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Algorithm for calculating scattering spectra on soliton potentials for the Zakharov-Shabat system
A.E. Chernyavsky1,2, L.L. Frumin1

1Institute of Automation and Electrometry SB RAS, Prospekt Akademika Koptyuga 1, Novosibirsk, 630090, Russia;
2Novosibirsk State University, Pirogova Str. 2, Novosibirsk, 630090, Russia

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DOI: 10.18287/COJ1716

Article ID: 1716

Abstract:
We present an algorithm for solving the direct scattering problem for a discrete spectrum of the Zakharov-Shabat system corresponding to soliton potentials, which are solutions of the nonlinear Schrödinger equation. The algorithm is designed to decode information in soliton optical communication lines. Soliton potentials are nonreflective. Scattering on them generates evanescent (decaying) waves. A transfer-matrix method is used to calculate the scattering spectra. An implicit difference scheme is constructed using the integral method of discrete approximation of matrices. The stability of the scheme is confirmed by estimates of the operator norm of the product of the transfer matrices. The algorithm finds spectral scattering coefficients and complex normalization parameters of the N-soliton solution of the nonlinear Schrödinger equation, reducing the problem to calculating products of polynomials that depend on the spectral parameter. The doubling strategy, convolution theorem, and fast Fourier transform are used to speed up the calculations. Numerical simulation for an N-soliton discrete spectrum of the Zakharov-Shabat system confirmed the stability, high speed of calculations and second order accuracy of the algorithm.

Keywords:
Zakharov-Shabat system, scattering problem, implicit scheme, discrete spectrum, soliton, algorithm.

Citation:
Chernyavsky AE, Frumin LL. Algorithm for calculating scattering spectra on soliton potentials for the Zakharov-Shabat system. Computer Optics 2026; 50(2): 1716. DOI: 10.18287/COJ1716.

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