## 3 The Linear Equations (4) and (5)
### Solutions of Equation (5)
On the other hand, in the case of Equation (5), depending on the values of the coefficients \(c_{0}\), \(c_{2}\) and \(c_{4}\), the function \(S(\omega)\) defined by Equation (48) _may_ be positive over certain frequency intervals, and in these cases the restriction  
Figure 21: Modulus \(|u(z=1.5,\ t)|\) of the solution of Equation (5) with \(c_{0}=1/15\), \(c_{2}=1/2\), \(c_{4}=1/256\) and the initial condition (42) with \(A=1\) and \(B=7\). Figure 22: Same as Figure 21, but with \(B=2\). \(a(\omega)=0\) is necessary, and the Equation (41) again defines a necessary relation between \(u(0,t)\) and \(u_{z}(0,t)\). However, for certain coefficients, the function \(S(\omega)\) might be always negative, and then the values of the functions \(\lambda_{a,b}(\omega)\) would be imaginary for all frequencies. In these cases there is no reason to impose the restriction \(a(\omega)=0\), and with the disappearance of this restriction also the relations (40) and (41) disappear. Therefore, in the case of Equation (5), when \(\lambda_{a}\) and \(\lambda_{b}\) are both imaginary, we face again the question: how do we specify the initial condition \(u_{z}(0,t)\)? In the following paragraph we address this question. When \(a(\omega)\
eq 0\) the Equations (40) and (41) no longer hold, but we can obtain an approximate relation between \(u_{z}(0,t)\) and \(u(0,t)\) by considering what happens when we send a light pulse at the beginning of an optical fiber. As the pulse enters into the fiber, the wavelength of the carrier wave changes (since the light's velocity is different inside the fiber), and this change introduces a factor \(\exp(i\Delta k\,z)\) in the mathematical form of the pulse. In other words, at the beginning of the fiber (inside the fiber, near the point \(z=0\)) the form of the light pulse will be:  
\[u(z,t)=f(t)\;e^{i\Delta k\,z} \tag{49}\]  
where \(f(t)\) defines the temporal profile of the pulse. Consequently, from Equation (49) it follows that:  
\[u_{z}(0,t)=i\Delta k\;u(0,t) \tag{50}\]  
and this equation, whose FT is similar to Equation (40), implies that the form of the function \(u(0,t)\) determines the form of \(u_{z}(0,t)\). It should be noticed, however, that we do not know the value of the parameter \(\Delta k\). This parameter depends on the exact form of the refractive index of the fiber as a function of the light intensity. However, if we approximate the wavelength of the light inside the fiber in the form \(\lambda\cong\lambda_{0}/n_{0}\), where \(\lambda_{0}\) is the wavelength outside the fiber, and \(n_{0}\) is the intensity-independent part of the refractive index, then we can approximate \(\Delta k\) in the form:  
\[\Delta k\cong 2\pi(n_{0}-1)/\lambda_{0} \tag{51}\]  
Therefore, when there is no justification for using Equation (41) to calculate \(u_{z}(0,t)\), this function might be calculated by means of the approximate Equations (50) and (51). In fact, in the following section we will see that the exact soliton solutions of Equation (6) are indeed consistent with a linear relationship between \(u_{z}(0,t)\) and \(u(0,t)\), such as that given by Equation (50). To close this section, it is important to emphasize that the peculiar radiation behavior observed in Figures 9, 10, 19 and 21 is a direct consequence of the presence of _bands of forbidden frequencies_ in the dispersion relations of Equations (4), (5) and (47), and these forbidden bands are the result of the interplay between the non-SVEA tem \(u_{zz}\) and higher-order dispersive terms such as \(-iu_{ttt}\) or \(u_{4t}\). Therefore, the principal lesson to be learnt from the study of Equations (4), (5) and (47) is the following: when a very short pulse is launched along an optical fiber, some of the frequencies which are contained in the initial pulse may not be permitted to travel along the fiber, since _bands of forbidden frequencies_ may appear in the dispersion relation of the equation which controls the propagation of the pulse.