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High-Order Block Toeplitz Inner-Bordering method for solving the Gelfand-Levitan-Marchenko equation (2405.00529v1)

Published 1 May 2024 in math.NA and cs.NA

Abstract: We propose a high precision algorithm for solving the Gelfand-Levitan-Marchenko equation. The algorithm is based on the block version of the Toeplitz Inner-Bordering algorithm of Levinson's type. To approximate integrals, we use the high-precision one-sided and two-sided Gregory quadrature formulas. Also we use the Woodbury formula to construct a computational algorithm. This makes it possible to use the almost Toeplitz structure of the matrices for the fast calculations.

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References (43)
  1. Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in non-linear media. Journal of Experimental and Theoretical Physics, 34(1):62–69, 1972.
  2. Solitons and the inverse scattering transform. SIAM, 1981.
  3. George L Lamb. Elements of soliton theory. Wiley, 1980.
  4. Information Transmission Using the Nonlinear Fourier Transform, Part III: Spectrum Modulation. IEEE Transactions on Information Theory, 60(7):4346–4369, 2014.
  5. Nonlinear inverse synthesis for high spectral efficiency transmission in optical fibers. Optics Express, 22(22):26720, 11 2014.
  6. High-order modulation on a single discrete eigenvalue for optical communications based on nonlinear Fourier transform. Optics Express, 25(17):20286, 8 2017.
  7. Sander Wahls. Generation of Time-Limited Signals in the Nonlinear Fourier Domain via b-Modulation. In 2017 European Conference on Optical Communication (ECOC), pages 1–3. IEEE, 9 2017.
  8. Nonlinear frequency division multiplexing with b-modulation: shifting the energy barrier. Optics Express, 26(21):27978, 10 2018.
  9. Polarization-multiplexed nonlinear inverse synthesis with standard and reduced-complexity NFT processing. Optics Express, 26(13):17360, 2018.
  10. Bound State Soliton Gas Dynamics Underlying the Spontaneous Modulational Instability. Physical Review Letters, 123(23):234102, 12 2019.
  11. Direct scattering transform of large wave packets. Optics Letters, 44(21):5298, 11 2019.
  12. Raman Kashyap. Fiber bragg gratings. Academic press, 1999.
  13. Exactly solvable profiles of quasi-rectangular bragg filter with dispersion compensation. Journal of Optics A: Pure and Applied Optics, 8(9):788, 2006.
  14. Vladimir M Akulin. Coherent dynamics of complex quantum systems. Springer Science & Business Media, 2005.
  15. L Carmel and A Mann. Geometrical approach to two-level hamiltonians. Physical review A, 61(5):052113, 2000.
  16. Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. i. anomalous dispersion. Applied Physics Letters, 23(3):142–144, 1973.
  17. Optical solitons in fibers. In Optical Solitons in Fibers, pages 41–59. Springer, 2003.
  18. Solitons in optical fibers: fundamentals and applications. Elsevier, 2006.
  19. Physics and mathematics of dispersion-managed optical solitons. Comptes Rendus Physique, 4(1):145–161, 2003.
  20. Eigenvalue communication. Journal of lightwave technology, 11(3):395–399, 1993.
  21. Multieigenvalue communication. Journal of Lightwave Technology, 34(13):3110–3117, 2016.
  22. Fast inverse nonlinear fourier transforms for continuous spectra of zakharov-shabat type. arXiv preprint arXiv:1607.01305, 2016.
  23. Full-spectrum periodic nonlinear fourier transform optical communication through solving the riemann-hilbert problem. Journal of Lightwave Technology, 38(14):3602–3615, 2020.
  24. G.B. Xiao and K. Yashiro. An efficient algorithm for solving zakharov-shabat inverse scattering problem. IEEE Trans. Antennas Propag, 50:807–811, 2002.
  25. Finite bragg grating synthesis by numerical solution of hermitian gel’fand-levitan-marchenko equations. JOSA B, 23(10):2040–2045, 2006.
  26. Efficient numerical method of the fiber bragg grating synthesis. JOSA B, 24(7):1451–1457, 2007.
  27. Efficient numerical method for solving the direct zakharov–shabat scattering problem. JOSA B, 32(2):290–296, 2015.
  28. Block toeplitz inner-bordering method for the gelfand–levitan–marchenko equations associated with the zakharov–shabat system. Journal of Inverse and Ill-posed Problems, 31(2):191–202, 2023.
  29. An efficient inverse scattering algorithm for the design of nonuniform fiber bragg gratings. IEEE journal of quantum electronics, 35(8):1105–1115, 1999.
  30. A. Rosenthal and M. Horowitz. Inverse scattering algorithm for reconstructing strongly reflecting fiber bragg gratings. IEEE J. Quantum. Electron., 39:1018–1026, 2003.
  31. F. Ahmad and M. Razzagh. A numerical solution to the gel’fand-levitan-marchenko equation. Appl. Math. Comput., 89:31–39, 1998.
  32. M. Yousefi and X. Yangzhang. Linear and nonlinear frequency-division multiplexing. IEEE Transaction on Information Theory, 66(1):478–495, 2020.
  33. Vahid Aref. Control and detection of discrete spectral amplitudes in nonlinear fourier spectrum. arXiv preprint arXiv:1605.06328, 2016.
  34. Modulation over nonlinear fourier spectrum: Continuous and discrete spectrum. Journal of Lightwave Technology, 36(6):1289–1295, 2018.
  35. Nonlinear fourier transform for optical data processing and transmission: advances and perspectives. Optica, 4(3):307–322, 2017.
  36. Comparison of inverse scattering algorithms for designing ultrabroadband fibre bragg gratings. Optics express, 17(3):1995–2004, 2009.
  37. Richard Hamming. Numerical methods for scientists and engineers. McGraw-Hill, 1962.
  38. GM Phillips. Gregory’s method for numerical integration. The American Mathematical Monthly, 79(3):270–274, 1972.
  39. An improved gregory-like method for 1-d quadrature. Numerische Mathematik, 141(1):1–19, 2019.
  40. Richard E Blahut. Fast algorithms for signal processing. Cambridge University Press, 2010.
  41. William W Hager. Updating the inverse of a matrix. SIAM review, 31(2):221–239, 1989.
  42. The toeplitz package users’ guide. Technical report, Argonne Nat. Lab., 1983.
  43. Exponential fourth order schemes for direct Zakharov-Shabat problem. Optics Express, 28(1):20, 1 2020.
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