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: $i{\partial U \over \partial z} + {1 \over 2} {\partial^{2} U \over \partial t^{2}} + \vert U \vert^{2} U -i \left(\sigma U + \alpha {\partial^{2}U \over \partial t^{2}} + \delta \vert U \vert^{2} U \right) = 0,$.