Maria V. Perel

Highly localized solutions of the wave equation

Simple explicit solutions of the linear wave equation in three dimensions are presented which describe wave packets exponentially localized near a point moving with the wave speed. For large values of a certain free parameter these new solutions are localized in Gaussian manner with respect to longitudinal and transverse variables and time. This agrees with considerations by Babich–Ulin and Ralston who have presented an asymptotic description of solutions exhibiting such local behavior. Global estimates and large-time asymptotics of these solutions are given.

New physical wavelet 'Gaussian wave packet'

An exact solution of the homogeneous wave equation, which was found previously, is treated from the point of view of continuous wavelet analysis (CWA). If time is a fixed parameter, the solution represents a new multidimensional mother wavelet for the CWA. Both the wavelet and its Fourier transform are given by explicit formulae and are exponentially localized. The wavelet is directional. The widths of the wavelet and the uncertainty relation are investigated numerically. If a certain parameter is large, the wavelet behaves asymptotically as the Morlet wavelet. The solution is a new physical wavelet in the definition of Kaiser, it may be interpreted as a sum of two parts: an advanced and a retarded part, both being fields of a pulsed point source moving at a speed of wave propagation along a straight line in complex spacetime.

Wavelet Analysis in Solving the Cauchy Problem for the Wave Equation in Three-Dimensional Space

We propose a new method for solving the Cauchy problem for the wave equation in three dimensional space. The method is based on continuous waveletanalysis. We show that the exact non-stationary solution of the wave equation with finite energy found in [1] at any fixed moment of time should be regarded as a mother wavelet. This solution was named in [1] as a “Gaussian wave packet”. It is a new three-dimensional axially symmetric wavelet which is given by a simple explicit formula as well as its Fourier transformation. This wavelet has an infinite number of vanishing moments. It is a smooth function, i.e. it has derivatives of any order with respect to spatial coordinates and time. We show that using the wavelet decomposition of the initial data we can find the exact formula for the solution of the Cauchy problem as a linear superposition of “Gaussian packets”.

Exponentially Localized Solutions to the Klein–Gordon Equation

Exponentially localized solutions of the Klein–Gordon equation for two and three space variables are presented. The solutions depend on four free parameters. For some relations between the parameters, the solutions describe wave packets filled with oscillations whose amplitudes decrease in the Gaussian way with distance from a point running with group velocity along a ray. The solutions are constructed by using exact complex solutions of the eikonal equation and may be regarded as ray solutions with amplitudes involving one term. It is also shown that the multidimensional nonlinear Klein–Gordon equation can be reduced to an ordinary differential equation with respect to the complex eikonal. Bibliography: 12 titles.

Wavelet analysis for the solutions of the wave equation

The new approach to the wavelet analysis for the solutions of the homogeneous wave equation in three spatial dimensions is presented. The approach is based on the ideas suggested by G. Kaiser but has different implementation and has some advantages versus the known approach. A new physical wavelet for this wavelet analysis is also presented, with the brief discussion of its main properties. The wavelet analysis has become widely used during the last twenty years and it has a lot of applications nowadays. However most of them are in the field of the numerical processing of the experimental data, digital images, astronomical, geophysical and medical data and other applications of that kind (see, for example, [1], [2]). The amount of the results in the application of the methods of the continuous wavelet analysis to the solutions of the differential equations is not large. In particular the continuous wavelet analysis for the solutions of the three-dimensional homogeneous wave equations with a constant wave speed was first developed by G. Kaiser in his book [3]. He suggests a method for decomposition of the solutions of the wave equation in terms of the localized solutions of the same equation based on the analytic signal transform and on the theory of the analytic functions of several variables. The wavelet for such decomposition was also suggested, and the class of such wavelets was named ’physical wavelets’. The sort of the wavelet analysis developed by Kaiser is close to the holomorphic wavelet transform (see, for example, [2]). However, this approach may be found unfamiliar by the people who deal with the wavelet analysis within the framework of the signal and image processing. The aims of this paper are as follows. First we develop the wavelet analysis for the solutions of the wave equation not involving the analytic signal transform. The ideas, which we base on, were suggested by Kaiser in [3], however their implementation here differs from his approach. The method we use is intrinsic to the common continuous wavelet transform and we hope will be more familiar to the people who work in the area of the signal and image processing. Our approach also provides some advantages in comparison to that, suggested by Kaiser. We enlarge the class of solutions which can be used as the mother wavelets for the analysis. The second aim is to find a new physical wavelet for our method, i.e., to find the solution of the wave equation which will be an admissible wavelet. The new wavelet is constructed by means of the field of point sources and of proxy wavelets using the technique suggested by G. Kaiser. This new spherically symmetric physical wavelet has good properties such as exponential localization in both the coordinate and the Fourier

Effects associated with a saddle point of the dispersion surface of a photonic crystal

We consider a one-dimensional photonic crystal consisting of alternating dielectric layers of two types. The dispersion relation for such a crystal gives the dependence of the frequency on the transverse wave vector and the quasi-momentum. If the frequency of the incident wave coincides with the frequency of the saddle point, the behavior of the envelope of the wave field is determined by the hyperbolic equation, where the longitudinal coordinate plays the role of time. Depending on the parameters of the isofrequency contour, the canalization or localization of the wave field may occur. If the parameters correspond to the localization, it can be achieved by a proper choice of the field distribution on the surface of the crystal.

The Poincar´e wavelet transform: Implementation and interpretation

Numerical implementation and examples of calculation of the Poincaré wavelet transform for model space-time signals are presented. This transform is a coefficient in the decomposition of solutions of the wave equation in terms of elementary localized solutions found in [1]. Elementary localized solutions are shifted and scaled versions of some chosen solution in a given reference frame, as well as in frames moving with respect to given the one with different constant speeds. We discuss what information about the wave field can be extracted from the Poincaré wavelet transform.

Multiscale Investigation of Solutions of the Wave Equation

We consider here an initial value problem for the homogeneous wave equation with constant coefficients in three spatial dimensions, that is,

Asymptotic study of a two-scale electromagnetic field in a layered periodic structure

\begin{cases}u_{tt} - c^2 (u_{xx} + u_{yy} + u_{zz}) = 0,\\ u|_{t=0} = w({\bf r}), \quad \left.\frac{\partial u}{\partial t}\right|_{t=0} = v({\bf r}).\end{cases}