Nonlinear Fourier Transform for Analysis of Coherent Structures in Dissipative Systems

The conventional Fourier transform is widely used mathematical methods in science and technology. It allows representing the signal/field under study as a set of spectral harmonics, that it many situations simplify understanding of such signal/field. In some linear equations, where spectral harmonics evolve independently of each other, the Fourier transform provides a straightforward description of otherwise complex dynamics. Something similar is available for certain classes of nonlinear equations that are integrable using the inverse scattering transform [1,2], also known as the nonlinear Fourier transform (NFT). Here we discuss potential of its application in dissipative, non-integrable systems to characterize coherent structures. We present a new approach for describing the evolution of a nonlinear system considering the cubic Ginzburg-Landau Equation (CGLE) as a particularly important example in the context of laser system modeling: i[(∂U)/(∂z)] +
¹
/2 [(∂
²
U)/(∂t
²
)] + |U |
²
U -i (σU + α[(∂
²
U)/(∂t
²
)] + δ|U |
²
U ) = 0.