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Article information
2023 , Volume 28, ¹ 3, p.60-83
Medvedev S.B., Vaseva I.A., Chekhovskoy I.S., Fedoruk M.P.
On numerical algorithms for solving the direct and inverse scattering problems of the Zakharov -Shabat system
The paper presents an overview of the current numerical algorithms for implementing the inverse scattering transform (IST). The IST attracted special attention since it was proposed to use solitons as a bit of information for data transmission. Solitons are capable to compensate for nonlinear signal distortions and go beyond the limitations of linear data transmission methods. Accurate and fast numerical methods are needed to correctly describe and analyze the structure of complex multisoliton signals. Solitons are special solutions to the nonlinear Schr¨odinger equation (NLSE), which describes the propagation of pulses in an optical fiber. The IST allows one to integrate the NLSE.The paper considers methods for the direct Zakharov – Shabat (ZS) problem, which is a part of the IST. It allows calculating the nonlinear Fourier spectrum for a given potential. In particular, the paper presents the fast methods, needed for solving the problem for a large number of values of the spectral parameter. Special attention is given to the conservation property of the quadratic invariant of the ZS system. Since the integration takes place over a large area, the conservation of the quadratic invariant in computational algorithms must be exact. The paper also considers methods to restore the NLSE solution from a known nonlinear spectrum (inverse problem). Computational problems arising for a soliton spectrum are described.The methods described in the paper will be useful for more accurate and realistic calculations for the construction of telecommunication data transmission systems based on NLSE soliton solutions
Keywords: inverse scattering transform, nonlinear Schrodinger equation, Zakharov-Shabat problem, Gelfand-Levitan-Marchenko equations, nonlinear Fourier transform
doi: 10.25743/ICT.2023.28.3.005
Author(s):
Medvedev Sergey Borisovich
Dr.
Position: Leading research officer
Office: Inctitute of Computational Technologies SB RAS
Address: 630090, Russia, Novosibirsk, Ac. Lavrentyev ave., 6
Phone Office: (383) 330-73-73
E-mail: serbormed@gmail.com
SPIN-code: 2140-1726Vaseva Irina Arkadievna
PhD.
Position: Research Scientist
Office: Institute of Computational Technologies SB RAS
Address: 630090, Russia, Novosibirsk, pr. Lavrentjeva, 6
Phone Office: (383) 330 73 73
E-mail: vaseva.irina@gmail.com
Chekhovskoy Igor Sergeevich
Position: Student
Office: Novosibirsk State University, Institute of Computational Technologies SB RAS
Address: 630090, Russia, Novosibirsk, pr. Lavrentjeva, 6
E-mail: igor428m@gmail.com
Fedoruk Mikhail Petrovich
Dr. , Academician RAS, Professor
Position: Chancellor
Office: Novosibirsk State University, Federal Research Center for Information and Computational Technologies
Address: 630090, Russia, Novosibirsk, str. Pirogova, 2
Phone Office: (3832) 349105
E-mail: mife@net.ict.nsc.ru
SPIN-code: 4929-8753 References:
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Bibliography link:
Medvedev S.B., Vaseva I.A., Chekhovskoy I.S., Fedoruk M.P. On numerical algorithms for solving the direct and inverse scattering problems of the Zakharov -Shabat system // Computational technologies. 2023. V. 28. ¹ 3. P. 60-83
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