Fast Algorithms for Solving the Inverse Scattering Problem

A. L. Delitsyn

Matematičeskie zametki, Tome 112 (2022) no. 2, pp. 198-217.

Résumé

The problem of numerical solution of a nonlinear Schrödinger equation is considered from the point of view of applications to the compensation of signal distortions in a fiber optic communication line. The problem of constructing fast algorithms for the direct and inverse scattering problems for the Zakharov–Shabat system of equations is studied. An overview of the main methods used currently is given. The time complexity of the algorithms is described together with their applicability to realistic signals.

Détail

Keywords: inverse scattering problem, nonlinear Schrödinger equation, Zakharov–Shabat equations, fast algorithms.

Bibliographie

[1] M. I. Yousefi, F. R. Kschischang, “Information transmission using the nonlinear Fourier transform. Part I: Mathematical tools”, IEEE Trans. Inform. Theory, 60:7 (2014), 4312–4328 | DOI | MR | Zbl

[2] M. I. Yousefi, F. R. Kschischang, “Information transmission using the nonlinear Fourier transform. Part II: Numerical Methods”, IEEE Trans. Inform. Theory, 60:7 (2014), 4329–4345 | DOI | MR | Zbl

[3] M. I. Yousefi, F. R. Kschischang, “Information transmission using the nonlinear Fourier transform. Part III: Spectrum Modulation”, IEEE Trans. Inform. Theory, 60:7 (2014), 4346–4369 | DOI | MR | Zbl

[4] J. Ginibre, G. Velo, “On a class of nonlinear Schrodinger equations”, J. Funct. Anal., 32 (1979), 1–32 | DOI | MR | Zbl

[5] W. A. Strauss, “Mathematical aspects of classical nonlinear field equations”, Nonlinear Problems in Theoretical Physics, Lect. Notes in Phys., 98, Springer, Berlin, 1979, 123–149 | DOI | MR

[6] M. Tsutsumi, “Sobolev spaces and repeadly decreased solutions of some nonlinear dispersive wave equations”, J. Differential Equations, 42:2 (1981), 260–281 | DOI | MR | Zbl

[7] L. A. Takhtadzhyan, L. D. Faddeev, Gamiltonov podkhod v teorii solitonov, Nauka, M., 1986 | MR

[8] T. Kormen, Ch. Leizerson, R. Rivest, K. Shtain, Algoritmy. Postroenie i analiz, Vilyams, M., 2013 | MR

[9] D. Rafique, “Fiber nonlinearity compensation: commercial applications and complexity analysis”, J. Lightwave Technology, 34:2 (2016), 544–553 | DOI

[10] Dzh. L. Lem, Vvedenie v teoriyu solitonov, Mir, M., 1983

[11] V. E. Zakharov, A. B. Shabat, Tochnaya teoriya dvumernoi samofokusirovki i odnomernoi avtomodulyatsii voln, IYaF, Novosibirsk, 1971

[12] V. A. Marchenko, “Vosstanovlenie potentsialnoi energii po fazam rasseyannykh voln”, Dokl. AN, 104:5 (1955), 695–698 | MR | Zbl

[13] I. M. Gelfand, B. M. Levitan, “Ob opredelenii differentsialnogo uravneniya po ego spektralnoi funktsii”, Izv. AN SSSR. Ser. matem., 15:4 (1951), 309–360 | MR | Zbl

[14] S. K. Turitsyn, J. E. Prilepsky, S. T. Le, S. Wahls, L. L. Frumin, M. Kamalian, S. A. Derevyanko, “Nonlinear Fourier transform for optical data processing and transmission: advances and perspectives”, Optica, 2017, no. 4, 307–322 | DOI

[15] V. Vaibhav, “Fast inverse nonlinear Fourier transformation using exponential one-step methods: Darboux transformation”, Phys. Rev. E, 96:6 (2017), 063302-1–063302-35 | DOI | MR

[16] W. K. McClary, “Fast seismic inversion”, Geophysics, 48:10 (1983), 1371–1372 | DOI

[17] V. Vaibhav, “Nonlinear Fourier transform of time-limited and one-sided signals”, J. Phys. A, 51:42 (2018), Art no. 425201 | DOI | MR

[18] S. Wahls, H. V. Poor, “Fast numerical nonlinear Fourier transforms”, IEEE Trans. Inform. Theory, 61:12 (2015), 6957–6974 | DOI | MR | Zbl

[19] S. Chimmalgi, P. J. Prins, S. Wahls, “Fast nonlinear Fourier transform algorithms using higher order exponential integrators”, IEEE Access, 7:1 (2019), 145161–145176 | DOI

[20] M. G. Krein, “Ob opredelenii potentsiala chastitsy po ee $S$-funktsii”, Dokl. AN, 105:4 (1955), 433–436 | MR | Zbl

[21] O. V. Belai, L. L. Frumin, E. V. Podivilov, D. A. Shapiro, “Efficient numerical method of the fiber Bragg grating synthesis”, J. Opt. Soc. Amer. B Opt. Phys., 24:7 (2007), 1451–1457 | DOI | MR

[22] V. V. Voevodin, E. E. Tyrtyshnikov, Vychislitelnye protsessy s teplitsevymi matritsami, Nauka, M., 1987 | MR | Zbl

[23] F. de Hoog, “A New Algorithm for Solving Toeplitz Systems of Equations”, Linear Algebra Appl., 88/89 (1987), 123–138 | DOI | MR | Zbl

[24] Dzh. Golub, E. Ch. Van-Loun, Matrichnye vychisleniya, Mir, M., 1999

[25] D. A. Bini, L. Gemignani, V. Y. Pan, “Fast and stable QR eigenvalue algorithms for generalized companion matrices and secular equations”, Numer. Math., 100:3 (2005), 373–408 | DOI | MR | Zbl

[26] V. Y. Pan, “Solving a polynomial equation: some history and recent progress”, SIAM Rev., 39:2 (1997), 187–220 | DOI | MR | Zbl

[27] L. M. Delves, J. N. Lyness, “A Numerical Method for Locating the Zeros of an Analytic Function”, Math. Comp., 21:100 (1967), 543–560 | DOI | MR | Zbl

[28] A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential methods in inverse scattering”, SIAM J. Appl. Math., 45:2 (1985), 312–335 | DOI | MR | Zbl

[29] A. M. Bruckstein, I. Koltracht, T. Kailath, “Inverse scattering with noisy data”, SIAM J. Sci. Statist. Comput., 7:4 (1986), 1331–1349 | DOI | MR | Zbl

[30] A. M. Bruckstein, T. Kailath, “Inverse scattering for discrete transmission-line models”, SIAM Rev., 29:3 (1987), 359–389 | DOI | MR | Zbl

[31] J. Skaar, L. Wang, T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling”, IEEE J. Quant. Electron., 37:2 (2001), 165–173 | DOI

[32] L. Poladian, “On the synthesis of fiber Bragg gratings by layer peeling”, IEEE J. Quant. Electron., 37:2 (2001), 165–173 | DOI

[33] L. Poladian, “Simple grating synthesis algorithm”, Optics Lett., 25:11 (2000), 787–789 | DOI