## 4 The Nonlinear Equations (6)-(8)
### The Solitons of Equation (8)  
Equation (8) is worth being studied because, in spite of being a particular case of Equation (6), it has soliton solutions which are _different_ from those of Equation (6). Direct substitution shows that Equation (8) has exact soliton solutions of the following form:  
\[u(z,t)=D\;sech^{2}(Et)\;exp(iH_{1,2}z) \tag{81}\]  
where the parameters \(D,E\) and \(H\) are defined by following equations:  
\[D^{2}=-\frac{3}{10}\frac{c_{2}^{2}}{c_{4}\gamma_{1}} \tag{82}\]  
\[E^{2}=-\frac{c_{2}}{20c_{4}} \tag{83}\]  
\[H_{1,2}=(2c_{0})^{-1}\left(-1\pm\sqrt{1-\frac{16}{25}\frac{c_{0}c_{2}^{2}}{c_{ 4}}}\right) \tag{84}\]  
Optical solitons with profiles given by _squared_ hyperbolic secants are not unknown in optics [50; 51], but they are far less common than the typical _sech-type_ solitons. Consequently, it is interesting that the solitons of Equation (8) have the form given by Equation (81). Moreover, it is also worth observing that even though Equation (6) reduces to Equation (8) when \(\gamma_{2}=0\), neither of the two soliton solutions of Equation (6) (given either by Equations (52)-(55) or by Equations (60)-(65)) reduces to the soliton solution of Equation (8) given by Equations (81)-(84) in the limit when \(\gamma_{2}\to 0\). This is not an unusual situation. It occurs in many systems with embedded solitons. For example, the equation:  
\[iu_{z}+\varepsilon_{1}u_{tt}+\varepsilon_{2}u_{4t}+\gamma_{1}|u|^{2}u-\gamma_ {2}|u|^{4}u=0 \tag{85}\]  
reduces to the standard NLS equation when \(\varepsilon_{2}\to 0\) and \(\gamma_{2}\to 0\), but it is known that the solitons of Equation (85) do not reduce to NLS solitons in this limit [11]. In a similar way, Equation (8) reduces to the NLS equation when \(c_{0}=c_{4}=0\), but the solution defined by Equations (81)-(84) does not reduce to a NLS soliton in this case. In fact, as \(c_{4}\to 0\) the soliton's height (given by the parameter \(D\) defined by Equation (82)) tends to infinity, and the soliton's width (given by the parameter \(E^{-1}\) defined by Equation (83)) tends to zero.  
Are the soliton solutions of Equation (8) _embedded solitons_? This is a question which deserves some attention. In the case when the inequality (17) holds, it is obvious that the solitons defined by Equations (81)-(84) are embedded, since in this case the wavenumbers permitted by the linear dispersion relation (15) cover the entire real axis (as explained in Section 2), and therefore the solitons' wavenumbers \(H_{1,2}\) will necessarily be contained in the range of this dispersion relation. On the other hand, when the inequality (18) holds, the range of the dispersion relation will contain a band of forbidden wavenumbers, and the boundaries of this band are the values \(k_{1}\) and \(k_{2}\) given by Equation (20). In this case it is not evident if the solitons defined by Equations (81)-(84) are embedded or not. However, if we choose values of \(c_{0}\) and \(c_{4}\) which lead to real values of \(H_{1,2}\) (when substituted in Equation (84)), an algebraic exercise shows that:  
* if \(c_{4}>0\) then: \(H_{1}<k_{1}\) and \(H_{2}>k_{2}\Rightarrow\) the solitons are embedded
* if \(c_{4}<0\) then: \(k_{1}<H_{1}<H_{2}<k_{2}\Rightarrow\) the solitons are NOT embedded  
Therefore, the soliton solutions of Equation (8) are interesting because, depending on the values of \(c_{0}\), \(c_{2}\) and \(c_{4}\), they may be embedded solitons or standard ones.