## 3 The Linear Equations (4) and (5)  
### The Issue of the Initial Conditions  
In the study of optical solitons many variants of the nonlinear Shrodinger (NLS) equation have been considered in the past in order to describe the propagation of light pulses in optical fibers under different conditions. However, different as they are, most of these variants share a common denominator: they are _first-order equations with respect to the evolution variable_\(z\). This characteristic implies that in all these cases the initial condition \(u(z=0,\ t)\) is sufficient to determine particular solutions of the corresponding equations. This fact seems to be consistent with physical reality: if we have an optical fiber (with well-defined characteristics: shape, composition, refraction index, type of cladding, etc. ), and a laser beam (of known frequency) is sent into the fiber, the light intensity (as a function of time) at the beginning of the fiber (i.e., at \(z=0\)) is all we need to determine completely the behavior of light along the fiber. On the other hand, the situation with Equations (4)-(8) is completely different. These equations are of _second order in_\(z\), and this characteristic has two important consequences:  
CONSEQUENCE 1:  
With these equations we require _two initial conditions,_\(u(z=0,\ t)\) and \(u_{z}(z=0,\ t)\), in order to determine a particular solution. This requirement is somewhat perplexing since physical intuition suggests that \(u(z=0,\ t)\) should be enough to determine the evolution of light along the fiber. Therefore, it seems as if we were in the presence of a contradiction: while mathematics indicates that Equations (4)-(8) require two initial conditions to define a particular solution, physics suggests that the knowledge of \(u(0,\ t)\) should be enough to define completely the behavior of the system. CONSEQUENCE 2:  
Although formal proofs are still lacking, several arguments suggest that the Cauchy problems for Equations (4)-(8) might be _ill-posed_ (in certain regions of the space of coefficients \(c_{n}\)). The following results permit us to understand why this ill-posedness is to be expected:  
1. The presence of the non-paraxial term \(u_{zz}\) in the equation: \[iu_{z}+\varepsilon u_{zz}+\frac{1}{2}\big{(}u_{xx}+u_{yy}\big{)}+|u|^{2}u=0\] (25) makes the Cauchy problem for this equation ill-posed [21,41]. 2. The presence of higher-order nonlinearities in equations of the form: \[iu_{z}+u_{tt}+\lambda|u|^{m}u=0\] (26) makes their Cauchy problems ill-posed if \(m\geq 4\) and \(\lambda>0\)[42]. 3. The presence of higher-order derivative \(u_{4t}\) in the equation: \[u_{zz}=u_{tt}+2u_{t}^{2}+2 uu_{tt}+u_{4t}\] (27) makes the Cauchy problem for this equation linearly ill-posed [43]. 4. The solutions of the linear Equations (4) and (5) and the linear parts of Equations (6)-(8) contain decaying as well as growing modes of the form \(\exp(\pm\sigma z-i\omega t)\), as we shall see in Sections 3.2 and 3.3. The existence of these modes makes the corresponding Cauchy problems linearly ill-posed [43]. 5. Attemps of solving initial value problems for Equations (4)-(8) by finite differences in \(t\), and Runge-Kutta algorithms in \(z\), encounter difficulties that are typical of ill-posed problems. At first sight these two consequences seem to be unrelated. However, as we explain in the following, there is a connection between them. Let us begin by paying attention to Consequence 2. If the Cauchy problems associated to Equations (4)-(8) are indeed ill-posed, then (as [PERSON] first observed [44]) it would be necessary to impose certain compatibility conditions among the initial conditions, for these problems to have global solutions (i.e., solutions defined for all \(t\)). Consequently, these compatibility conditions will establish a link between the two initial conditions \(u(z=0,\ t)\) and \(u_{z}(z=0,\ t)\), thus implying that, essentially, the initial condition \(u(z=0,\ t)\) does indeed determine the future of the system, as physical intuition suggested.