# Soliton Content of Fiber-Optic Light Pulses
## 2 The Nonlinear Schrodinger Equation and Some of Its Solutions  
The glass of optical fiber is a dispersive material. With the index being a function of wavelength or optical frequency, i.e., \(n=n(\omega)\), the group velocity is also frequency dependent. The glass is also a nonlinear material. This chiefly means that the refractive index follows the instantaneous intensity, i.e., \(n=n(I)\). Across the duration of a short pulse of light, the index experiences a slight modulation, which gives rise to intensity-dependent phase shifts known as self-phase modulation [3].  
A propagation equation for light pulses in fibers must contain these two influences, at least in leading order. The equation that fits this description is known as the nonlinear Schrodinger equation (NLSE). It plays an absolutely central role in all of the nonlinear fiber optics. It describes the evolution of the pulse amplitude envelope (the fast oscillation at some central optical frequency \(\omega_{0}\) is removed) in a frame of reference comoving with the traveling light, i.e., with the group velocity at \(\omega_{0}\). Written in physical quantities, it is given by:  
\[i\frac{\partial}{\partial z}A-\frac{\beta_{2}}{2}\frac{\partial^{2}}{\partial t ^{2}}A+\gamma|A|^{2}A=0\quad. \tag{1}\]  
Here, \(A=A(z,t)\) is the complex amplitude with \(z\) the position along the fiber and \(t\) the retarded time. \(\beta_{2}\) is the coefficient of group velocity dispersion, and \(\gamma\) is the coefficient of nonlinearity. Depending on the situation, it may become necessary to add corrective terms for loss, higher-order dispersion or other nonlinear effects, as we shall see in Section 3.1. The equation received its name due to its formal similarity to the Schrodinger equation of quantum mechanics fame: The quantum mechanical version typically describes how a wave function spreads out in space as time goes by; here, the equation describes how a short pulse gets broadened temporally as it propagates down the fiber. Therefore, time and space coordinates switch roles. The potential, in the nonlinear case, comes from the self-phase modulation, which is captured in the nonlinear term.  
The effort to find solutions to this equation benefited from the fact that the NLSE is integrable. Integrable equations have very special mathematical properties like an infinite number of preserved quantities. A first solution was found by [PERSON] and [PERSON] in 1972 [4] using the inverse scattering technique, developed by [PERSON] et al. in 1967 [5] (see Section 4.1). It exists for a particular combination of algebraic signs of the two parameters; with \(\gamma\) always positive, one would have the correct signs for \(\beta_{2}<0\), i.e., for what is known as anomalous dispersion. Two years later, [PERSON] and [PERSON] [6] treated the pertaining initial-value problem. Between them, these two papers already form a solid base of the understanding of solitons. In the time between those two, [PERSON] and [PERSON] suggested that the NLSE was applicable to pulse propagation in optical fibers, so that fiber-optic solitons would exist [7]. At that time, optical fibers were a novelty and not yet technologically matured. Attempts to transmit light pulses for data transmission suffered from dispersive distortions. It was therefore a bold proposal that the use of solitons as signaling pulses might overcome dispersive broadening, in other words that nonlinearity could be exploited to cancel distortions arising from a linear mechanism.  
Of course, the proposal required as a very first step that the existence of fiber-optic solitons be corroborated experimentally. In the anomalously dispersive regime, i.e., at wavelengths longer than ca. 1.3 \(\upmu\)m, they were still too lossy for any meaningful test. By 1980, this situation had improved, and [PERSON], [PERSON] and [PERSON] could experimentally verify the existence of solitons [8].