partial differential equation $$\begin{aligned}
\frac{\partial \bm{v}(t,\lambda)}{\partial t} = \begin{pmatrix}
-\jmath\lambda & q(t)\\ 
- q(t)^*&  \jmath\lambda
\end{pmatrix}\bm{v}(t,\lambda),\label{eqn:nft_dgl}\end{aligned}$$ where $\bm{v}(t,\lambda)=\begin{pmatrix} v_1(t,\lambda) & v_2(t,\lambda) \end{pmatrix}$ is the eigenvector of the auxiliary operator, with boundary conditions $$\begin{aligned}
\bm{v}^{(1)}(t,\lambda) &\rightarrow {\begin{pmatrix} 0 & 1 \end{pmatrix}^\mathsf{T}} {\operatorname{e}^{\jmath\lambda t}}, &\text{as } t &\rightarrow t_2\\
\bm{v}^{(2)}(t,\lambda) &\rightarrow {\begin{pmatrix} 1 & 0 \end{pmatrix}^\mathsf{T}} {\operatorname{e}^{-\jmath\lambda t}}, &\text{as } t &\rightarrow t_1,\end{aligned}$$ and $\lambda$ is the spectral component. Solving [()]{} gives rise to the continuous and discrete nonlinear spectrum $$\begin{aligned}
    \hat{q}(\lambda ) &= \frac{b(\lambda)}{a(\lambda)}, \lambda\in{\mathbb{R}}& \tilde{q}(\lambda_i) &= \frac{b(\lambda_i)}{\mathrm{d}a(\lambda) / \mathrm{d}\lambda|_{\lambda=\lambda_i}}, \lambda_i\in{\mathbb{C}}^{+},\end{aligned}$$ respectively, where $a(\lambda ) = \lim_{t\rightarrow t_2} v_1^{(2)}(t,\lambda)    {\operatorname{e}^{\jmath\lambda t}}$, $b(\lambda ) = \lim_{t\rightarrow t_2} v_2^{(2)}(t,\lambda)   {\operatorname{e}^{-\jmath\lambda t}}$, and $\lambda_i$ are the zeros of $a(\lambda)$, $\lambda_i\in{\mathbb{C}}^{+}$, a finite set of isolated complex zeros, referred to as eigenvalues. Hence, the represents the signal in the nonlinear spectral domain, where the influence of the channel on the signal is a multiplicative filter.

As a counterpart to the that transforms a signal from the time domain to the nonlinear spectral domain, the transforms a signal from the nonlinear spectral domain to the time domain. For an in-depth mathematical description of the , we refer the interested reader to [@yousefi2014nftI-III].

Soliton Transmission
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