Nonlinear Fourier Transform as a Tool for Analyzing the Soliton Dynamics in Systems Obeying the Haus–Ginzburg–Landau Equation

We propose using the nonlinear Fourier transform (NFT) not as a method for integrating equations but as a tool for studying the properties of localized coherent structures in dissipative nonlinear systems. The effectiveness of using a discrete nonlinear spectrum for quantitatively describing the dynamics of optical pulses is demonstrated, even when the original model is not integrable. The application of this method is substantiated using the solution to the cubic Haus–Ginzburg–Landau equation (HGLE), which describes the generation of optical pulses from noise in passively mode-locked lasers. It is shown that stabilization of the discrete nonlinear spectrum serves as a reliable indicator of stable soliton generation. Additionally, algorithms for tracing individual discrete eigenvalues are proposed, including a machine-learning-based algorithm, which expands the toolbox for analyzing and automating the processing of data obtained using NFT.