# Soliton Content of Fiber-Optic Light Pulses
## 4 Methods to Verify Soliton Content
### The Inverse and Direct Scattering Transform
#### 4.1.3 Applications and Limitations of DST  
The DST method as described so far is quite generally applicable to arbitrary pulse profiles provided two conditions are met: One is integrability, which is not fulfilled for many structures in the generalized NLSE in Section 3.1 and for DM systems in Section 3.2. The other is that for time to infinity, the structure must decay to zero sufficiently fast [65; 66]. This is not the case for the structures described in Section 2.2. If integrability is violated only weakly either by loss or through the Raman effect, the concept of solitons remains valid in an approximate sense as pulses that maintain an (near-) equilibrium between dispersive and nonlinear influences. It might be more accurate to speak then of solitary pulses. Strictly speaking, though, all preserved quantities are no more. Perturbation theory has shown [29; 30] that solitons rearrange their shape in terms of width and peak power; in the process, they acquire  
Figure 9: Direct scattering analysis of an \(N=1\) soliton with different time window widths \(T_{\text{W}}/T_{0}\). The difference between the soliton energy found by direct scattering transform (DST) and the analytical value is shown. Labels on the right indicate the number of sample points used. Solid curve: expected truncation error, Equation (28) of the input pulse. a new energy value, as well. In the sense that this may still be interpreted as solitons, some researchers have adopted the following strategy: In a mildly non-integrable system with loss [70] or Raman effect [71; 72; 73], the pulse structure is propagated numerically up to some position \(z_{i}\), where DST is then applied to find what one may call 'local eigenvalues'. Then, one lets \(z_{i}\) slide and repeats DST, in order to find the position dependence, or spatial evolution, of the eigenvalues. An example of such a case is shown in Figure 10 where the Raman effect, third-order dispersion and loss are treated separately. The most conspicuous effects are the energy reduction by the loss term and the frequency downshift by the Raman effect. When it comes to complex situations like in the context of optical supercontinuum generation where many interactions and collisions of several pulses take place, energy can be transferred between pulses, and more changes to local eigenvalues will occur. In any event, one has to interpret the results produced in this way with caution. If one considers infinitely-extended structures, one first needs to acknowledge that they have infinite energy. One possible approach is to start with a finite piece ('window') of the infinite structure and determine eigenvalues, then increment the window width and follow the eigenvalues; finally, let the window width tend to infinity [74]. In the limit, one obtains an infinite number of eigenvalues, but with a finite density (number of eigenvalues per unit time). These eigenvalues cover a continuum of energies from zero up to some maximum; if one considers the simplest case of an unmodulated continuous wave, this maximum depends only on its power \(P_{0}\) and is given by \(E_{\max}=2\sqrt{\left|\beta_{2}\right|P_{0}/\gamma}\). With respect to the interpretation of these eigenvalues as signatures of solitons, the following was shown in [74]: Consider a finite window, which contains \(m\) eigenvalues, and allow it to propagate. On account of integrability, it contains a constant number of \(m\) solitons. If one then applies a perturbation, e.g., by Raman effect, precisely \(m\) separate pulses emerge; once they are sufficiently separated from each other after some propagation, it can be checked that each fulfills the soliton criterion Equation (4). The other approach to infinitely extended fields is to consider periodic problems, like the Akhmediev breather.