## 3 The Linear Equations (4) and (5)
### Solutions of Equation (5)  
Now let us investigate how the solutions of Equation (5) behave. As we saw in Figure 4, when the coefficients \(c_{0}\), \(c_{2}\) and \(c_{4}\) satisfy the condition (17), the dispersion relation presents two finite bands of forbidden frequencies (defined in (22)). Therefore, we expect that narrow solitons will exhibit a behavior similar to that of the solutions of the Equation (4). More precisely, we expect that if take an initial condition whose Fourier transform (FT) is wide enough to contain significant frequency components in the intervals \(\omega<-\omega_{2}\) and \(\omega>\omega_{2}\), the pulse will start radiating, but the radiation will stop at some time, when the pulse widens and its FT becomes too narrow. To verify this expectation we obtained the numerical solution of Equation (5) with coefficients \(c_{0}=1/15\), \(c_{2}=1/2\) and \(c_{4}=1/256\), and using an initial condition \(u(0,t)\) of the form (42) with \(A=1\) and \(B=7\). As in the case of Equation (4), we can obtain the solution of Equation (5) by calculating the Fourier transform (FT) of this equation, and the FT of the solution also has the form shown in Equation (30), with \(\lambda_{a,b}(\omega)\) given by Equation (31), but with the function \(S(\omega)\) which appears within the square root now defined as follows:  
\[S(\omega)=-1-4c_{0}\Big{(}c_{4}\omega^{4}-c_{2}\omega^{2}\Big{)} \tag{48}\]  
As this function is positive on two regions of the \(\omega\) axis, the function \(U(z,\omega)\) (given in Equation (30)) would grow exponentially in \(z\), unless we impose the restriction \(a(\omega)=0\). This restriction implies that Equation (41) is also valid in this case (but using Equation (48) in the expression (31) which defines the function \(\lambda_{b}(\omega)\)). Therefore, to solve Equation (5) we used \(u(0,t)\) as defined in Equation (42), and we calculated \(u_{z}(0,t)\) with Equation (41). In Figure 21 we can see the profile of \(|u(z=1.5,t)|\) corresponding to the initial condition (42) with \(A=1\) and \(B=7\). As expected, we see that the pulse starts emitting radiation, but afterwards the radiation stops. The only important difference in comparison with the behavior of the solutions of Equation (4) is that the solution of  
Figure 20: 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 \(A=1\) and \(B=1.5\). Equation (5) emits radiation in both directions, due to the fact that its dispersion relation, shown in Figure 4, is now symmetrical with respect to the wavenumber axis. On the other hand, in Figure 22 we can see the profile of another solution of Equation (5) corresponding to an initial condition of the form (42), but now using \(A=1\) and \(B=2\) (and the same coefficients \(c_{0}\), \(c_{2}\) and \(c_{4}\)). In this case no radiation is observed because the initial pulse was very wide, and its FT was very narrow, and therefore the initial FT has no significant frequency components in the intervals \(\omega<-\omega_{2}\) and \(\omega>\omega_{2}\) (where the resonant frequencies are located). We have thus seen that the behavior of the radiating pulses of Equations (4) and (5) is similar, except that the solutions of Equation (5) emit radiation in both directions, and those of Equation (4) only radiate to the left. However, Equations (4) and (5) differ in a significant aspect. In the case of Equation (4) the function \(S(\omega)\) defined by Equation (32) is always positive in a portion of the real axis, and consequently is always necessary to impose the restriction \(a(\omega)=0\) which leads to the relationship between \(u(0,t)\) and \(u_{z}(0,t)\) given by Equation (41).