## 2 Dispersion Relations and Elliptic Curves
### Dispersion Relation of Equation (5)  
Now let us direct our attention to Equation (5). If we substitute the plane wave (9) into Equation (5) we arrive at the following dispersion relation:  
\[c_{0}k^{2}+k+\Big{(}c_{2}\omega^{2}-c_{4}\omega^{4}\Big{)}=0 \tag{15}\]  
Let us now study the form of this dispersion relation in more detail. To determine the range of wavenumbers permitted by this relation it is convenient to write it in the form:  
\[\omega^{2}=(2c_{4})^{-1}\bigg{\{}c_{2}\pm\sqrt{c_{2}^{2}+4c_{4}(c_{0}k^{2}+k)} \bigg{\}} \tag{16}\]  
Figure 2: Dispersion relation of Equation (4), given by Equation (12), with \(c_{0}=1/40\), \(c_{2}=1/2\) and \(c_{3}=1/50\). This equation implies that every real value of \(k\) will be permitted (i.e., every real value of \(k\) corresponds to a positive \(\omega^{2}\)) if and only if:  
\[c_{0}c_{2}^{2}>c_{4} \tag{17}\]  
In the opposite case, when:  
\[c_{0}c_{2}^{2}<c_{4} \tag{18}\]  
the dispersion relation contains a band of forbidden wavenumbers. When the condition (18) is satisfied, the analysis of the function defined by Equation (16) shows that \(\omega^{2}\) turns out to be negative if \(k\) is in the interval:  
\[k_{1}<k<k_{2} \tag{19}\]  
where:  
\[k_{1,2}=(2c_{0})^{-1}\biggl{\{}-1\pm\sqrt{1-c_{0}c_{2}^{2}/c_{4}}\biggr{\}} \tag{20}\]  
Therefore, when the condition (18) holds, the inequalities (19) define a band of forbidden wavenumbers. In Figure 3 we can see the shape of the dispersion relation \(k(\omega)\) defined by Equation (15) when \(c_{0}=1/30\), \(c_{2}=1/2\) and \(c_{4}=1/80\). As these values satisfy the condition (18), the gap of forbidden wavenumbers seen in the figure is explained. The dashed horizontal lines indicate the wavenumbers \(C_{1}\) and \(C_{2}\) of the soliton solutions of Equation (6) (with \(\gamma_{1}=5\) and \(\gamma_{2}=1\)) that will be determined in Section 4. Now let us study in more detail what happens when the condition (17) holds. In this case the linear dispersion relation presents _two bands of forbidden frequencies_, which is a novel situation in the study of optical solitons. To convince ourselves that such forbidden frequency gaps really exist, it is convenient to write the linear dispersion relation (15) in the form:  
\[k=(2c_{0})^{-1}\biggl{\{}-1\pm\sqrt{1-4c_{0}(c_{2}\omega^{2}-c_{4}\omega^{4}) }\biggr{\}} \tag{21}\]  
This equation implies that if \(c_{0}c_{2}^{2}>c_{4}\) holds, there are frequencies for which \(k\) turns out to be complex. Analyzing the radicand which appears on the right-hand-side of Equation (21) we find that these forbidden frequencies are in the intervals:  
\[-\omega_{2}<\omega<-\omega_{1}\text{ and }\omega_{1}<\omega<\omega_{2} \tag{22}\]  
Figure 3: Dispersion relation of Equations (5) and (6), defined by Equation (15), with \(c_{0}=1/30\), \(c_{2}=1/2\) and \(c_{4}=1/80\), which satisfy the condition (18). The dashed lines \(k=C_{1,2}\) indicate the wavenumbers \(C_{1}\) and \(C_{2}\) of the solution of Equation (6) given by Equations (52)–(55) with \(\gamma_{1}=5\) and \(\gamma_{2}=1\). and the frequencies \(\omega_{1}\) and \(\omega_{2}\) are defined by:  
\[\omega^{2}=(2c_{4})^{-1}\bigg{\{}c_{2}\pm\sqrt{c_{2}^{2}+4c_{4}(c_{0}k^{2}+k)} \bigg{\}} \tag{16}\]0  
where the _minus_ sign corresponds to \(\omega_{1}\), and the _plus_ sign defines \(\omega_{2}\). In Figure 4 we can see the shape of the dispersion relation (21) in the particular case when \(c_{0}=1/15\), \(c_{2}=1/2\) and \(c_{4}=1/256\). These coefficients satisfy the condition (17), and therefore there are two bands of forbidden frequencies, as we can see in the figure. The consequences of the presence of these forbidden frequency gaps will be investigated in the following section. The dashed horizontal lines shown in this figure indicate the values of the wavenumbers of the soliton solutions of Equation (6) (with \(\gamma_{1}=5\) and \(\gamma_{2}=1\)) that will be found in Section 4.