Abstract

Nowadays, improving the accuracy of computational methods to solve the initial value problem of the Zakharov-Shabat system remains an urgent problem in optics. In particular, increasing the approximation order of the methods is important, especially in problems where it is necessary to analyze the structure of complex waveforms. In this work, we propose two finite-difference algorithms of fourth order of approximation in the time variable. Both schemes have the exponential form and conserve the quadratic invariant of Zakharov-Shabat system. The second scheme allows applying fast algorithms with low computational complexity (fast nonlinear Fourier transform).

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Conservative multi-exponential scheme for solving the direct Zakharov–Shabat scattering problem

Sergey Medvedev, Igor Chekhovskoy, Irina Vaseva, and Mikhail Fedoruk
Opt. Lett. 45(7) 2082-2085 (2020)

Numerical algorithm with fourth-order accuracy for the direct Zakharov-Shabat problem

Sergey Medvedev, Irina Vaseva, Igor Chekhovskoy, and Mikhail Fedoruk
Opt. Lett. 44(9) 2264-2267 (2019)

Efficient numerical method for solving the direct Zakharov–Shabat scattering problem

Leonid L. Frumin, Oleg V. Belai, Eugeny V. Podivilov, and David A. Shapiro
J. Opt. Soc. Am. B 32(2) 290-296 (2015)

References

  • View by:
  • |
  • |
  • |

  1. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of wave in nonlinear media,” J. Exp. Theor. Phys. 34, 62–69 (1972).
  2. M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (Society for Industrial and Applied Mathematics, 1981).
  3. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2013), 4th ed.
  4. V. I. Karpman, Non-Linear Waves in Dispersive Media: International Series of Monographs in Natural Philosophy, vol. 71 (Elsevier, 2016).
  5. M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, part III: Spectrum modulation,” IEEE Trans. Inf. Theory 60(7), 4346–4369 (2014).
    [Crossref]
  6. S. T. Le, J. E. Prilepsky, and S. K. Turitsyn, “Nonlinear inverse synthesis for high spectral efficiency transmission in optical fibers,” Opt. Express 22(22), 26720 (2014).
    [Crossref]
  7. T. Gui, C. Lu, A. P. T. Lau, and P. K. A. Wai, “High-order modulation on a single discrete eigenvalue for optical communications based on nonlinear Fourier transform,” Opt. Express 25(17), 20286 (2017).
    [Crossref]
  8. S. Wahls, “Generation of time-limited signals in the nonlinear Fourier domain via b-modulation,” in 2017 European Conference on Optical Communication (ECOC), (IEEE, 2017), 6, pp. 1–3.
  9. T. Gui, G. Zhou, C. Lu, A. P. T. Lau, and S. Wahls, “Nonlinear frequency division multiplexing with b-modulation: shifting the energy barrier,” Opt. Express 26(21), 27978 (2018).
    [Crossref]
  10. S. Civelli, S. K. Turitsyn, M. Secondini, and J. E. Prilepsky, “Polarization-multiplexed nonlinear inverse synthesis with standard and reduced-complexity NFT processing,” Opt. Express 26(13), 17360 (2018).
    [Crossref]
  11. G. Boffetta and A. Osborne, “Computation of the direct scattering transform for the nonlinear Schroedinger equation,” J. Comput. Phys. 102(2), 252–264 (1992).
    [Crossref]
  12. S. Burtsev, R. Camassa, and I. Timofeyev, “Numerical algorithms for the direct spectral transform with applications to nonlinear Schrödinger type systems,” J. Comput. Phys. 147(1), 166–186 (1998).
    [Crossref]
  13. M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, part II: Numerical methods,” IEEE Trans. Inf. Theory 60(7), 4329–4345 (2014).
    [Crossref]
  14. 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).
    [Crossref]
  15. A. Vasylchenkova, J. Prilepsky, D. Shepelsky, and A. Chattopadhyay, “Direct nonlinear Fourier transform algorithms for the computation of solitonic spectra in focusing nonlinear Schrödinger equation,” Commun. Nonlinear Sci. Numer. Simul. 68, 347–371 (2019).
    [Crossref]
  16. S. Wahls and H. V. Poor, “Introducing the fast nonlinear Fourier transform,” in International Conference on Acoustics, Speech and Signal Processing, (IEEE, Vancouver, 2013), pp. 5780–5784.
  17. S. Wahls and H. V. Poor, “Fast inverse nonlinear Fourier transform for generating multi-solitons in optical fiber,” in International Symposium on Information Theory (ISIT), (IEEE, Hong Kong, 2015), pp. 1676–1680.
  18. V. Vaibhav, “Higher order convergent fast nonlinear Fourier transform,” IEEE Photonics Technol. Lett. 30(8), 700–703 (2018).
    [Crossref]
  19. S. Wahls, S. Chimmalgi, and P. J Prins, “FNFT: a software library for computing nonlinear Fourier transforms,” J. Open Source Softw. 3(23), 597 (2018).
    [Crossref]
  20. S. Chimmalgi, P. J. Prins, and S. Wahls, “Fast nonlinear Fourier transform algorithms using higher order exponential integrators,” IEEE Access 7, 145161 (2019).
    [Crossref]
  21. E. V. Sedov, A. A. Redyuk, M. P. Fedoruk, A. A. Gelash, L. L. Frumin, and S. K. Turitsyn, “Soliton content in the standard optical OFDM signal,” Opt. Lett. 43(24), 5985 (2018).
    [Crossref]
  22. P. J. Prins and S. Wahls, “Higher order exponential splittings for the fast non-linear Fourier transform of the Korteweg-De Vries equation,” in ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings, (IEEE, 2018), 4, pp. 4524–4528.
  23. S. Medvedev, I. Chekhovskoy, I. Vaseva, and M. Fedoruk, “Fast computation of the direct scattering transform by fourth order conservative multi-exponential scheme,” arXiv preprint arXiv:1909.13228 (2019).
  24. S. Medvedev, I. Vaseva, I. Chekhovskoy, and M. Fedoruk, “Numerical algorithm with fourth-order accuracy for the direct Zakharov-Shabat problem,” Opt. Lett. 44(9), 2264 (2019).
    [Crossref]
  25. 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(2), 290–296 (2015).
    [Crossref]
  26. S. Blanes, F. Casas, and M. Thalhammer, “High-order commutator-free quasi-Magnus exponential integrators for non-autonomous linear evolution equations,” Comput. Phys. Commun. 220, 243–262 (2017).
    [Crossref]
  27. M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, vol. 4 (Siam, 1981).
  28. G. G. Dahlquist, “A special stability problem for linear multistep methods,” BIT 3(1), 27–43 (1963).
    [Crossref]
  29. E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems (Springer-Verlag, 1987).
  30. G. A. Baker Jr and P. Graves-Morris, Padé Approximants (Cambridge University, 1996), 2nd ed.
  31. W. Magnus, “On the exponential solution of differential equations for a linear operator,” Comm. Pure Appl. Math. 7(4), 649–673 (1954).
    [Crossref]
  32. S. Blanes, F. Casas, J. Oteo, and J. Ros, “The Magnus expansion and some of its applications,” Phys. Rep. 470(5-6), 151–238 (2009).
    [Crossref]
  33. I. Puzynin, A. Selin, and S. Vinitsky, “A high-order accuracy method for numerical solving of the time-dependent Schrödinger equation,” Comput. Phys. Commun. 123(1-3), 1–6 (1999).
    [Crossref]
  34. I. Puzynin, A. Selin, and S. Vinitsky, “Magnus-factorized method for numerical solving the time-dependent Schrödinger equation,” Comput. Phys. Commun. 126(1-2), 158–161 (2000).
    [Crossref]
  35. O. Chuluunbaatar, V. Derbov, A. Galtbayar, A. Gusev, M. Kaschiev, S. Vinitsky, and T. Zhanlav, “Explicit Magnus expansions for solving the time-dependent Schrödinger equation,” J. Phys. A: Math. Theor. 41(29), 295203 (2008).
    [Crossref]
  36. S. Wahls and V. Vaibhav, “Fast inverse nonlinear Fourier transforms for continuous spectra of Zakharov-Shabat type,” arXiv preprint arXiv:1607.01305 pp. 1–15 (2016).
  37. R. Mullyadzhanov and A. Gelash, “Direct scattering transform of large wave packets,” Opt. Lett. 44(21), 5298–5301 (2019).
    [Crossref]
  38. S. Medvedev, I. Vaseva, I. Chekhovskoy, and M. Fedoruk, “Exponential fourth order schemes for direct Zakharov-Shabat problem,” arXiv preprint arXiv:1908.11725 (2019).
  39. R. M. Wilcox, “Exponential operators and parameter differentiation in quantum physics,” J. Math. Phys. 8(4), 962–982 (1967).
    [Crossref]
  40. G. Engeln-Mullges and F. Uhlig, Numerical Algorithms with C (Springer-Verlag, 1996).
  41. F. A. Grunbaum, “The scattering problem for a phase-modulated hyperbolic secant pulse,” Inverse Probl. 5(3), 287–292 (1989).
    [Crossref]
  42. S. Hari and F. R. Kschischang, “Bi-directional algorithm for computing discrete spectral amplitudes in the NFT,” J. Lightwave Technol. 34(15), 3529–3537 (2016).
    [Crossref]
  43. V. Vaibhav, “Efficient nonlinear Fourier transform algorithms of order four on equispaced grid,” IEEE Photonics Technol. Lett. 31(15), 1269–1272 (2019).
    [Crossref]

