# Soliton Content of Fiber-Optic Light Pulses
## 2 The Nonlinear Schrodinger Equation and Some of Its Solutions
### Breather Solutions  
There is a different family of solutions of the NLSE, which is best explained by starting with the continuous wave (cw) case. The NLSE is solved by the cw ansatz:  
\[A=\sqrt{P_{0}}\,\mathrm{e}^{i\gamma P_{0}z} \tag{8}\]  
with constant power \(P_{0}\). It turns out that this solution is stable for normal (\(\beta_{2}>0\)) and unstable for anomalous dispersion (\(\beta_{2}<0\)). The instability implies growth of small perturbations of the cw; this is known as modulation instability [1]. The frequency of maximum gain is offset by \(\pm\omega_{\mathrm{max}}\) from the carrier frequency, with:  
\[\omega_{\mathrm{max}}=\sqrt{\frac{2\gamma P_{0}}{|\beta_{2}|}}\quad. \tag{9}\]  
If the cw is perturbed by white noise, there is always energy in these Fourier components, and a ripple with frequency \(\omega_{\mathrm{max}}\) will begin to grow with gain:  
\[g_{\mathrm{max}}=2\gamma P_{0}\quad. \tag{10}\]  
The further fate was widely appreciated only a couple of years ago even though the mathematical results had all been worked out as early as 1986 [15]. There is a family of solutions of the NLSE characterized by an infinitely extended cw background and some modulation on top. Its general form can be written as [16]:  
\[A(a,z,t)=\sqrt{P_{0}}\,\left[1+M(a,z,t)\right]\exp(i\gamma P_{0}z)\quad. \tag{11}\]  
where \(M(a,z,t)\) describes the modulation term. In the case of what is now known as the Akhmediev breather, it takes the form:  
\[M(a,z,t)=\frac{2(1-2a)\cosh\left[b(a)\gamma P_{0}z\right]+ib(a)\sinh\left[b( a)\gamma P_{0}z\right]}{\sqrt{2a}\,\cos\left[\omega_{\mathrm{mod}}t\right]- \cosh\left[b(a)\gamma P_{0}z\right]}\quad. \tag{12}\]  
with parameters \(0<a<\
icefrac{{1}}{{2}}\) and \(b(a)=\sqrt{8a-16a^{2}}\). The modulation frequency is \(\omega_{\mathrm{mod}}=\omega_{\mathrm{c}}\sqrt{1-2a}\), and:  
\[\omega_{\mathrm{c}}=\sqrt{\frac{4\gamma P_{0}}{|\beta_{2}|}}\geq|\omega_{ \mathrm{mod}}| \tag{13}\]  
is the frequency range where gain is possible. Gain maximum occurs at \(a=\
icefrac{{1}}{{4}}\) with \(\omega_{\mathrm{max}}=\omega_{\mathrm{mod}}/\sqrt{2}\), and Equations (9) and (10) are recovered. This particular case is illustrated in Figure 4. From a cw background, the modulation grows, until at \(z=0\), it reaches a maximum amplitude. Here, one has a periodic train of pulses. During further evolution, the modulation decays until for \(z\to\infty\), the cw is recovered.  
Figure 4: Evolution of an Akhmediev breather at \(a=1/4\). \(T_{\mathrm{mod}}=2\pi/\omega_{\mathrm{mod}}\).  
The first experiments to demonstrate the Akhmediev breather in optical fibers were described in [17] and in more detail in [18]. The process was initiated either from random perturbation [17] or from a suitable weak modulation as suggested in [19] and performed experimentally in [18].  
If we let \(a\to\
icefrac{{1}}{{2}}\), this implies that \(\omega_{\text{mod}}\to 0\), and the temporal separation between the pulses in the train diverges. The modulation then takes the form:  
\[M(z,t)=-\frac{4(1+i2\gamma P_{0}z)}{1+\omega_{\text{c}}^{2}t^{2}+4(\gamma P_{0 }z)^{2}}\quad. \tag{14}\]  
This is a structure that is localized in both time and space, known as the Peregrine soliton [20]. It is shown in Figure 5 (left); a first experimental demonstration was given in [21]. A further member of this family of NLSE solutions was studied in [22] and is known as the Kuznetsov-Ma soliton. Mathematically, that solution is described by the modulation term:  
\[M(a,z,t)=\frac{2(1-2a)\text{cos}\big{[}\left|b(a)\right|\gamma P_{0}z\big{]}-i \left|b(a)\right|\text{ sin}\big{[}\left|b(a)\right|\gamma P_{0}z\big{]}}{\sqrt{2a} \text{ cosh}\big{[}\left|\omega_{\text{m}}\right|t\big{]}-\text{cos}\big{[} \left|b(a)\right|\gamma P_{0}z\big{]}} \tag{15}\]  
with a parameter range of \(a>\
icefrac{{1}}{{2}}\). It consists of a periodically-recurring peak; see Figure 5 (right).