# Nonlinear Fourier Transform for Analysis of Coherent Structures in Dissipative Systems

I.S. Chekhovskoy, Olga V. Shtyrina, Mikhail P. Fedoruk, Sergey B. Medvedev, Sergei K. Turitsyn

Research output: Chapter in Book/Report/Conference proceedingConference contribution

### Abstract

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: $i{\partial U \over \partial z} + {1 \over 2} {\partial^{2} U \over \partial t^{2}} + \vert U \vert^{2} U -i \left(\sigma U + \alpha {\partial^{2}U \over \partial t^{2}} + \delta \vert U \vert^{2} U \right) = 0,$.
Original language English 2019 Conference on Lasers and Electro-Optics Europe and European Quantum Electronics Conference, CLEO/Europe-EQEC 2019 IEEE 978-1-7281-0469-0 978-1-7281-0470-6 https://doi.org/10.1109/CLEOE-EQEC.2019.8872485 Published - 17 Oct 2019 2019 Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC) - Munich, GermanyDuration: 23 Jun 2019 → 27 Jun 2019

### Publication series

Name 2019 Conference on Lasers and Electro-Optics Europe and European Quantum Electronics Conference, CLEO/Europe-EQEC 2019

### Conference

Conference 2019 Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC) Germany Munich 23/06/19 → 27/06/19

### Fingerprint

harmonics
Landau-Ginzburg equations
inverse scattering
linear equations
nonlinear systems
nonlinear equations
lasers

### Cite this

Chekhovskoy, I. S., Shtyrina, O. V., Fedoruk, M. P., Medvedev, S. B., & Turitsyn, S. K. (2019). Nonlinear Fourier Transform for Analysis of Coherent Structures in Dissipative Systems. In 2019 Conference on Lasers and Electro-Optics Europe and European Quantum Electronics Conference, CLEO/Europe-EQEC 2019 [8872485] (2019 Conference on Lasers and Electro-Optics Europe and European Quantum Electronics Conference, CLEO/Europe-EQEC 2019). IEEE. https://doi.org/10.1109/CLEOE-EQEC.2019.8872485
Chekhovskoy, I.S. ; Shtyrina, Olga V. ; Fedoruk, Mikhail P. ; Medvedev, Sergey B. ; Turitsyn, Sergei K. / Nonlinear Fourier Transform for Analysis of Coherent Structures in Dissipative Systems. 2019 Conference on Lasers and Electro-Optics Europe and European Quantum Electronics Conference, CLEO/Europe-EQEC 2019. IEEE, 2019. (2019 Conference on Lasers and Electro-Optics Europe and European Quantum Electronics Conference, CLEO/Europe-EQEC 2019).
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title = "Nonlinear Fourier Transform for Analysis of Coherent Structures in Dissipative Systems",
abstract = "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: $i{\partial U \over \partial z} + {1 \over 2} {\partial^{2} U \over \partial t^{2}} + \vert U \vert^{2} U -i \left(\sigma U + \alpha {\partial^{2}U \over \partial t^{2}} + \delta \vert U \vert^{2} U \right) = 0,$.",
author = "I.S. Chekhovskoy and Shtyrina, {Olga V.} and Fedoruk, {Mikhail P.} and Medvedev, {Sergey B.} and Turitsyn, {Sergei K.}",
year = "2019",
month = "10",
day = "17",
doi = "10.1109/CLEOE-EQEC.2019.8872485",
language = "English",
isbn = "978-1-7281-0470-6",
series = "2019 Conference on Lasers and Electro-Optics Europe and European Quantum Electronics Conference, CLEO/Europe-EQEC 2019",
publisher = "IEEE",
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}

Chekhovskoy, IS, Shtyrina, OV, Fedoruk, MP, Medvedev, SB & Turitsyn, SK 2019, Nonlinear Fourier Transform for Analysis of Coherent Structures in Dissipative Systems. in 2019 Conference on Lasers and Electro-Optics Europe and European Quantum Electronics Conference, CLEO/Europe-EQEC 2019., 8872485, 2019 Conference on Lasers and Electro-Optics Europe and European Quantum Electronics Conference, CLEO/Europe-EQEC 2019, IEEE, 2019 Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC) , Munich, Germany, 23/06/19. https://doi.org/10.1109/CLEOE-EQEC.2019.8872485

Nonlinear Fourier Transform for Analysis of Coherent Structures in Dissipative Systems. / Chekhovskoy, I.S.; Shtyrina, Olga V.; Fedoruk, Mikhail P.; Medvedev, Sergey B.; Turitsyn, Sergei K.

2019 Conference on Lasers and Electro-Optics Europe and European Quantum Electronics Conference, CLEO/Europe-EQEC 2019. IEEE, 2019. 8872485 (2019 Conference on Lasers and Electro-Optics Europe and European Quantum Electronics Conference, CLEO/Europe-EQEC 2019).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

TY - GEN

T1 - Nonlinear Fourier Transform for Analysis of Coherent Structures in Dissipative Systems

AU - Chekhovskoy, I.S.

AU - Shtyrina, Olga V.

AU - Fedoruk, Mikhail P.

AU - Medvedev, Sergey B.

AU - Turitsyn, Sergei K.

PY - 2019/10/17

Y1 - 2019/10/17

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

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

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U2 - 10.1109/CLEOE-EQEC.2019.8872485

DO - 10.1109/CLEOE-EQEC.2019.8872485

M3 - Conference contribution

SN - 978-1-7281-0470-6

T3 - 2019 Conference on Lasers and Electro-Optics Europe and European Quantum Electronics Conference, CLEO/Europe-EQEC 2019

BT - 2019 Conference on Lasers and Electro-Optics Europe and European Quantum Electronics Conference, CLEO/Europe-EQEC 2019

PB - IEEE

ER -

Chekhovskoy IS, Shtyrina OV, Fedoruk MP, Medvedev SB, Turitsyn SK. Nonlinear Fourier Transform for Analysis of Coherent Structures in Dissipative Systems. In 2019 Conference on Lasers and Electro-Optics Europe and European Quantum Electronics Conference, CLEO/Europe-EQEC 2019. IEEE. 2019. 8872485. (2019 Conference on Lasers and Electro-Optics Europe and European Quantum Electronics Conference, CLEO/Europe-EQEC 2019). https://doi.org/10.1109/CLEOE-EQEC.2019.8872485