A Super-Fast Algorithm for Solving the Direct Scattering Problem for the Manakov System

The paper presents an accelerated algorithm for solving the direct scattering problem for the continuous spectrum of the Manakov system associated with the vector nonlinear Schrödinger equation of the Manakov model. The numerical formulation requires fast calculation of the products of polynomials depending on the spectral parameter of the problem. For localized solutions, a so-called super-fast algorithm for solving the direct scattering problem of the second order of accuracy is presented, based on the convolution theorem and the fast Fourier transform; for a discrete grid of size N, it requires asymptotically \(O(N~{{\log }^{2}}N)\) arithmetic operations. To speed up the calculation of the spectra of reflection coefficients, a matrix version of the fast Fourier transform is proposed and tested, in which the coefficients of the discrete Fourier transform are non-commuting matrices. Numerical simulation using the example of the exact solution of the Manakov system (hyperbolic secant) confirmed the high speed of calculations and the second order of accuracy of the algorithm.