2019 (5)

A. Vasylchenkova, J. Prilepsky, D. Shepelsky, and A. Chattopadhyay, “Direct nonlinear Fourier transform algorithms for the computation of solitonic spectra in focusing nonlinear Schrödinger equation,” Commun. Nonlinear Sci. Numer. Simul. 68, 347–371 (2019).
[Crossref]

S. Chimmalgi, P. J. Prins, and S. Wahls, “Fast nonlinear Fourier transform algorithms using higher order exponential integrators,” IEEE Access 7, 145161 (2019).
[Crossref]

S. Medvedev, I. Vaseva, I. Chekhovskoy, and M. Fedoruk, “Numerical algorithm with fourth-order accuracy for the direct Zakharov-Shabat problem,” Opt. Lett. 44(9), 2264 (2019).
[Crossref]

R. Mullyadzhanov and A. Gelash, “Direct scattering transform of large wave packets,” Opt. Lett. 44(21), 5298–5301 (2019).
[Crossref]

V. Vaibhav, “Efficient nonlinear Fourier transform algorithms of order four on equispaced grid,” IEEE Photonics Technol. Lett. 31(15), 1269–1272 (2019).
[Crossref]

2018 (5)

2017 (3)

2016 (1)

2015 (1)

2014 (3)

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, part III: Spectrum modulation,” IEEE Trans. Inf. Theory 60(7), 4346–4369 (2014).
[Crossref]

S. T. Le, J. E. Prilepsky, and S. K. Turitsyn, “Nonlinear inverse synthesis for high spectral efficiency transmission in optical fibers,” Opt. Express 22(22), 26720 (2014).
[Crossref]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, part II: Numerical methods,” IEEE Trans. Inf. Theory 60(7), 4329–4345 (2014).
[Crossref]

2009 (1)

S. Blanes, F. Casas, J. Oteo, and J. Ros, “The Magnus expansion and some of its applications,” Phys. Rep. 470(5-6), 151–238 (2009).
[Crossref]

2008 (1)

O. Chuluunbaatar, V. Derbov, A. Galtbayar, A. Gusev, M. Kaschiev, S. Vinitsky, and T. Zhanlav, “Explicit Magnus expansions for solving the time-dependent Schrödinger equation,” J. Phys. A: Math. Theor. 41(29), 295203 (2008).
[Crossref]

2000 (1)

I. Puzynin, A. Selin, and S. Vinitsky, “Magnus-factorized method for numerical solving the time-dependent Schrödinger equation,” Comput. Phys. Commun. 126(1-2), 158–161 (2000).
[Crossref]

1999 (1)

I. Puzynin, A. Selin, and S. Vinitsky, “A high-order accuracy method for numerical solving of the time-dependent Schrödinger equation,” Comput. Phys. Commun. 123(1-3), 1–6 (1999).
[Crossref]

1998 (1)

S. Burtsev, R. Camassa, and I. Timofeyev, “Numerical algorithms for the direct spectral transform with applications to nonlinear Schrödinger type systems,” J. Comput. Phys. 147(1), 166–186 (1998).
[Crossref]

1992 (1)

G. Boffetta and A. Osborne, “Computation of the direct scattering transform for the nonlinear Schroedinger equation,” J. Comput. Phys. 102(2), 252–264 (1992).
[Crossref]

1989 (1)

F. A. Grunbaum, “The scattering problem for a phase-modulated hyperbolic secant pulse,” Inverse Probl. 5(3), 287–292 (1989).
[Crossref]

1972 (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of wave in nonlinear media,” J. Exp. Theor. Phys. 34, 62–69 (1972).

1967 (1)

R. M. Wilcox, “Exponential operators and parameter differentiation in quantum physics,” J. Math. Phys. 8(4), 962–982 (1967).
[Crossref]

1963 (1)

G. G. Dahlquist, “A special stability problem for linear multistep methods,” BIT 3(1), 27–43 (1963).
[Crossref]

1954 (1)

W. Magnus, “On the exponential solution of differential equations for a linear operator,” Comm. Pure Appl. Math. 7(4), 649–673 (1954).
[Crossref]

Ablowitz, M. J.

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, vol. 4 (Siam, 1981).

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (Society for Industrial and Applied Mathematics, 1981).

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2013), 4th ed.

Baker Jr, G. A.

G. A. Baker Jr and P. Graves-Morris, Padé Approximants (Cambridge University, 1996), 2nd ed.

Belai, O. V.

Blanes, S.

S. Blanes, F. Casas, and M. Thalhammer, “High-order commutator-free quasi-Magnus exponential integrators for non-autonomous linear evolution equations,” Comput. Phys. Commun. 220, 243–262 (2017).
[Crossref]

S. Blanes, F. Casas, J. Oteo, and J. Ros, “The Magnus expansion and some of its applications,” Phys. Rep. 470(5-6), 151–238 (2009).
[Crossref]

Boffetta, G.

