## 2 Dispersion Relations and Elliptic Curves  
### Dispersion Relation of Equation (4)  
Let us begin by paying attention to the form of the dispersion relation of Equation (4). If we substitute the tentative solution:  
\[u(z,t)=\varepsilon\;exp(i(kz-\omega t)) \tag{9}\]  
in Equation (4), the following dispersion relation is easily found:  
\[c_{0}k^{2}+k=c_{3}\omega^{3}-c_{2}\omega^{2} \tag{10}\]  
and this equation can be rescaled by defining \(\widetilde{k}=c_{0}^{1/2}k\) and \(\widetilde{\omega}=c_{3}^{1/3}\omega\), thus obtaining (after supressing the tildes):  
\[k^{2}+a_{3}k=\omega^{3}+a_{2}\omega^{2} \tag{11}\]  
where \(a_{3}=c_{0}^{-1/2}\) and \(a_{2}=-c_{2}c_{3}^{-2/3}\). This equation describes an _elliptic curve_ (if \(a_{2}\
eq 0\) and \(a_{3}\
eq 0\)), which is an interesting result since elliptic curves have important roles in areas which might seem completely unrelated to the study of light pulses in optical fibers, such as the abstruse theory of _modular forms_[39], or the development of _cryptographic protocols_[40]. For the sake of simplicity in the following we will also say that Equation (10) itself is an elliptic curve, although, being rigorous, we should say that Equation (10) is an elliptic curve _up to scaling_. In other words, it is an elliptic curve distorted by a non-uniform scaling (i.e., a scale transformation which applies different scale factors in the \(k\) and \(\omega\) axes).  
Now let us pay more attention to the shape of the curves described by the dispersion relation (10). From Equation (10) it follows that the wavenumber \(k\) is given by the following function of the frequency:  
\[k(\omega)=\frac{-1\pm\sqrt{1-4c_{0}(c_{2}\omega^{2}-c_{3}\omega^{3})}}{2c_{0}} \tag{12}\]  
and the form of this function is quite different from the dispersion relations found in other optical systems. If we choose, for example, the coefficients \(c_{0}=1/30\), \(c_{2}=1/2\) and \(c_{3}=1/15\), the form of the curve \(k(\omega)\) is shown in Figure 1.  
Figure 1: Dispersion relation of Equation (4), given by Equation (12), with \(c_{0}=1/30\), \(c_{2}=1/2\) and \(c_{3}=1/15\).  
We can see that the curve is symmetrical with respect to a horizontal line, but completely asymmetrical with respect to the \(k\)-axis, which is an unusual behavior. However, the behavior of \(k(\omega)\) can be more bizarre. If we choose the coefficients \(c_{0}=1/40\), \(c_{2}=1/2\) and \(c_{3}=1/50\), \(k(\omega)\) has the form shown in Figure 2.  
We can see that in this case the dispersion relation exhibits two bands of forbidden frequencies, thus implying that small-amplitude linear waves cannot propagate with frequencies in the ranges:  
\[-\infty<\omega<\omega_{1}\text{ and }\omega_{2}<\omega<\omega_{3}, \tag{13}\]  
where:  
\[\omega_{1}=10\Big{(}1-\sqrt{2}\Big{)},\ \omega_{2}=5\text{ and }\omega_{3}=10 \Big{(}1+\sqrt{2}\Big{)} \tag{14}\]  
are the frequencies which make the radicand that appears in Equation (12) equal to zero. This radicand becomes negative for frequencies in the intervals shown in (13), and therefore \(k(\omega)\) becomes complex for these frequencies. In the next section we will investigate the consequences of the existence of these bands of forbidden frequencies.  
The fact that the linear dispersion relation of Equation (4) is an elliptic curve suggests that this equation might be related to other subjects where these curves also appear. In particular, a relationship may exist between optical solitons and the theory of _modular forms_, which played a central role in the proof of [PERSON]'s last theorem. For this reason in Appendix A we explain more precisely what an elliptic curve is, and in Appendix B we briefly review what was the role played by elliptic curves and modular forms in the proof of [PERSON]'s last theorem.  
To close this sub-section it is worth observing that if we introduced the mixed derivative \(u_{zt}\) in Equation (4), the resulting dispersion relation would still be an elliptic curve.