In the context of _spatial solitons_ the _paraxial_ approximation plays a role analogous to the SVEA in temporal solitons, and the effects of nonparaxiality have also been studied. In particular, the following non-paraxial extensions of the spatial NLS equation have been studied [34; 35; 36]:  
\[iu_{z}+\alpha u_{zz}+\frac{1}{2}u_{xx}+|u|^{2}u=0 \tag{2}\]  
\[iu_{z}+\alpha u_{zz}+\varepsilon_{2}u_{xx}-\gamma_{1}|u|^{2}u+\gamma_{2}|u|^{4 }u=0 \tag{3}\]  
The study of Equation (2) addresses a basic question which might be of interest to any reader interested in optical solitons: what is the effect of introducing the non-paraxial term \(u_{zz}\) in the standard NLS equation? On the other hand, the study of Equation (3) considers an equally interesting question, albeit a slightly more specialized one: what is the effect of taking into account _simultaneously_ nonparaxiality and higher-order nonlinearities? This question suggests that in the field of _temporal solitons_ it would be interesting to investigate the effect of taking into account _simultaneously_ non-SVEA terms, higher-order dispersion and higher-order nonlinearities such as \(|u|^{4}u\). A study of this type was carried out in Ref. [28], where a model with these characteristics was proposed (except that the nonlinearity \(|u|^{4}u\) was not considered). Complex models of this type, which include many different terms, may provide adequate descriptions for the behavior of very short pulses. However, in such models it might be difficult to appreciate the individual effect of each of the terms which have been taken into account. In the present communication we are interested in studying approximate linear and nonlinear pulse propagation models which incorporate higher-order dispersion, higher-order nonlinearities and one of the terms that appear when we drop the SVEA (a second-order derivative with respect to the propagation direction). When we abandon the SVEA, the terms \(u_{zz}\), \(i(|u|^{2}u)_{\tau}\) and \((|u|^{2}u)_{\tau\tau}\) must be introduced in the NLS equation (\(\tau\) being the standard laboratory time). However, if we use the retarded time to describe the propagation of the optical pulses, in addition to the term \(u_{zz}\), it is also necessary to introduce a mixed derivative of the form \(u_{zt}\) in the resulting equation [37]. To study the complete equation containing these two derivatives (\(u_{zz}\) and \(u_{zt}\)), the nonlinear terms \(i(|u|^{2}u)_{t}\) and \((|u|^{2}u)_{tt}\), an additional nonlinearity of the form \(|u|^{4}u\), and higher-order dispersive terms (\(-iu_{ttt}\) and/or \(u_{4t}\)) might be interesting and important, but it is not the objective of the present work. In this communication we are only interested in studying simplified models where the terms \(u_{zt}\), \(i(|u|^{2}u)_{t}\) and \((|u|^{2}u)_{tt}\) have been discarded. It is clear that these models do not pretend to provide a quantitative accurate description of a particular real system, but we will see that the study of these simplified models reveals that the _simultaneous_ presence of the non-SVEA term \(u_{zz}\) and higher-order dispersive terms such as \(-iu_{ttt}\) or \(u_{4t}\) may produce interesting results. In particular, we will show that the combined effects of these terms generate bands of forbidden frequencies, which may be related to the gaps which have been experimentally observed in the spectral profiles of very narrow pulses propagating in photonic crystal fibers [38]. Moreover, we will see that just the mathematical structure of the equations considered in this paper is interesting by itself, since we will show that these equations are related to _elliptic curves_, thus suggesting that a relationship may exist between the study of optical solitons and other fields where these curves play an important role, such as the abstruse theory of modular forms and Weierstrass \(\wp\) functions, or the design of cryptographic protocols.