G. Boffetta and A. Osborne, “Computation of the direct scattering transform for the nonlinear Schroedinger equation,” J. Comput. Phys. 102(2), 252–264 (1992).
[Crossref]

Burtsev, S.

S. Burtsev, R. Camassa, and I. Timofeyev, “Numerical algorithms for the direct spectral transform with applications to nonlinear Schrödinger type systems,” J. Comput. Phys. 147(1), 166–186 (1998).
[Crossref]

Camassa, R.

S. Burtsev, R. Camassa, and I. Timofeyev, “Numerical algorithms for the direct spectral transform with applications to nonlinear Schrödinger type systems,” J. Comput. Phys. 147(1), 166–186 (1998).
[Crossref]

Casas, F.

S. Blanes, F. Casas, and M. Thalhammer, “High-order commutator-free quasi-Magnus exponential integrators for non-autonomous linear evolution equations,” Comput. Phys. Commun. 220, 243–262 (2017).
[Crossref]

S. Blanes, F. Casas, J. Oteo, and J. Ros, “The Magnus expansion and some of its applications,” Phys. Rep. 470(5-6), 151–238 (2009).
[Crossref]

Chattopadhyay, A.

A. Vasylchenkova, J. Prilepsky, D. Shepelsky, and A. Chattopadhyay, “Direct nonlinear Fourier transform algorithms for the computation of solitonic spectra in focusing nonlinear Schrödinger equation,” Commun. Nonlinear Sci. Numer. Simul. 68, 347–371 (2019).
[Crossref]

Chekhovskoy, I.

S. Medvedev, I. Vaseva, I. Chekhovskoy, and M. Fedoruk, “Numerical algorithm with fourth-order accuracy for the direct Zakharov-Shabat problem,” Opt. Lett. 44(9), 2264 (2019).
[Crossref]

S. Medvedev, I. Chekhovskoy, I. Vaseva, and M. Fedoruk, “Fast computation of the direct scattering transform by fourth order conservative multi-exponential scheme,” arXiv preprint arXiv:1909.13228 (2019).

S. Medvedev, I. Vaseva, I. Chekhovskoy, and M. Fedoruk, “Exponential fourth order schemes for direct Zakharov-Shabat problem,” arXiv preprint arXiv:1908.11725 (2019).

Chimmalgi, S.

S. Chimmalgi, P. J. Prins, and S. Wahls, “Fast nonlinear Fourier transform algorithms using higher order exponential integrators,” IEEE Access 7, 145161 (2019).
[Crossref]

S. Wahls, S. Chimmalgi, and P. J Prins, “FNFT: a software library for computing nonlinear Fourier transforms,” J. Open Source Softw. 3(23), 597 (2018).
[Crossref]

Chuluunbaatar, O.

O. Chuluunbaatar, V. Derbov, A. Galtbayar, A. Gusev, M. Kaschiev, S. Vinitsky, and T. Zhanlav, “Explicit Magnus expansions for solving the time-dependent Schrödinger equation,” J. Phys. A: Math. Theor. 41(29), 295203 (2008).
[Crossref]

Civelli, S.

Dahlquist, G. G.

G. G. Dahlquist, “A special stability problem for linear multistep methods,” BIT 3(1), 27–43 (1963).
[Crossref]

Derbov, V.

O. Chuluunbaatar, V. Derbov, A. Galtbayar, A. Gusev, M. Kaschiev, S. Vinitsky, and T. Zhanlav, “Explicit Magnus expansions for solving the time-dependent Schrödinger equation,” J. Phys. A: Math. Theor. 41(29), 295203 (2008).
[Crossref]

Derevyanko, S. A.

Engeln-Mullges, G.

G. Engeln-Mullges and F. Uhlig, Numerical Algorithms with C (Springer-Verlag, 1996).

Fedoruk, M.

S. Medvedev, I. Vaseva, I. Chekhovskoy, and M. Fedoruk, “Numerical algorithm with fourth-order accuracy for the direct Zakharov-Shabat problem,” Opt. Lett. 44(9), 2264 (2019).
[Crossref]

S. Medvedev, I. Chekhovskoy, I. Vaseva, and M. Fedoruk, “Fast computation of the direct scattering transform by fourth order conservative multi-exponential scheme,” arXiv preprint arXiv:1909.13228 (2019).

S. Medvedev, I. Vaseva, I. Chekhovskoy, and M. Fedoruk, “Exponential fourth order schemes for direct Zakharov-Shabat problem,” arXiv preprint arXiv:1908.11725 (2019).

Fedoruk, M. P.

Frumin, L. L.

Galtbayar, A.

O. Chuluunbaatar, V. Derbov, A. Galtbayar, A. Gusev, M. Kaschiev, S. Vinitsky, and T. Zhanlav, “Explicit Magnus expansions for solving the time-dependent Schrödinger equation,” J. Phys. A: Math. Theor. 41(29), 295203 (2008).
[Crossref]

Gelash, A.

Gelash, A. A.

Graves-Morris, P.

G. A. Baker Jr and P. Graves-Morris, Padé Approximants (Cambridge University, 1996), 2nd ed.

Grunbaum, F. A.

F. A. Grunbaum, “The scattering problem for a phase-modulated hyperbolic secant pulse,” Inverse Probl. 5(3), 287–292 (1989).
[Crossref]

Gui, T.

Gusev, A.

O. Chuluunbaatar, V. Derbov, A. Galtbayar, A. Gusev, M. Kaschiev, S. Vinitsky, and T. Zhanlav, “Explicit Magnus expansions for solving the time-dependent Schrödinger equation,” J. Phys. A: Math. Theor. 41(29), 295203 (2008).
[Crossref]

Hairer, E.

E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems (Springer-Verlag, 1987).

Hari, S.

Kamalian, M.

Karpman, V. I.

V. I. Karpman, Non-Linear Waves in Dispersive Media: International Series of Monographs in Natural Philosophy, vol. 71 (Elsevier, 2016).

Kaschiev, M.

O. Chuluunbaatar, V. Derbov, A. Galtbayar, A. Gusev, M. Kaschiev, S. Vinitsky, and T. Zhanlav, “Explicit Magnus expansions for solving the time-dependent Schrödinger equation,” J. Phys. A: Math. Theor. 41(29), 295203 (2008).
[Crossref]

Kschischang, F. R.

S. Hari and F. R. Kschischang, “Bi-directional algorithm for computing discrete spectral amplitudes in the NFT,” J. Lightwave Technol. 34(15), 3529–3537 (2016).
[Crossref]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, part III: Spectrum modulation,” IEEE Trans. Inf. Theory 60(7), 4346–4369 (2014).
[Crossref]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, part II: Numerical methods,” IEEE Trans. Inf. Theory 60(7), 4329–4345 (2014).
[Crossref]

Lau, A. P. T.

Le, S. T.

Lu, C.

Magnus, W.

W. Magnus, “On the exponential solution of differential equations for a linear operator,” Comm. Pure Appl. Math. 7(4), 649–673 (1954).
[Crossref]

Medvedev, S.

S. Medvedev, I. Vaseva, I. Chekhovskoy, and M. Fedoruk, “Numerical algorithm with fourth-order accuracy for the direct Zakharov-Shabat problem,” Opt. Lett. 44(9), 2264 (2019).
[Crossref]

S. Medvedev, I. Chekhovskoy, I. Vaseva, and M. Fedoruk, “Fast computation of the direct scattering transform by fourth order conservative multi-exponential scheme,” arXiv preprint arXiv:1909.13228 (2019).

