Fast Computation of the Direct Scattering Transform by Fourth Order Conservative Multi-Exponential Scheme

A fourth-order multi-exponential scheme is proposed for the Zakharov-Shabat system. The scheme represents a product of 13 exponential operators. The construction of the scheme is based on a fourth-order three-exponential scheme, which contains only one exponent with a spectral parameter. This exponent is factorized to the fourth-order with the Suzuki formula of 11 exponents. The obtained scheme allows the use of a fast algorithm in calculating the initial problem for a large number of spectral parameters and conserves the quadratic invariant exactly for real spectral parameters.

[1]  Frank R. Kschischang,et al.  Information Transmission Using the Nonlinear Fourier Transform, Part III: Spectrum Modulation , 2013, IEEE Transactions on Information Theory.

[2]  Vishal Vaibhav,et al.  Higher Order Convergent Fast Nonlinear Fourier Transform , 2017, IEEE Photonics Technology Letters.

[3]  R. H. Hardin.  Application of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations , 1973 .

[4]  H. Vincent Poor,et al.  Fast Numerical Nonlinear Fourier Transforms , 2014, IEEE Transactions on Information Theory.

[5]  V. Zakharov,et al.  Exact Theory of Two-dimensional Self-focusing and One-dimensional Self-modulation of Waves in Nonlinear Media , 1970 .

[6]  Sander Wahls,et al.  Higher Order Exponential Splittings for the Fast Non-Linear Fourier Transform of the Korteweg-De Vries Equation , 2018, 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[7]  Akira Hasegawa,et al.  Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion , 1973 .

[8]  Masuo Suzuki,et al.  General Nonsymmetric Higher-Order Decomposition of Exponential Operators and Symplectic Integrators , 1992 .

[9]  Sander Wahls,et al.  Generation of Time-Limited Signals in the Nonlinear Fourier Domain via b-Modulation , 2017, 2017 European Conference on Optical Communication (ECOC).

[10]  Sander Wahls,et al.  FNFT: A Software Library for Computing Nonlinear Fourier Transforms , 2018, J. Open Source Softw..

[11]  Sergei K. Turitsyn,et al.  Nonlinear Fourier Transform for Optical Data Processing and Transmission: Advances and Perspectives , 2017, 2018 European Conference on Optical Communication (ECOC).

[12]  H. Vincent Poor,et al.  Introducing the fast nonlinear Fourier transform , 2013, 2013 IEEE International Conference on Acoustics, Speech and Signal Processing.

[13]  S. Wahls,et al.  Fast Inverse Nonlinear Fourier Transforms for Continuous Spectra of Zakharov-Shabat Type , 2016 .

[14]  G. Dahlquist.  A special stability problem for linear multistep methods , 1963 .

[15]  A. Osborne,et al.  Computation of the direct scattering transform for the nonlinear Schroedinger equation , 1992 .

[16]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[17]  Sergey Medvedev,et al.  Exponential Fourth Order Schemes for Direct Zakharov-Shabat problem , 2019, Optics express.