Abstract
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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REFERENCES
S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38 (2), 248–253 (1974).
G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, 1989).
V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” J. Exp. Theor. Phys. 34 (1), 62–69 (1972).
S. P. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons: The Inverse Scattering Method (Nauka, Moscow, 1980; Consultants Bureau, New York, 1984).
A. I. Maimistov, A. M. Basharov, S. O. Elyutin, and Y. M. Sklyarov, “Present state of self-induced transparency theory,” Phys. Rep. 191 (1), 1 (1990).
A. I. Maimistov and A. M. Basharov, Nonlinear Optical Waves (Springer Science and Business Media, Dordrecht, 2013).
L. L. Frumin, “Algorithms for solving scattering problems for the Manakov model of nonlinear Schrödinger equations,” J. Inv. Ill-Posed Probl. 29 (2), 369 (2021).
O. V. Belai, L. L. Frumin, E. V. Podivilov, and D. A. Shapiro, “Efficient numerical method of the fiber Bragg grating synthesis,” J. Opt. Soc. Am. B 24 (7), 1451 (2007).
L. L. Frumin, O. V. Belai, E. V. Podivilov, and D. A. Shapiro, “Efficient numerical method for solving the direct Zakharov–Shabat scattering problem,” J. Opt. Soc. Am. B 32, 290 (2015).
R. Blahut, Fast Algorithms for Digital Signal Processing (Addison-Wesley, Reading, Mass., 1985).
A. Buryak, J. Bland-Hawthorn, and V. Steblina, “Comparison of inverse scattering algorithms for designing ultrabroadband fibre Bragg gratings,” Opt. Express 17 (3), 1995 (2009).
O. V. Belai, L. L. Frumin, E. V. Podivilov, and D. A. Shapiro, “Inverse scattering problem for gratings with deep modulation,” Laser Phys. 20 (2), 318 (2010).
O. V. Belai, L. L. Frumin, E. V. Podivilov, and D. A. Shapiro, “Inverse scattering for the one-dimensional Helmholtz equation: Fast numerical method,” Opt. Lett. 33 (18), 2101 (2008).
L. L. Frumin, A. A. Gelash, and S. K. Turitsyn, “New approaches to coding information using inverse scattering transform,” Phys. Rev. Lett. 118 (22), 223901 (2017).
S. K. Turitsyn, J. E. Prilepsky, S. T. Le, S. Wahls, L. L. Frumin, M. Kamalian, and S. A. Derevyanko, “Nonlinear Fourier transform for optical data processing and transmission: Advances and perspectives,” Optica 4 (3), 307 (2017).
E. E. Tyrtyshnikov, Toeplitz Matrices, Some of Their Analogues, and Applications (Akad. Nauk SSSR, Moscow, 1989) [in Russian].
E. E. Tyrtyshnikov, “New fast algorithms for systems with Hankel and Toeplitz matrices,” USSR Comput. Math. Math. Phys. 29 (3), 1–6 (1989).
H. Akaike, “Block Toeplitz matrix inversion,” SIAM J. Appl. Math. 24 (2), 234 (1973).
O. V. Belai, “Fast second-order accurate numerical method for solving an inverse scattering method,” Kvant. Elektron. 52 (11), 1039 (2022).
V. V. Voevodin and E. E. Tyrtyshnikov, Computational Processes with Toeplitz Matrices (Nauka, Moscow, 1987) [in Russian].
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This work was supported by the Russian Science Foundation, grant no. 22-22-00653.
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Translated by E. Chernokozhin
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Belai, O.V., Frumin, L.L. & Chernyavsky, A.E. Algorithms for Solving the Inverse Scattering Problem for the Manakov Model. Comput. Math. and Math. Phys. 64, 453–464 (2024). https://doi.org/10.1134/S0965542524030059
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DOI: https://doi.org/10.1134/S0965542524030059