S. Medvedev, I. Vaseva, I. Chekhovskoy, and M. Fedoruk, “Exponential fourth order schemes for direct Zakharov-Shabat problem,” arXiv preprint arXiv:1908.11725 (2019).

Mullyadzhanov, R.

Nørsett, S. P.

E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems (Springer-Verlag, 1987).

Osborne, A.

G. Boffetta and A. Osborne, “Computation of the direct scattering transform for the nonlinear Schroedinger equation,” J. Comput. Phys. 102(2), 252–264 (1992).
[Crossref]

Oteo, J.

S. Blanes, F. Casas, J. Oteo, and J. Ros, “The Magnus expansion and some of its applications,” Phys. Rep. 470(5-6), 151–238 (2009).
[Crossref]

Podivilov, E. V.

Poor, H. V.

S. Wahls and H. V. Poor, “Fast inverse nonlinear Fourier transform for generating multi-solitons in optical fiber,” in International Symposium on Information Theory (ISIT), (IEEE, Hong Kong, 2015), pp. 1676–1680.

S. Wahls and H. V. Poor, “Introducing the fast nonlinear Fourier transform,” in International Conference on Acoustics, Speech and Signal Processing, (IEEE, Vancouver, 2013), pp. 5780–5784.

Prilepsky, J.

A. Vasylchenkova, J. Prilepsky, D. Shepelsky, and A. Chattopadhyay, “Direct nonlinear Fourier transform algorithms for the computation of solitonic spectra in focusing nonlinear Schrödinger equation,” Commun. Nonlinear Sci. Numer. Simul. 68, 347–371 (2019).
[Crossref]

Prilepsky, J. E.

Prins, P. J

S. Wahls, S. Chimmalgi, and P. J Prins, “FNFT: a software library for computing nonlinear Fourier transforms,” J. Open Source Softw. 3(23), 597 (2018).
[Crossref]

Prins, P. J.

S. Chimmalgi, P. J. Prins, and S. Wahls, “Fast nonlinear Fourier transform algorithms using higher order exponential integrators,” IEEE Access 7, 145161 (2019).
[Crossref]

P. J. Prins and S. Wahls, “Higher order exponential splittings for the fast non-linear Fourier transform of the Korteweg-De Vries equation,” in ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings, (IEEE, 2018), 4, pp. 4524–4528.

Puzynin, I.

I. Puzynin, A. Selin, and S. Vinitsky, “Magnus-factorized method for numerical solving the time-dependent Schrödinger equation,” Comput. Phys. Commun. 126(1-2), 158–161 (2000).
[Crossref]

I. Puzynin, A. Selin, and S. Vinitsky, “A high-order accuracy method for numerical solving of the time-dependent Schrödinger equation,” Comput. Phys. Commun. 123(1-3), 1–6 (1999).
[Crossref]

Redyuk, A. A.

Ros, J.

S. Blanes, F. Casas, J. Oteo, and J. Ros, “The Magnus expansion and some of its applications,” Phys. Rep. 470(5-6), 151–238 (2009).
[Crossref]

Secondini, M.

Sedov, E. V.

Segur, H.

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (Society for Industrial and Applied Mathematics, 1981).

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, vol. 4 (Siam, 1981).

Selin, A.

I. Puzynin, A. Selin, and S. Vinitsky, “Magnus-factorized method for numerical solving the time-dependent Schrödinger equation,” Comput. Phys. Commun. 126(1-2), 158–161 (2000).
[Crossref]

I. Puzynin, A. Selin, and S. Vinitsky, “A high-order accuracy method for numerical solving of the time-dependent Schrödinger equation,” Comput. Phys. Commun. 123(1-3), 1–6 (1999).
[Crossref]

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of wave in nonlinear media,” J. Exp. Theor. Phys. 34, 62–69 (1972).

Shapiro, D. A.

Shepelsky, D.

A. Vasylchenkova, J. Prilepsky, D. Shepelsky, and A. Chattopadhyay, “Direct nonlinear Fourier transform algorithms for the computation of solitonic spectra in focusing nonlinear Schrödinger equation,” Commun. Nonlinear Sci. Numer. Simul. 68, 347–371 (2019).
[Crossref]

Thalhammer, M.

S. Blanes, F. Casas, and M. Thalhammer, “High-order commutator-free quasi-Magnus exponential integrators for non-autonomous linear evolution equations,” Comput. Phys. Commun. 220, 243–262 (2017).
[Crossref]

Timofeyev, I.

S. Burtsev, R. Camassa, and I. Timofeyev, “Numerical algorithms for the direct spectral transform with applications to nonlinear Schrödinger type systems,” J. Comput. Phys. 147(1), 166–186 (1998).
[Crossref]

Turitsyn, S. K.

Uhlig, F.

G. Engeln-Mullges and F. Uhlig, Numerical Algorithms with C (Springer-Verlag, 1996).

Vaibhav, V.

V. Vaibhav, “Efficient nonlinear Fourier transform algorithms of order four on equispaced grid,” IEEE Photonics Technol. Lett. 31(15), 1269–1272 (2019).
[Crossref]

V. Vaibhav, “Higher order convergent fast nonlinear Fourier transform,” IEEE Photonics Technol. Lett. 30(8), 700–703 (2018).
[Crossref]

S. Wahls and V. Vaibhav, “Fast inverse nonlinear Fourier transforms for continuous spectra of Zakharov-Shabat type,” arXiv preprint arXiv:1607.01305 pp. 1–15 (2016).

Vaseva, I.

S. Medvedev, I. Vaseva, I. Chekhovskoy, and M. Fedoruk, “Numerical algorithm with fourth-order accuracy for the direct Zakharov-Shabat problem,” Opt. Lett. 44(9), 2264 (2019).
[Crossref]

S. Medvedev, I. Chekhovskoy, I. Vaseva, and M. Fedoruk, “Fast computation of the direct scattering transform by fourth order conservative multi-exponential scheme,” arXiv preprint arXiv:1909.13228 (2019).

S. Medvedev, I. Vaseva, I. Chekhovskoy, and M. Fedoruk, “Exponential fourth order schemes for direct Zakharov-Shabat problem,” arXiv preprint arXiv:1908.11725 (2019).

Vasylchenkova, A.

A. Vasylchenkova, J. Prilepsky, D. Shepelsky, and A. Chattopadhyay, “Direct nonlinear Fourier transform algorithms for the computation of solitonic spectra in focusing nonlinear Schrödinger equation,” Commun. Nonlinear Sci. Numer. Simul. 68, 347–371 (2019).
[Crossref]

Vinitsky, S.

O. Chuluunbaatar, V. Derbov, A. Galtbayar, A. Gusev, M. Kaschiev, S. Vinitsky, and T. Zhanlav, “Explicit Magnus expansions for solving the time-dependent Schrödinger equation,” J. Phys. A: Math. Theor. 41(29), 295203 (2008).
[Crossref]

I. Puzynin, A. Selin, and S. Vinitsky, “Magnus-factorized method for numerical solving the time-dependent Schrödinger equation,” Comput. Phys. Commun. 126(1-2), 158–161 (2000).
[Crossref]

I. Puzynin, A. Selin, and S. Vinitsky, “A high-order accuracy method for numerical solving of the time-dependent Schrödinger equation,” Comput. Phys. Commun. 123(1-3), 1–6 (1999).
[Crossref]

Wahls, S.

