## 5 Conclusions
Although the possibility of finding such a relation might seem remote, it is interesting to observe that it has already been pointed out that ideas related to ill-posed problems might be useful in cryptography [52]. Therefore, as Equations (4), (A12) and (A13) are related to elliptic curves, and are also related to ill-posed problems, it might be worth investigating if these equations may be used to design new cryptographic algorithms. It is worth mentioning that the new methods to obtain exact solutions for nonlinear and evolution equations described in Refs. [47; 48; 49], and references therein, might be helpful to investigate if Equations (A12) and (A13) can be indeed related to modular forms, or to Weierstrass \(\wp\) functions, which are closely related to elliptic curves [39]. We would like to emphasize that the appearance of bands of forbidden frequencies in the dispersion relations of Equations (4)-(8), (47) (A12) and (A13) is the principal consequence of the interplay between \(u_{zz}\) and higher-order dispersive terms (such as \(-iu_{ttt}\) or \(u_{4t}\)). We have seen (at theend of Section 3.2), that these forbidden bands may be cancelled by the effect of the term \(cu_{zt}\), but there may exist systems where such bands appear. The experimental observation by [PERSON] et al. [38] of a gap of forbidden frequencies in the spectral profile of very short optical pulses propagating in a photonic crystal fiber may be an example of a process of this type. The main results of this communication are, therefore, the following:  
* the discovery of a relationship between Equations (4), (47), (A12) and (A13) and _elliptic curves_,
* the discovery that the interplay between \(u_{zz}\) and higher-order dispersive terms generates bands of forbidden frequencies or forbidden wavenumbers in the dispersion relations of Equations (4)-(8) and (47),
* the discovery that the Cauchy problems associated to Equations (4)-(8) are probably _ill-posed_,
* the discovery that short pulses who evolve according to Equations (4), (5) and (47) radiate in a novel and peculiar way,
* the discovery of four different types of soliton solutions in Equations (6)-(8),
* the discovery that some of the solitons of Equation (6)-(8) are _embedded solitons_,
* the discovery that Equations (A12) and (A13) might be related to _modular forms_ or Weierstrass \(\wp\) functions,
* the discovery that we might find a relation between the equations studied in this paper and cryptography. Several of these results suggest further topics of research, and consequently we hope that this article encourages the research along these lines. **Acknowledgments:** We thank DGTIC-UNAM (Direccion General de Computo y Tecnologias de Informacion y Comunicacion de la Universidad Nacional Autonoma de Mexico) for granting us access to the computer _Miztili_ through the Project SC16-1-S-6, in order to study the numerical solutions of Equations (4)-(8). We also thank [PERSON] for detecting an error in one of the equations in a preliminary version of this article (the error is already corrected). Finally, we thank the reviewers who revised this paper, who generously dedicated a considerable amount of time to read the preliminary versions of this article, detected weak points of the paper, and suggested ways of improving it. **Author Contributions:** [PERSON] conceived the goal and the structure of the paper, and obtained the analytical solutions. [PERSON] carried out the calculations needed to obtain all the figures. [PERSON] carried out numerical experiments to investigate if the Cauchy problems of Equations (4)-(8) are ill-posed. **Conflicts of Interest:** The authors declare no conflicts of interest.