# Soliton Content of Fiber-Optic Light Pulses
## 4 Methods to Verify Soliton Content
### Soliton Radiation Beat Analysis
In that situation, however, the applicability of DST even in the approximate sense as described above in Section 4.1.3 is ultimately lost, and interpretations based on that approach [70] are doubtful and at variance to the SRBA results. A further important example of the application of SRBA is the determination of the dispersion-managed soliton content. This case is just not accessible for DST, not even in the sense of 'local' analysis as in Section 4.1.3. Other than SRBA, no other numerical method to analyze DM structures is known. In [88], a suitably-scaled Gaussian pulse was used as a good approximation to an ideal DM soliton, and an SRBA pattern was obtained. Both the modification of the soliton energy and the characteristic minima of soliton radiation beat traces were found. In subsequent research, SRBA was also applied to analyze Akhmediev breathers and a sinusoidally modulated cw [91], as well as soliton crystals [92]. One can assume that in principle, the SRBA method can be applied to all non-integrable systems where a nonlinear phase shift appears. Potential applications are soliton molecules (see Section 3.3.1) or dissipative solitons (see Section 3.3.2). SRBA as described so far relies on the evaluation of the central spectral power \(\widetilde{A}^{2}(\Omega=0,z)\). In a generalization of the technique, the consideration is extended to the full spectrum, \(\widetilde{A}^{2}(\Omega,z)\)[93]. In this variant, the technique also yields information about soliton velocities. Figure 12 shows an example for the case of an \(N=2\) soliton. The pattern was generated using a value of \(N=2.01\), so that there is always a little bit of radiation copropagating with the second-order soliton, and a beat note is generated. Parabola-like curves are due to radiation; horizontal flat lines touching the parabolas are due to solitons. Once the traces are identified, the soliton energies can be obtained from the spatial frequency of the nonlinear phase rotation. To illustrate the interpretation of such data, we point out that the two solitons appear with spatial frequencies in the ratio of 9:1. According to Equation (3), this means that the peak powers are \(P_{0}(S_{1})=9P_{0}(S_{2})\). Equation (4) then dictates that \(T_{0}(S_{1})=\
icefrac{{1}}{{3}}T_{0}(S_{2})\). The ratio of energies obtained from Equation (5) is then 3:1, precisely as expected from Figure 3. Any frequency shifts would appear as horizontal displacements; in this particular example, there are none. There is no scan of \(N\) here; this extended procedure has the advantage that only a single propagation simulation is necessary. A particular difficulty with DM solitons is that their shape undergoes continuous changes. At various positions within \(L_{\text{map}}\), the shape varies considerably, but the pulse shape repeats after \(L_{\text{map}}\) (it is'stroboscopically' stable). Even then, when the power is modified, the shape varies from 'nearly sech' to 'nearly Gaussian', and depending on the dispersion modulation depth, oscillating tails may appear. Therefore, extra measures need to be taken when the energy is scanned for an SRBA pattern calculation. For the data in the left panel of Figure 13, at 0.5 pJ increments of energy, the best approximation to the DM soliton shape was found using [PERSON]'s method [40]. If the pulse shape were exact, there would be no beat note with radiation. This is why in 0.5 pJ increments, there seem to be white lines in the figure. Above and below each such step, the energy was scanned by simple scaling of the pulse shape to \(\pm 0.25\) pJ. The slight deviation from the best pulse shape suffices to produce beat notes, and a full survey of their positions is obtained. For better clarity, the data from this calculation are transferred to the right panel of Figure 13, and different line thicknesses are standardized into three classes in the process. The data show that a DM soliton consists of several constituent solitons; the pertaining traces are labeled \(S_{1}\)-\(S_{3}\).