# Soliton Content of Fiber-Optic Light Pulses
## 3 Extensions to the Nonlinear Schrodinger Equation
### Dispersion-Managed Fibers  
As a further complication in a realistic assessment of pulse propagation in fibers, one must acknowledge that in the telecommunications field, a special type of fiber has been preferred for 20 years now in which \(\beta_{2}\) is not a constant. Originally, an attempt was made to cancel dispersive effects by concatenating segments of fiber with alternatingly positive and negative group velocity dispersion. As it turned out, a perfect cancellation (path average \(\overline{\beta}_{2}=0\))--possible at a single wavelength only anyway--did not work as well as if a small negative value were left. This is on account of the fiber's nonlinearity.  
Figure 6 illustrates the structure of a dispersion-managed fiber (DM fiber). The dispersion map is described by the spatial period \(L_{\mathrm{map}}=L^{+}+L^{-}\) with which the alternating pattern is repeated [31]. Strictly speaking, the value of the nonlinearity coefficient \(\gamma\) will also alternate from one fiber segment to the other.  
It is not immediately clear whether in a dispersion-managed (DM) fiber something like solitons can exist, as they are solutions of the NLSE for anomalous dispersion only. However, as it turned out, a type of pulse exists that is stabilized by nonlinearity [32; 33; 34; 35; 36; 37]. These pulses were then called dispersion-managed (DM) solitons. Their shape is neither of the usual unchirped sech type, nor constant. Rather, it breathes over a full dispersion period so that after a full period, the shape in both the amplitude and phase profile is restored. An example of a DM sol  
Figure 6: Dispersion-managed fiber consists of alternatingly normally (dispersion compensating fiber (DCF)) and anomalously dispersive fiber (standard single mode fiber (SSMF)) segments. The dispersion parameter \(\beta_{2}\) alternates between values of \(\beta_{2}^{+}\) and \(\beta_{2}^{-}\); the lengths of segments are \(L^{+}\) and \(L^{-}\), respectively. The dispersion map period is \(L_{\mathrm{map}}=L^{+}+L^{-}\).  
Mathematically, dispersion management is captured by making fiber parameters \(z\)-dependent:  
\[i\frac{\partial}{\partial z}A-\frac{\beta_{2}(z)}{2}\frac{\partial^{2}}{ \partial t^{2}}A+\gamma(z)|A|^{2}A=0\quad. \tag{17}\]  
This modifies the NLSE into a non-integrable form so that it does not support solitons in the usual sense. Depending on the dispersion modulation strength (essentially, \(\beta_{2}^{+}-\beta_{2}^{-}\)), the DM soliton shape may be nearly sech-shaped or closer to a Gaussian [38]. In the strong modulation limit DM, solitons exhibit oscillating tails [38; 39]. There is no closed analytical expression for it, but good approximations to its shape can be found by numerical procedures like that given in [40]. During propagation, a DM soliton suffers from continuous radiative loss, which may be weak, but nevertheless must lead to its eventual decay. Fortunately, for applications, the decay distance typically exceeds any practically relevant fiber length.  
It was eventually understood that dispersion-management provides advantages over a constant-dispersion fiber. It suppresses four-wave mixing, a process occurring in a transmission system with many wavelength channels filled simultaneously with independent data streams, where it leads to cross-talk and, thus, errors. An extensive review of dispersion-management is given in [31].