# A Novel Optical-Based Methodology for Improving Nonlinear Fourier Transform
## 2 Scattering Coefficient for Nonlinear Fourier Transform
\begin{bmatrix}2i\partial_{t}^{2}+i\sigma|u|^{2}&-2 iu\partial_{t}-iu_{t}\\ -2i\sigma u^{*}\partial_{t}-i\sigma u_{t}^{*}&-2i\partial_{t}^{2}-i\sigma|u|^ {2}\end{bmatrix}  \tag{6}\]  
The first differential operator in (5) gives us the associated eigenvalue problem to the NLSE, while the second one is the temporal evolution of the eigenfunction, as explained in the following theorem. Theorem 1: _The Lax equation given in Equation (4) is equivalent to the NLSE equation in Equation (2) on the hypothesis that the following conditions are true: \(\mathcal{L}\Phi=\lambda\Phi\), \(\Phi_{z}=A\Phi\), \(\lambda_{z}=0\), where \(\Phi= \begin{bmatrix}\dot{\Phi}_{1}\\ \dot{\Phi}_{2}\end{bmatrix} \) is an autofunction of two components._  
Proof of Theorem 1.: We first apply the operator \(A\) to \(\Phi\) as  
\[\Phi_{z}=A\Phi= \begin{pmatrix}2i\phi_{1,tt}+i\sigma|u|^{2}\phi_{1}-2 iu\phi_{ 2,t}-iu_{t}\phi_{2}\\ -2i\sigma u^{*}\phi_{1,t}-i\sigma u_{t}^{*}\phi_{1}-2i\partial_{t}^{2}\phi_{2, t}-i\sigma|u|^{2}\phi_{2}\end{pmatrix}  \tag{7}\]  
and after this, we apply \(\mathcal{L}\) to obtain two components: \(\mathcal{L}A\Phi_{1}\) and \(\mathcal{L}A\Phi_{2}\). We perform the same thing, but this time we first apply \(\mathcal{L}\) and then \(A\) to obtain the first component \(A\mathcal{L}\Phi_{1}\). Finally, we can find: \((\mathcal{L}A-A\mathcal{L})\Phi_{1}=-u_{tt}\phi_{2}-2\sigma|u|^{2}u\phi_{2}\) and \(\{\mathcal{L}_{z}+(\mathcal{L}A-A\mathcal{L})\}\Phi_{1}=-\Big{(}iu_{z}+u_{tt}+2 \sigma|u|^{2}u\Big{)}\phi_{2}=0\), which is the NLSE in Equation (2). After choosing the Lax pair, we need to analyze the direct NFT associated with the following equation:  
\[\mathcal{L}\Phi=\lambda\Phi \tag{8}\]  
For the NFT method to work, we need to impose that at infinity, our solution \(u\) will decay to zero, \(u\to 0\) for \(|t|\to+\infty\). Equation (8) will be approximated with the following result: \(i \begin{bmatrix}1&0\\ 0&-1\end{bmatrix} \Phi_{t}=\lambda\Phi\), with \(\phi_{1,t}=-i\lambda\phi_{1}\) and \(\phi_{2,t}=+i\lambda\phi_{2}\), whose solutions are simple to calculate. To form the basis of the solutions space of Equation (8), we can choose from the following Jost functions, defined as limit [48]:  
\[\phi(t,\lambda) \sim \begin{pmatrix}1\\ 0\end{pmatrix} e^{-i\lambda t},\,\overline{\phi}(t,\lambda)\sim \begin{pmatrix} 0\\ 1\end{pmatrix} e^{i\lambda t},\,for\,t\to-\infty \tag{9}\] \[\phi(t,\lambda) \sim \begin{pmatrix}0\\ 1\end{pmatrix} e^{i\lambda t},\,\overline{\phi}(t,\lambda)\sim \begin{pmatrix} 1\\ 0\end{pmatrix} e^{-i\lambda t},\,for\,t\to+\infty \tag{10}\]  
The overbar sign means that two autofunctions are in straight relation with each other. For example, the existence of \(\phi(t,\lambda^{*})\) will imply also that of \(\overline{\phi}(t,\lambda)\), given that if the first function is a solution to \(\mathcal{L}\Phi=\lambda\Phi\), then the second will also be a solution that is linearly independent from the first one:  
\[\overline{\phi}(t,\lambda)= \begin{bmatrix}0&-\sigma\\ 1&0\end{bmatrix} \phi^{*}(t,\lambda) \tag{11}\]  
**Proposition 1**. : _The four solutions \(\phi,\overline{\phi},\,\varphi,\,\overline{\varphi}\) are independent in pairs._  
Proof of Proposition 1.: The Wronskian of two autofunctions can be easily found to be 1 as: \(W(\phi,\overline{\phi})=\lim_{x\to\infty}W(\phi,\overline{\phi})=e^{-i\lambda t }e^{i\lambda t}=1\).