Nonlinear Fourier transform for analysis of coherent structures in dissipative systems

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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: (equation presented), (1) where δ > 0 is a distributed saturable absorber action and α > 0 describes the effect of distributed optical filtering. The CGLE has special solutions in the form of chirped dissipative solitons: U(z,t) = U01+iC(t) exp {iφz}, U0(t) = A/cosh (t/τ). The CGLE is not integrable, but we can still formally calculate the nonlinear spectrum of the optical field U(z,t) at each point z and study the evolution of the nonlinear spectrum by z. We anticipate that under certain conditions NFT might allow us to gain useful information concerning the evolution of coherent structures with fewer parameters compared to conventional Fourier harmonics analysis.

Original languageEnglish
Title of host publicationEuropean Quantum Electronics Conference, EQEC_2019
PublisherOSA - The Optical Society
ISBN (Electronic)9781557528209
Publication statusPublished - 1 Jan 2019
EventEuropean Quantum Electronics Conference, EQEC_2019 - Munich, United Kingdom
Duration: 23 Jun 201927 Jun 2019

Publication series

NameOptics InfoBase Conference Papers
VolumePart F143-EQEC 2019


ConferenceEuropean Quantum Electronics Conference, EQEC_2019
CountryUnited Kingdom


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