# A Novel Optical-Based Methodology for Improving Nonlinear Fourier Transform
## 2 Scattering Coefficient for Nonlinear Fourier Transform  
The evolution of the slowly varying complex-valued envelope of the scalar electric field \(y(z,t)\) propagating in a single mode optical fiber, neglecting the effects of polarization and attenuation, but taking into account the chromatic dispersion and nonlinear refraction (Kerr effect), is the following:  
\[\frac{\partial y(z,t)}{\partial z}=i\frac{\beta_{2}}{2}\frac{\partial^{2}y(z,t )}{\partial t^{2}}-i\gamma|y(z,t)|^{2}y(z,t) \tag{1}\]  
where \(z\) is the space coordinate, \(t\) is the time reference frame co-moving with the group velocity of the signal envelope, \(\beta_{2}\)\(\in\mathbb{R}\) chromatic dispersion, and \(\gamma>0\) nonlinear parameter. In particular, \(\beta_{2}\) represents the dispersion of the group velocity, which causes an enlargement of the impulse during propagation; parameter \(\gamma\) models the nonlinear refraction, which is the dependence of the refractive index on the optical field intensity. To simplify Equation (1), we will first multiply it by the imaginary number \(i\), and then we will make the following substitution: \(t\to t_{0}t\), \(z\rightarrow-\sigma_{2}oz\), \(y\rightarrow\sqrt{P_{0}}u\), with \(P_{0}=\frac{|\beta_{2}|}{\left(7T_{0}^{2}\right)}\) in power units, where \(T_{0}\) is a free normalization parameter in units of time, \(\sigma=-sgn(\beta_{2})\), a constant number that represents the nature of the signal inside the optical fiber; the following parameters are all positive quantities: \(P_{0}\), \(t_{0}\), \(z_{0}>0\). By selecting all these parameters to have the following relation: \(\frac{z_{0}|\beta_{2}|}{2T_{0}^{2}}=1,z_{0}\gamma P_{0}=2\), for example by choosing \(P_{0}=1,z_{0}=\frac{2}{\gamma},t_{0}=\sqrt{\frac{|\beta_{2}|}{\gamma}}\), then we will obtain a simplified form of Equation (1) as the following:  
\[i\frac{\partial u}{\partial z}+\frac{\partial^{2}u}{\partial t^{2}}+2\sigma|u|^ {2}u=0 \tag{2}\]  
when \(\sigma=-1\), then the solution envelope of Equation (2) will be solitons. These are usually called deep water waves. In the case of \(\sigma=+1\), then the soliton cannot exist and the solution is called shallow water as it is similar to the Korteweg-de Vries (KdV) equation. In this work, we are interested in a deep water waves solution with \(\sigma=-1\). Next, we will explain how the NFT method works. Let us suppose that we have a differential equation that describes the behavior of \(u(t,z)\) in (2), and we want to solve it using NFT. First, we have to find two linear operators \(\mathcal{L}\) and \(A\) that are dependent on \((t,z,\{u(t,z)\})\) (and possibly the derivatives of \(u(t,z)\)) that will describe a scattering problem of the following system:  
\[\left\{ \begin{array}{l}\mathcal{L}w=\lambda w\\ w_{z}=Aw\end{array} \right. \tag{3}\]  
where \(\lambda\) is the eigenvalue, \(w\) is the eigenfunction, and the subscript denotes the derivative. The operators \(\mathcal{L}\) and \(A\) will have to be chosen according to a certain criteria as explained below: we look for an adequate evolutionary equation over the spatial dimension for \(w\), assuming that the Schwarz Theorem applies; that is, \(w_{tz}=w_{zt}\). We are going to take the derivative of the first equation in (3) with respect to space, supposing that: \(\lambda_{z}=0\). After some simple manipulation, we can find the following relation: \(|\mathcal{L}_{z}+(\mathcal{L}A-A\mathcal{L})|w=0\). If we want this to be valid for every \(w\), then we have to impose the following Lax equation [48]:  
\[\mathcal{L}_{z}+(\mathcal{L}A-A\mathcal{L})=0 \tag{4}\]  
This equation contains a condition of compatibility for the operators, which in this case are called Lax pairs and can be possibly solved using the PDE for \(u\). To use the NFT, we need to associate Equation (2) with some Lax pairs. Suppose we have the following Lax pairs [49]:  
\[\mathcal{L}= \begin{bmatrix}i\partial_{t}&-iu(t,z)\\ -i\sigma u^{*}(t,z)&-i\partial_{t}\end{bmatrix}  \tag{5}\]  
\[A=