S. Chimmalgi, P. J. Prins, and S. Wahls, “Fast nonlinear Fourier transform algorithms using higher order exponential integrators,” IEEE Access 7, 145161 (2019).
[Crossref]

S. Wahls, S. Chimmalgi, and P. J Prins, “FNFT: a software library for computing nonlinear Fourier transforms,” J. Open Source Softw. 3(23), 597 (2018).
[Crossref]

T. Gui, G. Zhou, C. Lu, A. P. T. Lau, and S. Wahls, “Nonlinear frequency division multiplexing with b-modulation: shifting the energy barrier,” Opt. Express 26(21), 27978 (2018).
[Crossref]

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).
[Crossref]

P. J. Prins and S. Wahls, “Higher order exponential splittings for the fast non-linear Fourier transform of the Korteweg-De Vries equation,” in ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings, (IEEE, 2018), 4, pp. 4524–4528.

S. Wahls, “Generation of time-limited signals in the nonlinear Fourier domain via b-modulation,” in 2017 European Conference on Optical Communication (ECOC), (IEEE, 2017), 6, pp. 1–3.

S. Wahls and H. V. Poor, “Introducing the fast nonlinear Fourier transform,” in International Conference on Acoustics, Speech and Signal Processing, (IEEE, Vancouver, 2013), pp. 5780–5784.

S. Wahls and H. V. Poor, “Fast inverse nonlinear Fourier transform for generating multi-solitons in optical fiber,” in International Symposium on Information Theory (ISIT), (IEEE, Hong Kong, 2015), pp. 1676–1680.

S. Wahls and V. Vaibhav, “Fast inverse nonlinear Fourier transforms for continuous spectra of Zakharov-Shabat type,” arXiv preprint arXiv:1607.01305 pp. 1–15 (2016).

Wai, P. K. A.

Wanner, G.

E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems (Springer-Verlag, 1987).

Wilcox, R. M.

R. M. Wilcox, “Exponential operators and parameter differentiation in quantum physics,” J. Math. Phys. 8(4), 962–982 (1967).
[Crossref]

Yousefi, M. I.

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, part II: Numerical methods,” IEEE Trans. Inf. Theory 60(7), 4329–4345 (2014).
[Crossref]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, part III: Spectrum modulation,” IEEE Trans. Inf. Theory 60(7), 4346–4369 (2014).
[Crossref]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of wave in nonlinear media,” J. Exp. Theor. Phys. 34, 62–69 (1972).

Zhanlav, T.

O. Chuluunbaatar, V. Derbov, A. Galtbayar, A. Gusev, M. Kaschiev, S. Vinitsky, and T. Zhanlav, “Explicit Magnus expansions for solving the time-dependent Schrödinger equation,” J. Phys. A: Math. Theor. 41(29), 295203 (2008).
[Crossref]

Zhou, G.

BIT (1)

G. G. Dahlquist, “A special stability problem for linear multistep methods,” BIT 3(1), 27–43 (1963).
[Crossref]

Comm. Pure Appl. Math. (1)

W. Magnus, “On the exponential solution of differential equations for a linear operator,” Comm. Pure Appl. Math. 7(4), 649–673 (1954).
[Crossref]

Commun. Nonlinear Sci. Numer. Simul. (1)

A. Vasylchenkova, J. Prilepsky, D. Shepelsky, and A. Chattopadhyay, “Direct nonlinear Fourier transform algorithms for the computation of solitonic spectra in focusing nonlinear Schrödinger equation,” Commun. Nonlinear Sci. Numer. Simul. 68, 347–371 (2019).
[Crossref]

Comput. Phys. Commun. (3)

S. Blanes, F. Casas, and M. Thalhammer, “High-order commutator-free quasi-Magnus exponential integrators for non-autonomous linear evolution equations,” Comput. Phys. Commun. 220, 243–262 (2017).
[Crossref]

I. Puzynin, A. Selin, and S. Vinitsky, “A high-order accuracy method for numerical solving of the time-dependent Schrödinger equation,” Comput. Phys. Commun. 123(1-3), 1–6 (1999).
[Crossref]

I. Puzynin, A. Selin, and S. Vinitsky, “Magnus-factorized method for numerical solving the time-dependent Schrödinger equation,” Comput. Phys. Commun. 126(1-2), 158–161 (2000).
[Crossref]

IEEE Access (1)

S. Chimmalgi, P. J. Prins, and S. Wahls, “Fast nonlinear Fourier transform algorithms using higher order exponential integrators,” IEEE Access 7, 145161 (2019).
[Crossref]

IEEE Photonics Technol. Lett. (2)

V. Vaibhav, “Higher order convergent fast nonlinear Fourier transform,” IEEE Photonics Technol. Lett. 30(8), 700–703 (2018).
[Crossref]

V. Vaibhav, “Efficient nonlinear Fourier transform algorithms of order four on equispaced grid,” IEEE Photonics Technol. Lett. 31(15), 1269–1272 (2019).
[Crossref]

IEEE Trans. Inf. Theory (2)

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, part II: Numerical methods,” IEEE Trans. Inf. Theory 60(7), 4329–4345 (2014).
[Crossref]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, part III: Spectrum modulation,” IEEE Trans. Inf. Theory 60(7), 4346–4369 (2014).
[Crossref]

Inverse Probl. (1)

F. A. Grunbaum, “The scattering problem for a phase-modulated hyperbolic secant pulse,” Inverse Probl. 5(3), 287–292 (1989).
[Crossref]

J. Comput. Phys. (2)

G. Boffetta and A. Osborne, “Computation of the direct scattering transform for the nonlinear Schroedinger equation,” J. Comput. Phys. 102(2), 252–264 (1992).
[Crossref]

S. Burtsev, R. Camassa, and I. Timofeyev, “Numerical algorithms for the direct spectral transform with applications to nonlinear Schrödinger type systems,” J. Comput. Phys. 147(1), 166–186 (1998).
[Crossref]

J. Exp. Theor. Phys. (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of wave in nonlinear media,” J. Exp. Theor. Phys. 34, 62–69 (1972).

J. Lightwave Technol. (1)

J. Math. Phys. (1)

R. M. Wilcox, “Exponential operators and parameter differentiation in quantum physics,” J. Math. Phys. 8(4), 962–982 (1967).
[Crossref]

J. Open Source Softw. (1)

S. Wahls, S. Chimmalgi, and P. J Prins, “FNFT: a software library for computing nonlinear Fourier transforms,” J. Open Source Softw. 3(23), 597 (2018).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Phys. A: Math. Theor. (1)

O. Chuluunbaatar, V. Derbov, A. Galtbayar, A. Gusev, M. Kaschiev, S. Vinitsky, and T. Zhanlav, “Explicit Magnus expansions for solving the time-dependent Schrödinger equation,” J. Phys. A: Math. Theor. 41(29), 295203 (2008).
[Crossref]

Opt. Express (4)

Opt. Lett. (3)

Optica (1)

Phys. Rep. (1)

S. Blanes, F. Casas, J. Oteo, and J. Ros, “The Magnus expansion and some of its applications,” Phys. Rep. 470(5-6), 151–238 (2009).
[Crossref]

Other (14)

E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems (Springer-Verlag, 1987).

