Efficient Nonlinear Fourier Transform Algorithms of Order Four on Equispaced Grid

Abstract

We explore two classes of exponential integrators in this letter to design nonlinear Fourier transform (NFT) algorithms with a desired accuracy-complexity trade-off and a convergence order of 4 on an equispaced grid. The integrating factor based method in the class of Runge-Kutta methods yield algorithms with complexity O(N log 2 N) (where N is the number of samples of the signal) which have superior accuracy-complexity trade-off than any of the fast methods known currently. The integrators based on Magnus series expansion, namely, standard and commutator-free Magnus methods yield algorithms of complexity O(N 2 ) that have superior error behavior even for moderately small step-sizes and higher signal strengths.

Simulation of nonuniform transmission lines

“…The first is solution in the spatial domain, x . For this purpose, the approach in Vaibhav (2019) and Blanes and Moan (2006) is used. It is a fourth-order algorithm based on an equispaced division of the x axis.…”

“…The method proposed by Blanes and Moan (2006) and used by Vaibhav (2019) involved a truncation of this series to obtain a fourth-order method. The first two terms in the Magnus series are: where A is as defined in equation (5).…”

“…Computing matrix exponentials is computationally expensive and hence, the structure of the Λ( x , x n ) matrix must be investigated for savings in computation as was done in Vaibhav (2019) and Blanes and Moan (2006).…”

A Chebyshev Spectral Method for Nonlinear Fourier Transform: Norming Constants

“…Such an algorithm would prove extremely useful for system design and benchmarking. Currently, there are primarily two successful approaches proposed in the literature for computing the continuous NF spectrum which are capable of achieving algebraic orders convergence at quasilinear complexity: (a) the integrating factor (IF) based exponential integrators [4]- [8] (b) exponential time differencing (ETD) method based exponential integrators [9]. Note that while the IF schemes uses fast polynomial arithmetic in the monomial basis, the ETD schemes use fast polynomial arithmetic in the Chebyshev basis.…”

“…The plateauing of the error seen in these plots are again on account of the lack of compact support of q(t). e θ [8] e δ [8] e θ [12] e δ [12] e θ [16] e δ [16]…”