Cathleen S. Morawetz

Solving the Helmholtz Equation for Exterior Problems with Variable Index of Refraction

A new technique for numerically solving the reduced wave equation on exterior domains is presented. The method is basically a relaxation scheme which exploits the limiting amplitude principle. A modified boundary condition at “infinity” is also given. The technique is tested on several model problems: the scattering of a plane wave off a metal cylinder, a metal strip, a Helmholtz resonator, an inhomogeneous cylinder (lens), and a nonlinear plasma column. The results are in good qualitative agreement with previously calculated values. In particular, the numerical solutions exhibit the correct refractive and diffractive effects at moderate frequencies.

A formulation for higher dimensional inverse problems for the wave equation

Abstract Two particular inverse scattering problems are of special interest. The first concerns the discovery of a perturbation in the speed of sound by analyzing the return signal from a blast wave set off above it. The second concerns the determination of a potential from the back scattering of plane waves. In one dimensional problems the two cases are very closely related. In higher dimensions the situation becomes much more complicated. We present here a new approach to four such classical higher dimensional inverse problems for determining the coefficients of a partial differential equation, including the two mentioned. The idea stems from the work of Deift and Trubowitz [1], in one space dimension. No conclusive theorem is found but the approach provides an algorithm for iteration which might lead to an existence theorem and which will be explored numerically elsewhere.

MODIFICATION FOR MAGNETOHYDRODYNAMIC SHOCK STRUCTURE WITHOUT COLLISIONS

It is shown that the steady‐state shock structure given earlier [C. S. Morawetz, Phys. Fluids 4, 988 (1961)] is valid only for zero‐temperature electrons. However, certain modifications of the asymptotic development of the electrons permit one to show that the same shock form will appear if the electrons have a relative thermal speed of the order e−ν, where e2 is the mass ratio and ν satisfies ½ < ν < 1. This corresponds to a ratio of electron to ion temperature of the order e2(1−ν).

Variations on conservation laws for the wave equation

The first part of this paper, presented as an Emmy Noether lecture in connection with the ICM in Berlin in August 1998, gives some examples of using Noether’s theorem for conservation laws for Tricomi-like equations and for the wave equation. It is also shown that equations which are semilinear variations of the wave equation can very often be handled similarly. The type of estimate obtained can even be used to get otherwise unobtainable local estimates for regularity. The fourth part is an introduction to the relation of black holes to the wave equation mainly showing the results of D. Christodoulou. His results use much more difficult estimates not corresponding at all to those in the first part of the paper.

THE CALCULATIONS OF AN INVERSE POTENTIAL PROBLEM

A method is developed for solving an inverse problem for the wave equation with potential where the object is to find the potential given Cauchy data on a time-like surface. The computation is carried out with one space variable by an iterative procedure. The point of this method is that it can be extended to higher dimensions in principle. A coarse mesh finite difference scheme is used which yields fair accuracy.

Time decay and relaxation schemes

A general method for concocting relaxation schemes for solving steady state boundary value problems is suggested by the method of multipliers. Its application to a problem in mixed equations is discussed.

Computations with the nonlinear Helmholtz equation

A new technique is presented for numerically studying the interaction of an electromagnetic wave with a nonlinear plasma lens. The method is basically a relaxation scheme that exploits the limiting amplitude principle. A modified boundary condition at infinity is also given.

Asymptotics of a nonlinear relativistic wave equation

with m ^ O , in all space-time. I. Segal in [3] has proved the existence of the free-to-perturbed wave operators and in [4] the existence of the scattering operator on numerically small solutions. He has conjectured that the scattering operator exists in general. Segals conjecture has been verified when m = 0 in [5]. We have succeeded in proving the conjecture when ra>0. DEFINITIONS. By a free solution we mean a solution of the associated linear equation (equation (*) without u). The norm

Energy flow: Wave motion and geometrical optics

Energy distribution for solutions of the wave equation in the presence of a reflecting body can be investigated with varying degrees of refinement by using quadratic inequalities, Huyghens principle and geometrical optics. The relations between these properties and their validity in general cases is discussed and some of the simpler proofs outlined.