The purpose of this chapter is to introduce another equation describing nonlinear wave phenomena: the nonlinear Schrödinger equation (NLS), which should not to be confused with the linear Schrödinger equation from quantum mechanics (see below).

As indicated in Chapter 18, this equation, like the KdV equation, has been discovered rather recently. The two equations appear in, and are used for, wave phenomena of various types. In particular, the NLS equation, like the KdV equation, can describe water-wave phenomena, and it can also be deduced from the Euler equation of perfect fluids under appropriate hypotheses.

However, the NLS equation also describes phenomena that are very important nowadays: the propagation of waves in wave guides in relation to the design of optical long-distance communications lines and all-optical signalprocessing devices for reliable and high-bit-rate transmission of information.

Owing to the importance of the subject, and to diversify the mathematical techniques developed in this book, we will in this chapter derive the NLS equation from the Maxwell equations in the context of wave guides rather than deduce them from the Euler equations in the context of fluid mechanics.

We start in Section 20.1 by recalling the Maxwell equations, and we introduce a new phenomenon that is essential for optic fibers, namely polarization, which corresponds to, and describes, the electromagnetically anisotropic behavior of the medium.