Algorithms for solving the inverse scattering problem for the Manakov model

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The paper considers algorithms for solving inverse scattering problems based on the discretization of the Gelfand–Levitan–Marchenko integral equations, associated with the system of nonlinear Schrödinger equations of the Manakov model. The numerical algorithm of the first order approximation for solving the scattering problem is reduced to the inversion of a series of nested block Toeplitz matrices using the Levinson-type bordering method. Increasing the approximation accuracy violates the Toeplitz structure of block matrices. Two algorithms are described that solve this problem for second order accuracy. One algorithm uses a block version of the Levinson bordering algorithm, which recovers the Toeplitz structure of the matrix by moving some terms of the systems of equations to the right-hand side. Another algorithm is based on the Toeplitz decomposition of an almost block-Toeplitz matrix and the Tyrtyshnikov bordering algorithm. The speed and accuracy of calculations using the presented algorithms are compared on an exact solution (the Manakov vector soliton).

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作者简介

O. Belaiа

Institute of Automation and Electrometry, Siberian Branch, RAS

编辑信件的主要联系方式.
Email: ovbelai@gmail.com
俄罗斯联邦, Acad. Koptyug Ave. 1, Novosibirsk, 630090

L. Frumin

Institute of Automation and Electrometry, Siberian Branch, RAS

Email: lfrumin@iae.nsk.su
俄罗斯联邦, Acad. Koptyug Ave. 1, Novosibirsk, 630090

A. Chernyavsky

Institute of Automation and Electrometry, Siberian Branch, RAS

Email: alexander.cher.99@gmail.com
俄罗斯联邦, Acad. Koptyug Ave. 1, Novosibirsk, 630090

参考

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2. Fig. 1.a is a graph of the dependence of the vector soliton reconstruction error; b is a graph of the dependence of the vector soliton reconstruction calculation time. The squares show the calculated values ​​for algorithm A, the circles for algorithm B.

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