## 4 The Nonlinear Equations (6)-(8)
### The Solitons of Equation (7)  
Direct substitution shows that Equation (7) has _moving_ soliton solutions of the following form:  
\[u(z,t)=R\ sech\left(\frac{t-Vz}{W}\right)\ e^{i(qz-rt)} \tag{66}\]  
where \(r\), \(q\), \(V\), \(R\) and \(W\) are constants whose values can be calculated as explained in the following. First we calculate \(r\):  
\[r=-\frac{c_{3}}{4c_{4}} \tag{67}\]and then we define the following parameters:  
\[Y=c_{4}r^{4}+c_{3}r^{3}-c_{2}r^{2} \tag{68}\]  
\[M=\frac{3c_{0}\gamma_{2}}{20c_{4}}\Bigl{(}-8c_{4}r^{3}-2c_{2}r\Bigr{)}^{2}\Bigl{(} \frac{24c_{4}}{\gamma_{2}}\Bigr{)}^{1/2} \tag{69}\]  
\[L=\frac{3\gamma_{2}}{40c_{4}}\biggl{(}\frac{24c_{4}}{\gamma_{2}}\biggr{)}^{1/2 }\biggl{[}\gamma_{1}\biggl{(}\frac{24c_{4}}{\gamma_{2}}\biggr{)}^{1/2}-\frac{3} {4}\frac{c_{3}^{2}}{c_{4}}-c_{2}\biggr{]} \tag{70}\]  
Then we can obtain the possible values of \(q\) from the following equation:  
\[(1+2c_{0}q)^{4}\Bigl{[}\gamma_{1}^{2}-6\gamma_{2}\Bigl{(}q+c_{0}q^{2}-Y\Bigr{)} \Bigr{]}=\Bigl{[}L(1+2c_{0}q)^{2}-M-\gamma_{1}(1+2c_{0}q)^{2}\Bigr{]}^{2} \tag{71}\]  
and with each of the possible values of \(q\) we can calculate \(V\), \(R\) and \(W\) with the following expressions:  
\[V=\frac{-8c_{4}r^{3}-2c_{2}r}{1+2c_{0}q} \tag{72}\]  
\[R^{2}=\frac{1}{20c_{4}}\biggl{(}\frac{24c_{4}}{\gamma_{2}}\biggr{)}^{1/2} \biggl{[}\gamma_{1}\biggl{(}\frac{24c_{4}}{\gamma_{2}}\biggr{)}^{1/2}-\frac{3} {4}\frac{c_{3}^{2}}{c_{4}}-2c_{2}-2c_{0}V^{2}\biggr{]} \tag{73}\]  
\[W=\biggl{(}\frac{24c_{4}}{\gamma_{2}}\biggr{)}^{1/4}\frac{1}{R} \tag{74}\]  
As Equation (71) is a sixth-order equation for \(q\), we may have up to six different values for \(q\), and consequently Equation (7) might have six different soliton solutions for a given set of coefficients \(\{c_{0}\), \(c_{2}\), \(c_{3}\), \(c_{4}\), \(\gamma_{1}\), \(\gamma_{2}\}\). This result suggests that Equation (7) might have a continuous family of soliton solutions, in which case the solitons defined by Equations (66)-(74) would only be particular elements of this family that can be expressed in terms of hyperbolic secants. Now let us investigate if the solitons defined by Equations (66)-(74) are standard or embedded. Apparently, in order to determine if the soliton (66) is embedded, we would only need to investigate if the soliton's wavenumber \(q\) (defined by Equation (71)) is contained in the range of wavenumbers permitted by the linear dispersion relation corresponding to Equation (7), which is given by:  
\[c_{0}k^{2}+k=c_{4}\omega^{4}+c_{3}\omega^{3}-c_{2}\omega^{2} \tag{75}\]  
However, this is not true. When we deal with a _moving soliton_ (i.e., a soliton which moves along the retarded time axis, as the soliton in Equation (66)), the determination if the soliton is embedded is more involved. As explained in Ref. [18], in this case we must investigate if the _intrinsic_ soliton's wavenumber defined as:  
\[r=-\frac{c_{3}}{4c_{4}} \tag{67}\]0  
is contained in the range of a _modified_ dispersion relation, which is obtained by substituting a linear wave written in the form \(u=\exp[kz-\omega(t-Vz)]\) into the linear part of the equation under study. In the case of Equation (7), this modified dispersion relation takes the form:  
\[r=-\frac{c_{3}}{4c_{4}} \tag{67}\]1  
This equation permits us to write k as function of \(\omega\), but now something unexpected occurs. As Equation (77) involves \(V\), and \(V\) depends on \(q\) (as shown in Equation (72)), the dispersion relation (77)will depend on the value of \(q\). In other words: _we will have a different dispersion relation for each of the different real values of \(q\) defined by Equation (71)_. Consider, for example, the following set of coefficients:  
\[r=-\frac{c_{3}}{4c_{4}} \tag{67}\]2  
Substituting these values in Equations (67)-(70) we obtain the values of \(Y\), \(M\) and \(L\).