OSCILLATORY SOLUTIONS OF PARTIAL DIFFERENTIAL AND DIFFERENCE EQUATIONS

Abstract

This chapter provides an overview of the oscillatory behavior of solutions of partial differential and differential equations. Partial differential equations typically contain parameters of physical significance, such as speed of light, coefficients of viscosity, of heat conduction, etc. Very often, it is the extreme values of these parameters, the very large or very small, that are of interest and are amenable to analysis. Such a parameter is dispersion; it is known, from theory and numerical experiments, that nonlinear equation with zero dissipation and a small amount of dispersion have solutions that are highly oscillatory. There are, at least, three sources of oscillatory behavior of solutions of partial differential equations. One source is the oscillatory behavior of the coefficients of the differential equation; these occur in the description of the propagation of waves through composite materials. Another source is oscillatory initial or boundary conditions. The chapter discusses the third source where the oscillations are not imposed but arise spontaneously.