# Soliton Content of Fiber-Optic Light Pulses
## 4 Methods to Verify Soliton Content
### The Inverse and Direct Scattering Transform  
#### 4.1.1 The Method  
The inverse scattering transform (IST) is a technique that can be used to solve certain nonlinear partial differential equations (PDE) like the Korteweg-de Vries equation, the sine-Gordon equation or the nonlinear Schrodinger equation. The basic concepts of this method were presented in 1967 in a groundbreaking paper by [PERSON] and coworkers dealing with the Korteweg-de Vries equation [5]. Shortly after, [PERSON] showed how to apply the method also to other nonlinear PDEs when certain conditions are met [61]. The IST was successfully adapted to the NLSE by [PERSON] and [PERSON] in 1972 [4]. The method of IST is based on transformations between the time-space domain and a nonlinear spectral domain, in which the nonlinear evolution of some given input field reduces to a linear problem. In the spectral domain, a given field is represented by its scattering spectrum, which is calculated by the so-called direct scattering transform (DST). The inverse operation is called the inverse scattering transform. As these transformations can be regarded as extensions to the well-known Fourier transform, they are also known as nonlinear Fourier transform and inverse nonlinear Fourier transform, respectively. Using the nonlinear Fourier transform, one can get insight into the linear and nonlinear components of a given field by analyzing its scattering spectrum. Generally, it consists of a continuous part, which represents the small amplitude radiation, and a discrete part, which can be attributed to the solitons contained. In recent years, several groups took up the idea of using multi-soliton pulses for nonlinear optical data transmission, which originally was suggested by [PERSON] and [PERSON] [62]. In the course of this renewed attention, several nonlinear transmission schemes and techniques for the nonlinear Fourier transform and its inverse transform have been (and still are) developed to overcome the linear capacity limits of fiber-optical transmission lines. An overview about the latest developments can be found in [63]. DST is a great tool to analyze given pulses, but its application is somewhat involved. The DST has been applied analytically to only a few simple pulse shapes like sech-shaped [6] or rectangular pulses [64]. Even for such a standard shape as a Gaussian,  
\[A(t)=A_{0}\sqrt{P_{0}}\exp\left[-\frac{1}{2}\left(\frac{t}{T_{0}}\right)^{2} \right]\quad, \tag{18}\]  
no analytical result is known to the authors' best knowledge. Instead, approximate solutions can be found numerically. As an example, when setting \(A_{0}=3\) and choosing \(P_{0}\) and \(T_{0}\) according to Equation (4), DST reveals that the pulse consists of two solitons and a small amplitude radiation part of nontrivial spectral shape (see Figure 8). Various schemes for numerical calculations of the DST are available [65; 66; 67; 68; 63]. They all solve the Zakharov-Shabat eigenvalue problem of a discretized input field with \(M\) sample points  
Figure 8: Nonlinear spectrum calculated for a Gaussian pulse shape (Equation (18)) with \(A_{0}=3\) and \(P_{0}\), \(T_{0}\) fulfilling Equation (4). (Left) The discrete spectrum, consisting of two solitons of the same center frequency. The reference energy here is chosen as \(E_{0}=2P_{0}T_{0}\). (Right) The power spectral density (PSD) of the linear radiation part. and determine the nonlinear spectrum. One can distinguish between matrix methods that calculate the eigenvalues directly from a \(M\times M\) matrix and search methods in which the eigenvalues are found in an iterative manner. For a recent discussion, see, e.g., [63; 66; 67; 68]. In the following, we will sketch briefly how to calculate the scattering spectrum using a search method. Using the formulation of [PERSON] et al.