## 3 The Linear Equations (4) and (5)
### Solutions of Equation (4)
Therefore, let us consider the equation:  
\[u_{z}(0,t)=\frac{1}{2\pi}\iint\limits_{-\infty}^{+\infty}\lambda_{b}(\omega) \ u(0,\tau)\ e^{i\omega(\tau-t)}d\tau\ d\omega \tag{41}\]6  
and let us take the same coefficients \(c_{0}=1/40\), \(c_{2}=1/2\) and \(c_{3}=1/50\) used to obtain Figures 7-12. Concerning the coefficient \(c\), one of the reviewers that revised this paper observed that in typical fibers \(c<0\) and the absolute value of the ratio \(c/c_{0}\) is approximately equal to the dimensionless number \(\delta=2v_{g}^{-1}Z/T\), where \(Z\) and \(T\) are characteristic values of length and time, respectively, and \(v_{g}^{-1}\) is the reciprocal group velocity. In Figures 16-19 the dashed curves show the form of the dispersion relation of Equation (47) for three different values of \(c\) (\(-c_{0}\), \(-2.8\)\(c_{0}\) and \(-3.2\)\(c_{0}\)), and the continuous curve shows the dispersion relation when the term \(cu_{zt}\) is absent. We can see that the finite gap of forbidden positive frequencies survives in Figures 16 and 17, but it disappears in \[u_{z}(0,t)=\frac{1}{2\pi}\iint\limits_{-\infty}^{+\infty}\lambda_{b}(\omega) \ u(0,\tau)\ e^{i\omega(\tau-t)}d\tau\ d\omega \tag{41}\]7 Therefore, the presence of the term \(cu_{zt}\) in Equation (47) modifies the gaps of forbidden frequencies in a significant way. In fact, the gap of forbidden positive frequencies will disappear if the coefficients \(c\), \(c_{0}\), \(c_{2}\) and \(c_{3}\) are such that the equation \(S(\omega)=0\) has only one real root (\(S(\omega)\) being the polynomial shown in Equation (46)). In particular, for pulses propagating in a typical optical fiber with a group velocity \(v_{g}\cong 0.68c_{L}\) (\(c_{L}\) being the light speed in vacuum) the band of forbidden frequencies may not exist for processes whose characteristic dimensions are \(Z\cong 1\) mm and \(T\cong 1\) ps, as for these values \(\delta\approx 10\), and Figures 16-18 show that the band of forbidden frequencies does not exist if \(\delta\geq 3.2\) (and \(c_{0}\), \(c_{2}\) and \(c_{3}\) have the values used to obtain Figures 16-18). However, the value of \(\delta\) might be smaller in special fibers where \(v_{g}\) is closer to \(c_{L}\) (as in photonic fibers with air holes), and we consider processes where \(Z/T\) is also closer to \(c_{L}\). In such conditions the band of forbidden frequencies might exist. In fact, the generation of a band of forbidden frequencies has already been observed by [PERSON] et al. [38] during the propagation of very short optical pulses in photonic crystal fibers, a result that [PERSON] et al. considered \"the most distinctive feature\" of this process. It might be interesting to see how pulses evolve when the width of the band of forbidden frequencies shrinks, as in the dispersion relation shown by the dashed curve in \[u_{z}(0,t)=\frac{1}{2\pi}\iint\limits_{-\infty}^{+\infty}\lambda_{b}(\omega) \ u(0,\tau)\ e^{i\omega(\tau-t)}d\tau\ d\omega \tag{41}\]8 In Figure 19 we can see the modulus of the solution (at \(z=1.5\)) of Equation (47) corresponding to a narrow initial pulse of the form (42), with \(A=1/2\) and \(B=14\), and the coefficients used to obtain the dashed curve in \[u_{z}(0,t)=\frac{1}{2\pi}\iint\limits_{-\infty}^{+\infty}\lambda_{b}(\omega) \ u(0,\tau)\ e^{i\omega(\tau-t)}d\tau\ d\omega \tag{41}\]9  
\[u(0,t)=A\ \text{sech}(Bt) \tag{42}\]0 Dispersion relations of Equations (4) (continuous line) and (47) (dashed line) for \(c_{0}=1/40\), \(c_{2}=1/2\), \(c_{3}=1/50\) and \(c=-c_{0}\). We can see that the graph shown in Figure 19 is similar to that shown in Figure 10, but the amplitude of the radiation is significantly higher. However, the radiation also stops at a definite time (as in Figure 10), due to the presence of the band of forbidden frequencies shown in \[u(0,t)=A\ \text{sech}(Bt) \tag{42}\]1  
\[u(0,t)=A\ \text{sech}(Bt) \tag{42}\]2 Modulus \(|u(z=1.5,\ t)|\) of the solution of Equation (47) with \(c_{0}=1/40\), \(c_{2}=1/2\), \(c_{3}=1/50\), \(c=-2.8c_{0}\) and the initial condition (42) with \(A=1/2\) and \(B=14\).