The field of nonlinear dispersive waves has developed rapidly over the past 50 years. Its roots go back to the work of Stokes in 1847, Boussinesq in the 1870s and Korteweg and de Vries (KdV) in 1895, all of whom studied water wave problems. In the early 1960s researchers developed effective asymptotic methods, such as the method of multiple scales, that allow one to obtain nonlinear wave equations such as the KdV equation and the nonlinear Schrödinger (NLS) equation, as leading-order asymptotic equations governing a broad class of physical phenomena. Indeed, we now know that both the KdV and NLS equations are “universal” models. It can be shown that KdV-type equations arise whenever we have weakly dispersive and weakly nonlinear systems as the governing system. On the other hand, NLS equations arise from quasi-monochromatic and weakly nonlinear systems.

The discovery of solitons associated with the KdV equation in 1965 by Zabusky and Kruskal was a major development. They employed a synergistic approach: computational methods and analytical insight. This was soon followed by a remarkable publication in 1967 by Gardner, Greene, Kruskal and Miura that described the analytical method of solution to the KdV equation, with rapidly decaying initial data. They employed concepts of direct and inverse scattering in the solution of the KdV equation that was perceived by researchers then as nothing short of astonishing.

In terms of the methods of asymptotic analysis, so far we have studied integral asymptotics associated with Fourier integrals that represent solutions of linear PDEs. Now, suppose we want to study physical problems like the propagation of waves in the ocean, or the propagation of light in optical fibers; the general equations obtained from first principles in these cases are the Euler or Navier–Stokes equations governing fluid motion on a free surface and Maxwell's electromagnetic (optical wave) equations with nonlinear induced polarization terms. These equations are too difficult to handle using linear methods or, in most situations, by direct numerical simulation. Loosely speaking, these physical equations describe “too much”.

Mathematical complications often arise when one has widely separated scales in the problem, e.g., the wavelength of a typical ocean wave is small compared to the ocean's depth and the wavelength of light in a fiber is much smaller than the fiber's length or transmission distance. For example, the typical wavelength of light in an optical fiber is of the order of 10-6 m, whereas the length (distance) of an undersea telecommunications fiber is of the order of 10,000 km or 107 m, i.e., 13 orders of magnitude larger than the wavelength! Therefore, if we were to try solving the original equations numerically – and resolve both the smallest scales as well as keep the largest ones – we would require vast amounts of computer time and memory.

In the previous chapter we have seen that both constant dispersion and dispersion-managed solitons are important pulses in the study of long-distance transmission/communications. It is noteworthy that experiments reveal that solitons or localized pulses are also present in mode-locked (ML) lasers. Femtosecond solid-state lasers, such as those based on the Ti:sapphire (Ti:S) gain medium, and fiber ring lasers have received considerable attention in the field of ultra-fast science. In the past decade, following the discovery of mode-locking, the improved performance of these lasers has led to their widespread use, cf. Cundiff et al. (2008). In most cases interest in ultra-short pulse mode-locking has been in the net anomalous dispersive regime. But mode-locking has also been demonstrated in fiber lasers operating in the normal regime. Mode-locking operation has been achieved with relatively large pulse energies (Ilday et al., 2004b,a; Chong et al., 2008b).

In our investigations we have employed a distributive model, termed the power-energy saturation (PES) equation (cf. Ablowitz et al., 2008; Ablowitz and Horikis, 2008, 2009b; Ablowitz et al., 2009c). This model goes beyond the well-known master laser equation, cf. Haus (1975, 2000), in that it contains saturable power (intensity) terms; i.e., terms that saturate due to large field amplitudes. This equation has localized pulses that propagate and mode-lock in both an anomalous and a normally dispersive laser for both in the constant as well as dispersion-managed system. This is consistent with recent experimental observations (Ilday et al., 2004b; Chong et al., 2008a).

Nonlinear optics is the branch of optics that describes the behavior of light in nonlinear media; such as, media in which the induced dielectric polarization responds nonlinearly to the electric field of the light. This nonlinearity is typically observed at very high light intensities such as those provided by pulsed lasers. In this chapter, we focus on the application of high bit-rate communications. We will see that the nonlinear Schrödinger (NLS) equation and the dispersion-managed nonlinear Schrödinger (DMNLS) equation play a central role.

Communications

In 1973 Hasegawa and Tappert (Hasegawa and Tappert, 1973a; Hasegawa and Kodama, 1995) showed that the nonlinear Schrödinger equation derived in Chapter 7 [see (7.26), and the subsequent discussion] described the propagation of quasi-monochromatic pulses in optical fibers. Motivated by the fact that the NLS equation supports special stable, localized, soliton solutions, Mollenauer et al. (1980) demonstrated experimentally that solitons can propagate in a real fiber. However, it was soon apparent that due to unavoidable damping in optical fibers, solitons lose most of their energy over relatively short distances. In the mid-1980s all-optical amplifiers (called erbium doped fiber amplifiers: EDFAs) were developed. However with such amplifiers there is always some additional small amount of noise. Gordon and Haus (1986) (see also Elgin, 1985) showed that solitons suffered seriously from these noise effects. The frequency and temporal position of the soliton was significantly shifted over long distances, thereby limiting the available transmission distance and speed of soliton-based systems.

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.