# Soliton Content of Fiber-Optic Light Pulses
## 3 Extensions to the Nonlinear Schrodinger Equation  
### Generalized Nonlinear Schrodinger Equation  
The NLSE neglects linear loss, approximates dispersion by the leading group velocity dispersion term and ignores nonlinear effects other than the intensity dependence of the refractive index. Depending on the situation and/or the required accuracy of the calculated results, terms must be added to include what is not represented. This leads to a generalized NLSE, which may take the form:  
\[i\frac{\partial}{\partial z}A-\frac{\beta_{2}}{2}\frac{\partial^{2 }}{\partial t^{2}}A - \underbrace{\frac{i\beta_{3}}{6}\frac{\partial^{3}}{\partial t^{3} }A+\frac{\beta_{4}}{24}\frac{\partial^{4}}{\partial t^{4}}A\ldots+\gamma|A|^{ 2}A}_{\text{higher-order dispersion}} \tag{16}\] \[+ \underbrace{\frac{i\gamma_{0}}{\partial t}\Big{(}|A|^{2}A\Big{)} }_{\text{self-steepening}}-\underbrace{T_{\text{R}}\gamma\,A\frac{\partial}{ \partial t}|A|^{2}}_{\text{Raman}}+\underbrace{i\frac{\alpha}{2}A=0}_{\text{ loss}}\quad.\]  
As the generalized NLSE is not integrable, it usually must be treated numerically. The impact of the new terms is as follows:  
The dispersion curve can be written as a Taylor expansion around the carrier frequency \(\omega_{0}\); the NLSE is written so that the \(\beta_{2}\) term appears. All higher-order terms (shown here up to the fourth order) provide corrections that become relevant when either the pulse spectrum gets quite wide or when the carrier frequency is close to a zero-dispersion point (wavelength where \(\beta_{2}\) passes through zero) of the fiber.  
Figure 5: Evolution of a Peregrine soliton (left) and a Kuznetsov–Ma soliton (right). \(T_{\text{c}}=2\pi/\omega_{\text{c}}\).  
The'self-steepening' term [1; 23] results from the intensity dependence of the group velocity and leads to an asymmetry in the pulse shape. It can produce shifts in spectral and temporal positions even in the absence of the Raman term.  
The Raman effect leads to a redistribution of spectral power to the advantage of the low-frequency slope of the pulse spectrum. A first study of the Raman gain in fibers was done by [PERSON] et al. [24]. The gain curve peaks at a frequency offset of \(\approx\)13 THz, but is nonzero all the way down to zero offset. As a result, the spectral 'center of mass' of a pulse is shifted to lower frequencies. This'redshift' scales with the inverse fourth power of the pulse width [25]. The term expressing this in the NLSE is shown here in a simplified version, which contains the Raman response time \(T_{\mathrm{R}}\). This model is valid for pulses being longer than \(\approx\)100 fs; more accurate models are described in [26; 27; 28].  
The loss term contains [PERSON]'s loss coefficient \(\alpha\); the impact of the loss can be characterized by the characteristic loss length \(L_{\alpha}=\
icefrac{{1}}{{\alpha}}\). As long as the loss is weak, i.e., \(L_{\alpha}\gg L_{\mathrm{D}},L_{\mathrm{NL}}\), the equilibrium of dispersive and nonlinear effects that characterizes the soliton is only mildly perturbed. In that situation, the soliton can rearrange its shape: With the peak power drooping and the width increasing slightly, Equation (4) can still be approximately fulfilled even when the energy is reduced [29; 30].