G. A. Baker Jr and P. Graves-Morris, Padé Approximants (Cambridge University, 1996), 2nd ed.

S. Wahls and V. Vaibhav, “Fast inverse nonlinear Fourier transforms for continuous spectra of Zakharov-Shabat type,” arXiv preprint arXiv:1607.01305 pp. 1–15 (2016).

S. Medvedev, I. Vaseva, I. Chekhovskoy, and M. Fedoruk, “Exponential fourth order schemes for direct Zakharov-Shabat problem,” arXiv preprint arXiv:1908.11725 (2019).

G. Engeln-Mullges and F. Uhlig, Numerical Algorithms with C (Springer-Verlag, 1996).

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, vol. 4 (Siam, 1981).

S. Wahls, “Generation of time-limited signals in the nonlinear Fourier domain via b-modulation,” in 2017 European Conference on Optical Communication (ECOC), (IEEE, 2017), 6, pp. 1–3.

P. J. Prins and S. Wahls, “Higher order exponential splittings for the fast non-linear Fourier transform of the Korteweg-De Vries equation,” in ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings, (IEEE, 2018), 4, pp. 4524–4528.

S. Medvedev, I. Chekhovskoy, I. Vaseva, and M. Fedoruk, “Fast computation of the direct scattering transform by fourth order conservative multi-exponential scheme,” arXiv preprint arXiv:1909.13228 (2019).

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (Society for Industrial and Applied Mathematics, 1981).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2013), 4th ed.

V. I. Karpman, Non-Linear Waves in Dispersive Media: International Series of Monographs in Natural Philosophy, vol. 71 (Elsevier, 2016).

S. Wahls and H. V. Poor, “Introducing the fast nonlinear Fourier transform,” in International Conference on Acoustics, Speech and Signal Processing, (IEEE, Vancouver, 2013), pp. 5780–5784.

S. Wahls and H. V. Poor, “Fast inverse nonlinear Fourier transform for generating multi-solitons in optical fiber,” in International Symposium on Information Theory (ISIT), (IEEE, Hong Kong, 2015), pp. 1676–1680.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1.
Fig. 1. The approximation order with respect to the spectral parameter $\xi$ .
Fig. 2.
Fig. 2. Continuous spectrum normalized mean squared errors Eq. (69) of $a(\xi )$ and $b(\xi )$ .
Fig. 3.
Fig. 3. (a, c, e) Errors Eq. (68) of continuous spectrum energy. (b, d, f) Absolute errors of quadratic invariants.
Fig. 4.
Fig. 4. Discrete spectrum errors for the maximum eigenvalue $\zeta _0$ .
Fig. 5.
Fig. 5. The errors Eq. (68) of phase coefficients for the maximum eigenvalue $\zeta _0$ Eq. (61).

Equations (92)

Equations on this page are rendered with MathJax. Learn more.

