## 3 The Linear Equations (4) and (5)
### Solutions of Equation (4)
As this growth is completely unphysical (as it would imply a fictitious energy growth), the coefficient \(a(\omega)\) has to be equal to zero, and consequently from Equation (36) it follows that there is an unexpected relationship between \(U_{z}(0,\omega)\) and \(U(0,\omega)\):  
\[U_{z}(0,\omega)=\lambda_{b}(\omega)U(0,\omega) \tag{40}\]  
thus implying that:  
\[u_{z}(0,t)=\frac{1}{2\pi}\iint\limits_{-\infty}^{+\infty}\lambda_{b}(\omega) \ u(0,\tau)\ e^{i\omega(\tau-t)}d\tau\ d\omega \tag{41}\]  
Therefore, as far as Equation (4) is concerned, the perplexing situation associated to the Consequence 1 (mentioned in the previous sub-section) is completely clarified: physically realistic solutions of Equation (4) are indeed completely determined by the initial condition \(u(0,t)\), as physics suggested. Once \(u(0,t)\) is chosen, the function \(u_{z}(0,\ t)\) is determined by Equation (41). Now let us calculate the solution of Equation (4) corresponding to a soliton-like initial condition of the form:  
\[u(0,t)=A\ \text{sech}(Bt) \tag{42}\]  
If we take the coefficients \(c_{0}=1/40\), \(c_{2}=1/2\) and \(c_{3}=1/50\) in Equation (4), and the parameters \(A=1\) and \(B=1.5\) in Equation (42), the evolution of the pulse can be seen in Figure 7, which shows the form of \(|u(z,t)|\) for \(z=0.5\), \(1.0\) and \(1.5\). We do not see anything special in this figure, which is somewhat disappointing, since we expected an unusual behavior due to the awkward form of the dispersion relation \(k(\omega)\) shown in Figure 2.  
If we now obtain the solution of Equation (4) (with the same coefficients indicated above) and using again an initial condition of the form (42), but with \(A=1/2\) and \(B=14\), the result is more interesting. In this case small ripples are generated on the leading edge of the pulse (i.e., the left hand side of the pulse), as shown in Figure 8.  
As the pulse advances along the fiber these ripples form a well-defined radiation wavetrain which propagates to the left of the pulse, as shown in Figures 9 and 10. Figure 8: Modulus \(|u(z=0.05,\ t)|\) of the solution of Equation (4) with \(c_{0}=1/40\), \(c_{2}=1/2\), \(c_{3}=1/50\) and the initial condition (42) with \(A=1/2\) and \(B=14\). Figure 7: Modulus \(|u(z,t)|\) of the solution of Equation (4) with \(c_{0}=1/40\), \(c_{2}=1/2\), \(c_{3}=1/50\) and the initial condition (42) with \(A=1\) and \(B=1.5\). The shape of \(|u(z,t)|\) is shown for \(z=0.5\) (continuous), \(1.0\) (dashed) and \(1.5\) (dotted). At first sight, the radiation seen in Figure 9 might seem similar to the small-amplitude radiation waves emitted by solitary pulses which obey equations such as the following [1;2;11]:  
\[iu_{z}+c_{2}u_{tt}-ic_{3}u_{tttt}+\gamma_{1}|u|^{2}u=0 \tag{43}\]  
\[iu_{z}+c_{2}u_{tt}+c_{4}u_{tt}+\gamma_{1}|u|^{2}u-\gamma_{2}|u|^{4}u=0 \tag{44}\]  
However, Figures 9 and 10 show that the radiation emitted by the solution of Equation (4) possesses a unique characteristic, that no other model of optical pulses has ever predicted: _the emission of radiation stops abruptly at a certain point along the retarded-time axis, and this point moves away from the pulse's peak at a constant velocity_. To understand clearly what this means we should remember that the independent variable \"\(t\)\" which appears in Equations (1), (4)-(8) and (26), and appears at the horizontal axes in Figures 8-10, is not the laboratory time, but the so-called _retarded time_. Therefore, the graphs shown in Figures 8-10 display the time-dependence of the square of the light intensity as measured by observers placed at different points (i.e., different values of \(z\)) along the optical fiber, but each of these observers has shifted his/her time origin in such a way that the time \(t=0\) corresponds to the instant when the pulse's maximum passes through the point \(z\).