schemes and superconvergent quadrature rules for isogeometric mass and stiffness matrices. A detailed analysis is carried out for the full-discrete isogeometric formulation of wave equations and an error measure for the full-discrete algorithm is established. It is shown that a desirable superconvergence regarding the isogeometric transient analysis of wave equations can be achieved by two ingredients, namely, the design of a superconvergent quadrature rule and the criteria to properly define the step size for temporal integration. It turns out that the semi-discrete and full-discrete isogeometric formulations of wave equations with quadratic splines share an identical quadrature rule for a sixth-order accurate superconvergent analysis. Meanwhile, the relationships between the time step size and the element size are presented for various typical Newmark time integration schemes, in order to ensure the sixth-order accuracy in transient analysis. Numerical results of the transient analysis of wave equations consistently reveal that the proposed superconvergent isogeometric formulation is sixth-order accurate with respect to spatial discretizations, in contrast to the fourth-order accuracy produced by the standard isogeometric approach with quadratic splines.

The existence of a pullback (and also a uniform forward) attractor is proved for a damped wave equation containing a delay forcing term which, in particular, covers the models of sine–Gordon type. The result follows from the existence of a compact set which is uniformly attracting for the two-parameter semigroup associated to the model.

A discussion and an improvement of the Neumann–Kelvin's model are suggested in this paper. This model is used in the simulation of progressive wave phenomenon. As mentioned by several authors ([7, 19, 23, 29]), this model is ill posed unless a capillary energy is introduced. The mathematical explanation is that a compactness inversions' property occurs if the capillary forces are omitted. Theoretical arguments and numerical simulations are used in the following which aims at giving an explanation of what happens from a mechanical point of view.

The Noether symmetry algebras admitted by wave equations on plane-fronted gravitational waves with parallel rays are determined. We apply the classification of different metric functions to determine generators for the wave equation, and also adopt Noether's theorem to derive conserved forms. For the possible cases considered, there exist symmetry groups with dimensions two, three, five, six and eight. These symmetry groups contain the homothetic symmetries of the spacetime.

Stimulating from the hyperbolic octonionic generalization of the Maxwell-type equations of compressible fluids, an alternative reformulation has been proposed for the analogous multifluid plasma equations. In this paper, using both the fluid and electromagnetic behavior of the plasma, the compact and elegant expressions have been derived in terms of hyperbolic octonions as previously given for electromagnetic theory, linear gravity and fluid mechanics. Using the advantages of proposed model, the field equations of multifluid plasma have been combined in a single equation. Furthermore, the plasma wave equations in terms of generalized vorticity and Lamp vector have been expressed in a form similar to electromagnetic, gravitational counterparts previously given in relevant literature.

In this paper we study how to approximate the Leray weak solutions of the incompressible Navier–Stokes equations. In particular we describe an hyperbolic version of the so-called artificial compressibility method investigated by J. L. Lions and Temam. By exploiting the wave equation structure of the pressure of the approximating system we achieve the convergence of the approximating sequences by means of dispersive estimates of Strichartz type. We prove that the projection of the approximating velocity fields on the divergence free vectors is relatively compact and converges to a Leray weak solution of the incompressible Navier–Stokes equation.

We analyze a method of approximation for the weak solutions of the incompressible magnetohydrodynamic equations (MHD) in unbounded domains. In particular we describe an hyperbolic version of the so-called artificial compressibility method adapted to the MHD system. By exploiting the wave equation structure of the approximating system we achieve the convergence of the approximating sequences by means of dispersive estimate of Strichartz type. We prove that the soleinoidal component of the approximating velocity and magnetic fields is relatively compact and converges strongly to a weak solution of the MHD equation.

In this paper, we consider the Cauchy problem for semi-linear wave equations with structural damping term $\nu {(-\mathrm{\Delta })}^2{u}_t$, where $\nu >0$ is a constant. As is now well known, the linear principal part brings both the diffusion phenomenon and the regularity loss of solutions. This implies that, for the nonlinear problems, the choice of solution spaces plays an important role to obtain the global solutions with the sharp decay properties in time. Our main purpose in this paper is to prove the global (in time) existence of solutions for the small data and their decay properties for the supercritical nonlinearities.

The wave equation with variable coefficients with a nonlinear dissipative boundary feedback is studied. By the Riemannian geometry method and the multiplier technique, it is shown that the closed loop system decays exponentially or asymptotically, and hence the relation between the decay rate of the system energy and the nonlinearity behavior of the feedback function is established.

Consider a system of nonlinear wave equations

This paper considers the exponential decay of the solution to a damped semilinear wave equation with variable coefficients in the principal part by Riemannian multiplier method. A differential geometric condition that ensures the exponential decay is obtained.

Backscattered wave acoustic tomography using wideband probing signals makes it possible to obtain three-dimensional (3D) images of scattering inhomogeneities. Signal processing based on the reverse time migration (RTM) method allows one to take into account the influence of background refractive obstacles of the medium to minimize distortions of reconstructed tomographic images. We propose a noniterative method of acoustic tomography in an immersion medium based on RTM approach supplemented with linear signal preprocessing to enhance resolution of reconstructing tomography images. The visualization of scattering objects is based on wave inversion from the measurement area considering the probing wave field specially distorted to perform regularized back convolution. The applicability of the proposed method for visualizing scattering objects in water is shown analytically, numerically and experimentally. The proposed method is resistant to noise according to regularization. The results obtained show the agreement between the numerical and analytical solution. Using the example of sounding with linear frequency modulation signals, it is demonstrated that the proposed method allows increasing the resolution of tomographic images in comparison with conventional RTM. The novelty of the proposed method is the preliminary filtration of the forward propagation wave in the course of solving the inverse problem. This approach improves the resolution of tomographic images and allows considering the influence of obstacles.

For determination of absorbing coefficient, one-dimensional wave is studied to give the analytical expression of absorbing term, and the analytical predictions are compared to available physical laboratory data in flume for wave propagation through bottom-mounted breakwaters. We consider the problem of wave partial/full reflection and transmission by a wave-absorbing layer with the focus on the understanding of wave reduction by wave-permeable structures. We obtain accurate predictions of reflected and transmitted results around the wave-permeable breakwater for various incident-wave conditions. The results show that wave permeable breakwaters with proper absorbing coefficient can be used as an effective alternative to massive gravity breakwaters in reduction of wave transmission in shallow water.

In this paper we consider the class of equations u_tt = f(x,t,u,u_x,u_t)u_xx + g(x,t,u,u_x,u_t). We derive equivalence transformations and we classify differential invariants.