## 4 The Nonlinear Equations (6)-(8)
### The Solitons of Equation (6)  
Equation (6) is an interesting generalization of Equation (5) because it accepts soliton solutions of two different types. The first of these solutions is the following:  
\[u(z,t)=A\;\text{sech}(Bt)\;e^{iCz} \tag{52}\]  
where \(A\), \(B\) and \(C\) are the following constants:  
\[B^{2}=\frac{\gamma_{1}}{20c_{4}}\bigg{(}\frac{24c_{4}}{\gamma_{2}}\bigg{)}^{1/ 2}-\frac{c_{2}}{10c_{4}} \tag{53}\]  
\[A=\bigg{(}\frac{24c_{4}}{\gamma_{2}}\bigg{)}^{1/4}B \tag{54}\]  
\[C=(2c_{0})^{-1}\bigg{[}-1\pm\sqrt{1+4c_{0}(c_{4}B^{4}+c_{2}B^{2})}\bigg{]} \tag{55}\]  
Equation (55) defines two values for \(C\), and consequently Equation (6) has two solutions of the form (52) with different wavenumbers \(C_{1}\) and \(C_{2}\) corresponding, respectively, to the signs \"plus\" and \"minus\" in front of the radical that appears in Equation (55). It is worth observing that the solution defined by Equations (52)-(55) is the particular solution of Equation (6) corresponding to the initial conditions:  
\[u(0,t)=A\;\text{sech}(Bt) \tag{56}\]  
\[u_{z}(0,t)=iC\;u(0,t) \tag{57}\]  
with \(A\), \(B\) and \(C\) given by Equations (53)-(55). We can see that Equation (57) has the same form as Equation (50), but with a particular value of the wavenumber. Therefore, the use of initial conditions of the form (50) in order to look for particular solutions of Equations (5) and (6) seems justified. We must now remember that in Section 2 we saw that if the inequality (17) holds, the range of wavenumbers permitted by the linear dispersion relation of Equation (6) (given by Equation (15) or, alternatively, by Equation (21)), covers the entire real axis, and consequently, in this case (when \(c_{0}c_{2}^{2}>c_{4}\)) the soliton wavenumbers \(C_{1}\) and \(C_{2}\) are obviously contained within the range of wavenumbers permitted by the linear dispersion relation of the system. This characteristic implies that these solitons are not standard ones (since standard solitons have wavenumbers lying _outside_ the linear spectrum of the system), but they are _embedded solitons_. It is worth remembering that prior to 1997 it was believed that soliton wavenumbers must necessarily lie _outside_ the linear spectra of the systems. Otherwise (it was believed) the solitons would resonate with the small-amplitude linear waves capable of propagating in the system, and the solitons would emit resonant radiation. However, in 1997 it was discovered that _exact_ radiationless soliton solutions with wavenumbers contained within the range of the linear dispersion relation of the system may indeed exist [11], and the term \"embedded soliton\" was coined two years later to distinguish these peculiar solitons from the standard ones [45; 46]. In Figure 4 we showed the form of the dispersion relation, given by Equation (21), in the particular case when \(c_{0}=1/15\), \(c_{2}=1/2\) and \(c_{4}=1/256\). This figure also shows the position of the soliton wavenumbers, \(C_{1}\) and \(C_{2}\), given by Equation (55) with \(\gamma_{1}=5\) and \(\gamma_{2}=1\). If we look at this figure an interesting question arises: how is the frequency spectrum of the exact soliton solution (52) in comparison to the dispersion relation? To answer this question, in Figure 23 we have superimposed the Fourier transform (FT) of the exact soliton (52) and the _scaled_ dispersion relation \(k(\omega)/8.6287\). The scaling factor (8.6287)\({}^{-1}\) was introduced for the amplitude of the FT and the scaled dispersion relation to have the same size at \(\omega=0\). This figure shows that the amplitude of the spectrum of the exact soliton solution is completely insignificant in the range of frequencies where the lateral branches of the dispersion relation are placed.