## 3 The Linear Equations (4) and (5)
### Solutions of Equation (4)
However, the pulse's width will begin to grow (and its spectrum will become narrower) as a result of the dispersive terms \(c_{2}u_{tt}\) and \(ic_{3}u_{tttt}\) which appear in Equation (4). As the dispersion of the pulse will continue indefinitely (since Equation (4) does not contain any nonlinear term which could cancel this dispersion), a moment will necessarily arrive when the pulse will be too wide, and its spectrum will be too narrow to have frequency components with \(\omega>\omega_{3}\). Moreover, the components of the pulse with frequencies in the interval \(\omega_{2}<\omega<\omega_{3}\) will die quickly, since the dispersion relation (shown in Figure 2) does not permit the propagation of linear waves with these frequencies. Therefore, a moment will arrive when the pulse's spectrum will be essentially contained in the interval \(\omega_{1}<\omega<\omega_{2}\) (where the dispersion relation allows the propagation of linear waves), and then the pulse will no longer be able to resonate with the wave with \(\omega=25\) (the resonant frequency), and the emission of radiation will stop. So, the interruption of the radiation is due to the fact that the pulse's spectrum is confined to live in the interval \(\omega_{1}<\omega<\omega_{2}\) (far from the resonant frequency \(\omega=25\)), and the presence of the forbidden frequency gap \(\omega_{2}<\omega<\omega_{3}\) seen in Figure 2 strongly contributes to this confinement. We have thus seen that a very original consequence of having a dispersion relation given by an elliptic curve such as that shown on Figure 2 is the abrupt interruption of the resonant radiation emitted by a narrow pulse which obeys the linear Equation (4). Therefore, this novel radiation process is the consequence of the _simultaneous_ presence of the non-slowly-varying-envelope term \(u_{zz}\) and the third-order temporal derivative. To close this sub-section it is worth observing that no qualitative changes are to be expected if we introduced the mixed derivative \(u_{zt}\) in Equation (4). The equation thus obtained could be solved by Fourier transforms, just as we solved Equation (4). The form of the Equation (30) would still be valid, although the form of the functions \(\lambda_{a,b}(\omega)\) and \(S(\omega)\) would be slightly different. In this case (when we take into account a term of the form \(cu_{zt}\) in Equation (4)), instead of Equations (31) and (32) we would have:  
\[\lambda_{a,b}=\frac{-i(1-c\omega)\pm\sqrt{S(\omega)}}{2c_{0}} \tag{45}\]  
\[u_{z}(0,t)=\frac{1}{2\pi}\iint\limits_{-\infty}^{+\infty}\lambda_{b}(\omega) \ u(0,\tau)\ e^{i\omega(\tau-t)}d\tau\ d\omega \tag{41}\]0 Fourier transform of the curve shown with the continuous line in \[u_{z}(0,t)=\frac{1}{2\pi}\iint\limits_{-\infty}^{+\infty}\lambda_{b}(\omega) \ u(0,\tau)\ e^{i\omega(\tau-t)}d\tau\ d\omega \tag{41}\]1  
\[u_{z}(0,t)=\frac{1}{2\pi}\iint\limits_{-\infty}^{+\infty}\lambda_{b}(\omega) \ u(0,\tau)\ e^{i\omega(\tau-t)}d\tau\ d\omega \tag{41}\]2 Initial conditions for Equation (4) used to obtain Figure 7 (continuous line) and Figures 8–10 (dashed line). \[u_{z}(0,t)=\frac{1}{2\pi}\iint\limits_{-\infty}^{+\infty}\lambda_{b}(\omega) \ u(0,\tau)\ e^{i\omega(\tau-t)}d\tau\ d\omega \tag{41}\]3 Fourier transform of the curve shown with the dashed line in \[u_{z}(0,t)=\frac{1}{2\pi}\iint\limits_{-\infty}^{+\infty}\lambda_{b}(\omega) \ u(0,\tau)\ e^{i\omega(\tau-t)}d\tau\ d\omega \tag{41}\]4  
\[u_{z}(0,t)=\frac{1}{2\pi}\iint\limits_{-\infty}^{+\infty}\lambda_{b}(\omega) \ u(0,\tau)\ e^{i\omega(\tau-t)}d\tau\ d\omega \tag{41}\]5  
It should be emphasized that the key ingredient which produces the peculiar radiation process shown in Figures 9 and 10 is the presence of gaps of forbidden frequencies in the dispersion relation of Equation (4). It is interesting to investigate if these gaps survive if we introduce a term of the form \(cu_{zt}\) in Equation (4).