TY - GEN

T1 - Nonlinear Fourier transform for analysis of coherent structures in dissipative systems

AU - Chekhovskoy, I. S.

AU - Shtyrina, O. V.

AU - Fedoruk, M. P.

AU - Medvedev, S. B.

AU - Turitsyn, S. K.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85084591774&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:85084591774

T3 - Optics InfoBase Conference Papers

BT - European Quantum Electronics Conference, EQEC_2019

PB - OSA - The Optical Society

T2 - European Quantum Electronics Conference, EQEC_2019

Y2 - 23 June 2019 through 27 June 2019

ER -