Linear and nonlinear wave equations

Abstract

In Chapter 1 we saw how the KdV equation can be derived from the FPU problem. We also mentioned that the KdV equation was originally derived for weakly nonlinear water waves in the limit of long or shallow water waves. Researchers have subsequently found that the KdV equation is “universal” in the sense that it arises whenever we have a weakly dispersive and a weakly quadratic nonlinear system. Thus the KdV equation has also been derived from other physical models, such as internal waves, ocean waves, plasma physics, waves in elastic media, etc. In later chapters we will analyze water waves in depth, but first we will discuss some basic aspects of waves. Broadly speaking, the study of wave propagation is the study of how signals or disturbances or, more generally, information is transmitted (cf. Bleistein, 1984). In this chapter we begin with a study of “dispersive waves” and we will introduce the notion of phase and group velocity. We will then briefly discuss: the linear wave equation, the concept of characteristics, shock waves in scalar first-order partial differential equations (PDEs), traveling waves of the viscous Burgers equation, classification of second-order quasilinear PDEs, and the concept of the well-posedness of PDEs.