i q z + σ 2 2 q t 2 + | q | 2 q = 0 ,
q ( t , z ) | z = z 0 = q 0 ( t ) .
L z = M L L M
L Ψ = ζ Ψ , Ψ z = M Ψ ,
L = i ( t q σ q t ) , M = i ( σ 2 t 2 + 1 2 | q | 2 σ q t 1 2 σ q t q t 1 2 q t σ 2 t 2 1 2 | q | 2 ) .
d Ψ ( t ) d t = Q ( t ) Ψ ( t ) ,
Ψ ( t ) = ( ψ 1 ( t ) ψ 2 ( t ) ) , Q ( t ) = ( i ζ q σ q i ζ ) .
( ψ 1 ψ 2 ) t = J ( ψ 1 σ ψ 2 ) = J ( H ψ 1 H ψ 2 ) , J = ( i ζ σ q σ q i σ ζ )
H = { ( Ψ , σ 0 Ψ ) , for  σ = 1 ( Ψ , σ 3 Ψ ) , for  σ = 1 , Q = { J σ 0 , for  σ = 1 J σ 3 , for  σ = 1 ,
Ψ = ( ψ 1 ψ 2 ) = ( e i ζ t 0 ) [ 1 + o ( 1 ) ] , t ,
Φ = ( ϕ 1 ϕ 2 ) = ( 0 e i ζ t ) [ 1 + o ( 1 ) ] , t ,
a ( ξ ) = lim t ψ 1 ( t , ξ ) e i ξ t , b ( ξ ) = lim t ψ 2 ( t , ξ ) e i ξ t .
r k = b ( ζ ) a ( ζ ) | ζ = ζ k , where a ( ζ ) = d a ( ζ ) d ζ ;
Ψ ( t , ζ k ) = Φ ( t , ζ k ) b ( ζ k ) .
C n = 1 π ( 2 i ξ ) n ln | a ( ξ ) | 2 d ξ + k = 0 K 1 1 n + 1 [ ( 2 i ζ k ) n + 1 ( 2 i ζ k ) n + 1 ] ,
C 0 = | q | 2 d t , C 1 = q q t d t , C 2 = ( q q t t + | q | 4 ) d t , C 3 = ( q q t t t + 4 | q | 2 q q t + | q | 2 q q t ) d t .
C 0 = 1 π ln | a ( ξ ) | 2 d ξ + k = 0 K 1 [ 2 i ( ζ k ζ k ) ]
E c = 1 π ln | a ( ξ ) | 2 d ξ .
D x = Q ( t ) x , D = d d t ,
x n + 1 = T x n ,
D x = Q x , D 2 x = ( D Q ) x + Q D x , D 3 x = ( D 2 Q ) x + 2 ( D Q ) ( D x ) + Q D 2 x , D 4 x = ( D 3 Q ) x + 3 ( D 2 Q ) ( D x ) + 3 ( D Q ) ( D 2 x ) + Q D 3 x , D 5 x = ( D 4 Q ) x + 4 ( D 3 Q ) ( D x ) + 6 ( D 2 Q ) ( D 2 x ) + 4 ( D Q ) ( D 3 x ) + Q D 4 x .
D k x = Q k x , Q 1 = Q .
Q 2 = Q ( 1 ) + Q 2 , Q 3 = Q ( 2 ) + 2 Q ( 1 ) Q + Q Q ( 1 ) + Q 3 , Q 4 = Q ( 3 ) + 3 Q ( 2 ) Q + Q Q ( 2 ) + 3 ( Q ( 1 ) ) 2 + 3 Q ( 1 ) Q 2 + 2 Q Q ( 1 ) Q + Q 2 Q ( 1 ) + Q 4 , Q 5 = Q ( 4 ) + 4 Q ( 3 ) Q + Q Q ( 3 ) + 6 Q ( 2 ) Q ( 1 ) + 4 Q ( 1 ) Q ( 2 ) + 6 Q ( 2 ) Q 2 + 3 Q Q ( 2 ) Q + Q 2 Q ( 2 ) + + 8 ( Q ( 1 ) ) 2 Q + 4 Q ( 1 ) Q Q ( 1 ) + 3 Q ( Q ( 1 ) ) 2 + 4 Q ( 1 ) Q 3 + 3 Q Q ( 1 ) Q 2 + 2 Q 2 Q ( 1 ) Q + Q 3 Q ( 1 ) + Q 5 .
x ( t n + 1 ) = x + s τ D x + ( s τ ) 2 2 ! D 2 x + ( s τ ) 3 3 ! D 3 x + ( s τ ) 4 4 ! D 4 x + ( s τ ) 5 5 ! D 5 x + O ( τ 6 ) ,
x ( t n ) = x + s ¯ τ D x + ( s ¯ τ ) 2 2 ! D 2 x + ( s ¯ τ ) 3 3 ! D 3 x + + ( s ¯ τ ) 4 4 ! D 4 x + ( s ¯ τ ) 5 5 ! D 5 x + O ( τ 6 ) .
L k = s k k ! Q k , R k = s ¯ k k ! Q k
( E + L 1 + L 2 + L 3 + L 4 + L 5 ) = ( T 0 + T 1 + T 2 + T 3 + T 4 + T 5 ) ( E + R 1 + R 2 + R 3 + R 4 + R 5 ) ,
L 1 = R 1 + T 1 ,
L 2 = R 2 + T 1 R 1 + T 2 ,
L 3 = R 3 + T 1 R 2 + T 2 R 1 + T 3 ,
L 4 = R 4 + T 1 R 3 + T 2 R 2 + T 3 R 1 + T 4 ,
L 5 = R 5 + T 1 R 4 + T 2 R 3 + T 3 R 2 + T 4 R 1 + T 5 .
T 1 = L 1 R 1 = s Q s ¯ Q = Q .
T 2 = L 2 R 2 T 1 R 1 = s 2 s ¯ 2 2 ! Q 2 s ¯ Q 2 = 2 s 1 2 ! Q 2 s ¯ Q 2 .
T 3 = L 3 R 3 T 1 R 2 T 2 R 1 = s 3 s ¯ 3 3 ! Q 3 Q s ¯ 2 2 ! Q 2 s ¯ T 2 Q = = 3 s 2 3 s + 1 3 ! Q 3 s ¯ 2 2 ! Q Q 2 ( 2 s 1 ) s ¯ 2 ! Q 2 Q + s ¯ 2 Q 3 .
s 3 s ¯ 3 = 3 s 2 3 s + 1 = 0
T 4 = L 4 R 4 T 1 R 3 T 2 R 2 T 3 R 1 = s 4 s ¯ 4 4 ! Q 4 s ¯ 3 3 ! Q Q 3 s ¯ 2 2 ! T 2 Q 2 s ¯ T 3 Q .
s 4 s ¯ 4 = ( 2 s 1 ) ( 2 s 2 2 s + 1 ) = 0
T 5 = L 5 R 5 T 1 R 4 T 2 R 3 T 3 R 2 T 4 R 1 = s 5 s ¯ 5 5 ! Q 4 + .
s 5 s ¯ 5 = 0
T = E + τ Q + τ 2 T 2 + τ 3 T 3 + τ 4 T 4 + O ( τ 5 ) ,
T 2 = 2 s 1 2 ! Q 2 s ¯ Q 2 ,
T 3 = 3 s 2 3 s + 1 3 ! Q 3 s ¯ 2 2 ! Q Q 2 ( 2 s 1 ) s ¯ 2 ! Q 2 Q + s ¯ 2 Q 3 ,
T 4 = ( 2 s 1 ) ( 2 s 2 2 s + 1 ) 4 ! Q 4 s ¯ 3 3 ! Q Q 3 s ¯ 2 2 ! T 2 Q 2 s ¯ T 3 Q
Q 2 = Q ( 1 ) + Q 2 ,
Q 3 = Q ( 2 ) + 2 Q ( 1 ) Q + Q Q ( 1 ) + Q 3 ,
Q 4 = Q ( 3 ) + 3 Q ( 2 ) Q + Q Q ( 2 ) + 3 ( Q ( 1 ) ) 2 + 3 Q ( 1 ) Q 2 + 2 Q Q ( 1 ) Q + Q 2 Q ( 1 ) + Q 4 .
T = E + τ Q + τ 2 2 ! Q 2 + τ 3 3 ! Q 3 + τ 4 4 ! Q 4 + O ( τ 5 ) .
T = E + τ Q + 1 2 τ 2 Q 2 + τ 3 3 ! Q 3 + τ 3 24 Q ( 2 ) + τ 3 12 ( Q ( 1 ) Q Q Q ( 1 ) ) + + τ 4 4 ! Q 4 + τ 4 48 ( Q Q ( 2 ) + Q ( 2 ) Q ) + τ 4 24 ( Q ( 1 ) Q 2 Q 2 Q ( 1 ) ) .
Q n + 1 2 ( 1 ) = Q n + 3 2 Q n 1 2 2 τ + O ( τ 2 ) , Q n + 1 2 ( 2 ) = Q n + 3 2 2 Q n + 1 2 + Q n 1 2 τ 2 + O ( τ 2 ) .
T n + 1 2 = E + τ Q n + 1 2 + τ 2 2 Q n + 1 2 2 + τ 3 3 ! Q n + 1 2 3 + τ 4 4 ! Q n + 1 2 4 + + τ 3 12 ( Q n + 1 2 ( 1 ) Q n + 1 2 Q n + 1 2 Q n + 1 2 ( 1 ) ) + τ 3 24 Q n + 1 2 ( 2 ) + + τ 4 48 ( Q n + 1 2 Q n + 1 2 ( 2 ) + Q n + 1 2 ( 2 ) Q n + 1 2 ) + τ 4 24 ( Q n + 1 2 ( 1 ) Q n + 1 2 2 Q n + 1 2 2 Q n + 1 2 ( 1 ) ) + O ( τ 5 ) .
