# A Novel Optical-Based Methodology for Improving Nonlinear Fourier Transform
## 2 Scattering Coefficient for Nonlinear Fourier Transform
#### Numerical Method for the Continuous Scattering Coefficient  
We have developed a numerical algorithm to find the continuous scattering coefficient of the Lax equation: \(\mathcal{L}\Phi=\lambda\Phi\), with the initial condition signal \(u(t,0)\). We followed the idea introduced in the article by [PERSON] and [PERSON] [PERSON] and [PERSON] (1996). The algorithm will use a discretization of \(u(t,0)\) in \(2M+1\) samples as: \(u_{n}=u(t_{n},0)=u(n\Delta t,0)=u(n\frac{T}{M},0)\), where \(2T\) is the total width of the function \(u\) centered at \(t=0\), such that \(u(t,0)=0\) for \(|t|\geq T\) and \(M>0\). The steps followed to find out the numerical algorithm for the continuous scattering coefficient when \(\lambda\in\mathbb{R}\) are given below:  
Step 1. We need to solve numerically the Lax equation: \(\mathcal{L}\Phi=\lambda\Phi\), and find an equation that provides us with a relationship between \(\phi(t,\lambda)\) and \(\phi(-t,\lambda)\) for \(|t|\geq T\). Instead of solving it directly, we will first assume that \(u(t,0)=u_{n}\) is constant in the interval \(\left[t_{n}-\frac{\Delta t}{2},t_{n}+\frac{\Delta t}{2}\right]\). After this, we are going to associate to it the equation, \(\Phi_{t}=B_{n}\Phi\), which is equivalent to \(\mathcal{L}\Phi=\lambda\Phi\). \(B_{n}\) is an approximation of \(\mathcal{L}\) with constant step. We are going to solve numerically the following Cauchy problem, in an iterative way:  
\[\left\{ \begin{array}{c}\Phi_{t}=B_{n}\Phi\\ \Phi\Big{(}t_{n}-\frac{\Delta t}{2},\lambda\Big{)}=\Phi_{n}\end{array} \right.,t\in(t_{n}-\frac{\Delta t}{2},t_{n}+\frac{\Delta t}{2}) \tag{21}\]  
with  
\[B_{n}(\lambda)= \begin{bmatrix}-i\lambda&u_{n}\\ -\sigma u_{n}^{*}&i\lambda\end{bmatrix} ,\forall t\in\left(t_{n}-\frac{\Delta t }{2},t_{n}+\frac{\Delta t}{2}\right) \tag{22}\]  
in order to find the new starting point \(\Phi\Big{(}t_{n+1}-\frac{\Delta t}{2},\lambda\Big{)}=\Phi_{n+1}\), having already previously found \(\Phi_{n}\). Solving equation Equation (21) until \(t_{n+1}\) will give us the following iteration:  
\[\Phi\bigg{(}t_{n+1}-\frac{\Delta t}{2}\bigg{)}=\Phi\bigg{(}t_{n  
An explicit expression of \(U(u_{n})\) can be calculated for our case through the definition of the exponential of a matrix:  
\[\exp(\Delta tB_{n})=\sum_{k=0}^{\infty}\frac{\Delta t^{k}}{k!}B_{n}^{k} \tag{25}\]  
and taking into account [PERSON]'s series of the \(\cosh\) and \(\sinh\) functions, for the matrix taken into consideration. The obtained result is:  
\[U(u_{n})= \begin{bmatrix}\cosh(q_{n}\Delta t)-\frac{i\lambda}{q_{n}}\sinh(q_{n} \Delta t)&\frac{u_{n}}{q_{n}}\sinh(q_{n}\Delta t)\\ -\frac{\rho\overline{u}_{n}}{q_{n}}\sinh(q_{n}\Delta t)&\cosh(q_{n}\Delta t)+ \frac{i\lambda}{q_{n}}\sinh(q_{n}\Delta t)\end{bmatrix}  \tag{26}\]  
with \(q_{n}^{2}=-\sigma|u_{n}|^{2}-\lambda^{2}\) being a complex number. If we introduce a new vector as:  
\[\Xi(t,\lambda)= \begin{bmatrix}\Phi\\ \Phi^{{}^{\prime}}\end{bmatrix}  \tag{27}\]  
with \(\Phi^{{}^{\prime}}=\frac{\partial\Phi}{\partial\lambda}\), Equation (23) will be transformed into:  
\[\Xi\bigg{(}t_{n+1}-\frac{\Delta t}{2}\bigg{)}=T(u_{n})\Xi\bigg{(}t_{n}-\frac {\Delta t}{2}\bigg{)} \tag{28}\]  
with  
\[T(u_{n})= \begin{bmatrix}U(u_{n})&0\\ U^{\prime}(u_{n})&U(u_{n})\end{bmatrix}  \tag{29}\]  
We take \(t_{-M}\) as the starting point and iterate it until we find the following relation for a generic \(t_{n}\):  
\[\Xi\bigg{(}t_{n+1}+\frac{\Delta t}{2}\bigg{)}=\prod_{i=n}^{-M}T(u_{i})\Xi \bigg{(}t_{-M}-\frac{\Delta t}{2}\bigg{)} \tag{30}\]  
having the relation:  
\[\prod_{i=M}^{-M}T(u_{i})\Xi(t_{-M})= \begin{bmatrix}\Sigma&0\\ \Sigma^{{}^{\prime}}&\Sigma\end{bmatrix} \Xi\bigg{(}t_{-M}-\frac{\Delta t}{2} \bigg{)}=S\Xi\bigg{(}t_{-M}-\frac{\Delta t}{2}\bigg{)} \tag{31}\]  
and define \(S\) the scattering matrix as:  
\[S= \begin{bmatrix}\Sigma&0\\ \Sigma^{{}^{\prime}}&\Sigma\end{bmatrix} ,\;\Sigma(\lambda)=\prod_{i=M}^{-M}U( u_{i}) \tag{32}\]  
In Equation (31), we found a relation that connects \(\phi(t,\lambda)\) and \(\phi(-t,\lambda)\) for \(|t|\geq T\).