Applied sciences  
Pulse Propagation Models with Bands of Forbidden Frequencies or Forbidden Wavenumbers: A Consequence of Abandoning the Slowly Varying Envelope Approximation and Taking into Account Higher-Order Dispersion  
[PERSON] 1,*, [PERSON] 2 and [PERSON] 3  
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as Equation (1), but with a spatial variable \(x\) (corresponding to a transversal coordinate) instead of the retarded time \(t\). In spite of its great importance, it should be remembered that Equation (1) is an _approximate_ equation. In order to arrive at this equation several terms have been neglected. Some of these neglected terms are the following:  
1. \(-iu_{ttt}\) and/or \(u_{4t}\): these higher-order derivatives are necessary to describe sub-picosecond pulses [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12]; in particular, conditions for including \(u_{4t}\) and discarding \(-iu_{ttt}\) are discussed in [7; 9],
2. \(|u|^{4}u\): this higher-order nonlinearity is used when we want to describe the propagation of pulses when the light intensity approaches the values which produce the \"saturation\" of the refractive index [13; 14; 15; 16; 17; 18; 19; 20],
3. \(i(|u|^{2}u)_{t}\): this term is necessary to describe the self-steepening of the optical pulses [21; 22; 23],
4. \(u(|u|^{2})_{t}\): this term is associated with the effect of Raman scattering [24; 25; 26; 27]. The effects of including these terms in generalized NLS equations have been thoroughly studied in the literature. Moreover, an additional approximation introduced in the deduction of the NLS equation is the slowly-varying-envelope approximation (SVEA). This approximation is adequate when the complex amplitude of an optical pulse varies slowly in space and time. However, when we deal with optical pulses whose widths only contain a few cycles of the carrier wave, it is necessary to improve the SVEA. Several alternatives have been proposed to improve this approximation. As early as 1985, [PERSON] and [PERSON] studied an extended NLS equation which contained higher-order dispersion and the additional terms \(u_{zz}\), \(i(|u|^{2}u)_{\tau}\) and \((|u|^{2}u)_{\tau\tau}\), where \(\tau\) was the standard (laboratory) time [28], and they found an interesting exact soliton solution by an algebraic _tour de force_. In more recent times completely different models have been proposed to describe ultra-short optical pulses, and the famous Kortweg-de Vries (KdV), modified KdV (mKdV) and sine-Gordon (sG) equations have been found useful for this purpose, as well as a combination of the last two of these equations [29; 30]. Another two equations which have been useful to describe ultra-short spatiotemporal pulses are a two-dimensional sG equation [31] and a cubic generalized Kadomtsev-Petviashvili (cgKP) equation [32; 33]. It is worth observing that in all these models (KdV, mKdV, sG, mKdV-sG, 2D-sG and cgKP) the temporal variable which appears in the equations is a _delayed time_ (not the laboratory time), similar to the retarded time that appears in the standard NLS equation.