T = exp { τ F 1 + τ 3 F 3 } + O ( τ 5 ) ,
F 1 = Q , F 3 = 1 24 Q ( 2 ) + 1 12 ( Q ( 1 ) Q Q Q ( 1 ) ) , F 2 = F 4 = 0.
Ψ ( t ) = U ( t , 0 ) Ψ ( 0 ) , U ( t , 0 ) = e Ω ( t ) ,
Ω 1 ( t ) = 0 t d t 1 Q ( t 1 ) , Ω 2 ( t ) = 1 2 0 t d t 1 0 t 1 d t 2 [ Q ( t 1 ) , Q ( t 2 ) ] ,
Ω 3 ( t ) = 1 6 0 t d t 1 0 t 1 d t 2 0 t 3 d t 3 ( [ Q ( t 1 ) , [ Q ( t 2 ) , Q ( t 3 ) ] ] + [ Q ( t 3 ) , [ Q ( t 2 ) , Q ( t 1 ) ] ] ) .
Ω 1 = t Q + t 3 24 Q ( 2 ) + O ( t 5 ) , Ω 2 = t 3 12 [ Q ( 1 ) Q Q Q ( 1 ) ] + O ( t 5 ) , Ω 3 = O ( t 5 ) .
T = e τ Q + τ 2 12 [ Q ( 1 ) e τ Q e τ Q Q ( 1 ) ] + τ 3 48 [ e τ Q Q ( 2 ) + Q ( 2 ) e τ Q ] + O ( τ 5 ) .
T = exp { τ 2 12 Q ( 1 ) + τ 3 48 Q ( 2 ) } exp { τ Q } exp { τ 2 12 Q ( 1 ) + τ 3 48 Q ( 2 ) } ,
Ψ n + 1 2 = T Ψ n 1 2 .
T = e τ 2 Q n [ I τ 48 ( M n + 1 + M n 1 ) ] 1 [ I + τ 48 ( M n + 1 + M n 1 ) ] e τ 2 Q n
M n + 1 = e τ Q n ( Q n + 1 Q n ) e τ Q n , M n 1 = e τ Q n ( Q n 1 Q n ) e τ Q n .
a ( ζ ) = ψ 1 ( L τ / 2 , ζ ) e i ζ ( L τ / 2 ) , b ( ζ ) = ψ 2 ( L τ / 2 , ζ ) e i ζ ( L τ / 2 ) .
d a d ζ = d ψ 1 d ζ e i ζ ( L τ / 2 ) + i ( L τ / 2 ) a ( ζ ) .
d d ζ Ψ n + 1 2 = T ζ Ψ n 1 2 + T d d ζ Ψ n 1 2 ,
d d ζ Ψ ( L τ / 2 , ζ ) = ( i ( L τ / 2 ) ψ 1 ( L τ / 2 , ζ ) 0 ) .
T (ES4) = e a 0 ( c + s a 3 + s a 3 s a 1 + s a 1 i s a 2 i s a 2 s a 1 + s a 1 + i s a 2 + i s a 2 c s a 3 s a 3 ) ,
c = cos ( ω ) , s = sin ( ω ) ω , ω = a 1 2 a 2 2 a 3 2 , c = sin ( ω ) ω , s = ω ω ( c s ) , ω = 1 ω ( a 1 a 1 + a 2 a 2 + a 3 a 3 ) , a 1 = i τ 2 24 ( d 12 d 21 ) , a 2 = τ 2 24 ( d 12 + d 21 ) , a 3 = i τ , d 12 = q n + 1 q n 1 , d 21 = σ ( q n + 1 q n 1 ) .
T (TES4) = exp { τ 2 12 Q ( 1 ) + τ 3 48 Q ( 2 ) } D exp { τ 2 12 Q ( 1 ) + τ 3 48 Q ( 2 ) } ,
( e τ Q n ) = τ ζ ω n sin ( ω n τ ) I + ζ ω n 3 [ τ ω n cos ( ω n τ ) sin ( ω n τ ) ] Q n i sin ( ω n τ ) ω n σ 3 ,
T (CT4) = ( A 1 B ) = A 1 Q A 1 A A 1 B ,
A = [ I τ 48 ( M n + 1 + M n 1 ) ] e τ 2 Q n = A 1 e τ 2 Q n ,
B = [ I + τ 48 ( M n + 1 + M n 1 ) ] e 3 τ 2 Q n = B 1 e 3 τ 2 .
A = τ 48 ( ( e 2 τ Q n ) ( Q n 1 Q n ) e 2 τ Q n + e 2 τ Q n ( Q n 1 Q n ) ( e 2 τ Q n ) ) e τ 2 Q n + A 1 ( e τ 2 Q n ) .
B = τ 48 ( ( e 2 τ Q n ) ( Q n 1 Q n ) e 2 τ Q n + e 2 τ Q n ( Q n 1 Q n ) ( e 2 τ Q n ) ) e 3 τ 2 Q n + B 1 ( e 3 τ 2 Q n ) .
q ( t ) = A [ sech ( t ) ] 1 + i C .
a ( ξ ) = Γ [ 1 / 2 i ( ξ + C / 2 ) ] Γ [ 1 / 2 i ( ξ C / 2 ) ] Γ [ 1 / 2 i ξ D ] Γ [ 1 / 2 i ξ + D ] , b ( ξ ) = 1 2 i C A Γ [ 1 / 2 i ( ξ + C / 2 ) ] Γ [ 1 / 2 + i ( ξ C / 2 ) ] Γ [ i C / 2 D ] Γ [ i C / 2 + D ] , D = σ A 2 C 2 / 4 .
ζ k = i ( A 2 C 2 / 4 1 / 2 k ) , k = 0 , , [ A 2 C 2 / 4 1 / 2 ] ,
a ( ζ ) = f ( ζ ) Γ [ 1 / 2 i ζ D ] , where f ( ζ ) = Γ [ 1 / 2 i ( ζ + C / 2 ) ] Γ [ 1 / 2 i ( ζ C / 2 ) ] Γ [ 1 / 2 i ζ + D ] .
a ( ζ ) | ζ = ζ k = i f ( ζ k ) φ k ,
φ k + 1 = ( k + 1 ) φ k , φ 0 = 1.
r k = b ( ζ ) a ( ζ ) | ζ = ζ k = b ( ζ ) f ( ζ ) | ζ = ζ k i φ k .
E d = 4 k = 0 K 1 η k = 2 ( K + δ 1 / 2 ) 2 2 ( δ 1 / 2 ) 2 .
E c = E E d = 2 ( C 2 / 4 + ( δ 1 / 2 ) 2 ) .
m = log τ 1 τ 2 error τ 1 ( Ψ ) 2 error τ 2 ( Ψ ) 2 = log 2 error τ 1 ( Ψ ) 2 error τ 2 ( Ψ ) 2 log 2 ( τ 1 / τ 2 ) ,
error [ ϕ ] = | ϕ c o m p ϕ e x a c t | | ϕ 0 | , ϕ 0 = { ϕ e x a c t ,  if  | ϕ e x a c t | > 1 1 ,  otherwise ,
N M S E [ ϕ ] = 1 N j = 1 N | ϕ c o m p ( ξ j ) ϕ e x a c t ( ξ j ) | 2 | ϕ 0 ( ξ j ) | 2 , ϕ 0 = { ϕ e x a c t ( ξ j ) ,  if  | ϕ e x a c t ( ξ j ) | > 1 1 ,  otherwise ,
L ξ = π / ( 2 τ ) .
d d t ( Ψ , D Ψ ) = ( d Ψ d t , D Ψ ) + ( Ψ , D d Ψ d t ) = ( K D Ψ , D Ψ ) + ( Ψ , D K D Ψ ) = = ( Ψ , ( D ) T ( K ) T D Ψ ) + ( Ψ , D K D Ψ ) = ( Ψ , D ( K + K ) D Ψ ) = 0.
( Ψ n + 1 2 , D Ψ n + 1 2 ) = ( e τ R n D Ψ n 1 2 , D e τ R n D Ψ n 1 2 ) = ( e τ R n D Ψ n 1 2 , e τ D R n D Ψ n 1 2 ) = = ( Ψ n 1 2 , e τ R n D e τ D R n D Ψ n 1 2 ) = ( Ψ n 1 2 , D Ψ n 1 2 ) .
σ 0 = ( 1 0 0 1 ) , σ 1 = ( 0 1 1 0 ) , σ 2 = ( 0 i i 0 ) , σ 3 = ( 1 0 0 1 )
e A = e a 0 [ c σ 0 + i s ( a 1 σ 1 + a 2 σ 2 + a 3 σ 3 ) ] = e a 0 ( c + s a 3 s ( a 1 i a 2 ) s ( a 1 + i a 2 ) c s a 3 ) ,

Metrics