Numerical algorithms for the direct spectral Zakharov–Shabat problem with application to the solution of nonlinear equations

The numerical implementation of the nonlinear Fourier transformation (NFT) for the nonlinear Shrödinger equation (NLSE) requires effective numerical algorithms for each stage of the method. The very first step in this scheme is the solution of the direct scattering problem for the Zakharov–Shabat system. One of the most efficient methods for the solution of this problem is the second-order Boffetta–Osborne algorithm [1].
In this report we propose a generalization of the Boffetta–Osborne method. A two-parametric family of one-step fourth-order difference schemes is constructed. It requires the potential values at three neighboring points and reduces to the Boffetta–Osborne scheme in case of constant potentials. The family contains a scheme (super-scheme) that preserves the integral of the system for the continuous spectrum.
[1] G.Boffetta and A.R.Osborne, Computation of the direct scattering transform for the nonlinear Schrödinger equation, J. Comput. Phys. 102(2), 252–264 (1992).