## 5 Conclusions  
In this paper we have studied five pulse propagation models (Equations (4)-(8)) which take into account the non-SVEA term \(u_{zz}\), higher order dispersion, and a nonlinearity of the form \(\left|u\right|^{4}\)\(u\). Several terms which also participate in the propagation of short optical pulses along optical fibers have been discarded, not because they are negligible or unimportant, but to get a better understanding of the interplay between the terms \(u_{zz}\), \(-iu_{ttt}\), \(u_{4t}\) and \(\left|u\right|^{4}\)\(u\). The dismissal of significant terms implies that the models considered in this communication cannot aspire to provide quantitatively accurate descriptions of short light pulses propagating along an optical fiber. However, the analysis of the equations here considered (Equations (4)-(8)) reveals interesting aspects of the propagation of short pulses under conditions which are beyond the SVEA, and shows that the mathematical structure of these equations is interesting by itself, because it suggests that the study of optical solitons might be connected to other fields which seemed to be completely unrelated to optics. The analysis of the nonlinear models (6)-(8) reveals that short soliton-like pulses of different forms might be able to propagate under non-SVEA conditions. Equation (6) has soliton solutions of two different types: one of them with a phase linear in the propagation distance (as usual), and the second one with a nonlinear phase shift (see Equation (60)). Equation (7) has _moving_ solitons of different heights and velocities, and the determination of the parameters that appear in these solitons involves the solution of a sixth-order algebraic equation (see Equations (66)-(74)). In the case of Equation (8), the profile of its solitons is given by a squared hyperbolic secant and, depending on the values of the coefficients of the equation, these solitons may be standard or _embedded_ (i.e., with a wavenumber that is contained in the linear spectrum of the system). On the other hand, the analysis of the linear models (4) and (5) shows that the interplay between \(u_{zz}\) and a higher-order dispersive term such as \(-iu_{ttt}\) or \(u_{4t}\) generates bands of forbidden frequencies or forbidden wavenumbers. As a consequence of the existence of these forbidden bands, the behavior of short optical pulses which propagate along an optical fiber is particularly interesting. At the beginning of their journey along the fiber, the pulses start emitting radiation, due to a resonance between the pulses and the small-amplitude continuous waves (radiation modes) capable of propagating in the fiber. However, after some time, the radiation stops quite abruptly. This abrupt interruption of the radiation is a phenomenon that had never been predicted before. This unusual behavior is a consequence of the presence of bands of forbidden frequencies in the dispersion relations of Equations (4) and (5), as shown in Figures 2 and 4, respectively. The dispersion relation of Equation (4) (given by Equation (11)) is particularly interesting, because it is an _elliptic curve_, and these curves have had a profound influence in other fields, as in the proof of [PERSON]'s last theorem, or in the development of new cryptographic protocols. As explained in Appendix B, in the process of proving [PERSON]'s last theorem it was found that every elliptic curve is associated with a _modular form_. Consequently there must exist a certain relationship between Equation (4) (and its nonlinear extensions such as Equations (A12) and (A13)) and _modular forms_. The investigation of this hidden relationship may deserve further studies. On the other hand, as elliptic curves are useful in encrypting information, it is natural to wonder if a relation may be found between cryptography and equations such as Equations (4), (A12) or Equation (A13).