Application of the Nonlinear Fourier Transform for Analysis of the Coherent Structures Generation

The conventional (linear) Fourier transform is a widely used mathematical method in science and engineering. It allows the signal of interest to be represented as a set of spectral harmonics, which, in many situations, helps in understanding the structure and properties of that signal. In certain linear equations, where the spectral harmonics evolve independently, the Fourier transform provides a straightforward description of complex temporal or spatial dynamics. A related approach exists for certain classes of integrable nonlinear equations that can be solved via the inverse scattering transform [1], [2], also known as the nonlinear Fourier transform (NFT). A particularly important example for optical applications is the nonlinear Schrödinger equation (NLSE), which is one of the fundamental equations of nonlinear science, relevant to various fields of physics and practical engineering. However, NFT can also be used to analyze localized waveforms, rather than strictly to solve integrable equations. In particular, it allows one to describe a waveform or the evolution of coherent structures with fewer parameters than traditional harmonic Fourier analysis.