Fast numerical method of the second order of accuracy for solving the inverse scattering problem

A method is proposed for the numerical solution of the inverse scattering problem formulated in the form of integral equations of the Gelfand–Levitan–Marchenko type. Approximation of integral equations is performed with the second order of accuracy, which leads to a system of equations with a non-Toeplitz matrix; however, the method requires O(N²) arithmetic operations. The method is applied to the problem of self-focusing of radiation with allowance for polarisation, which was solved by Manakov.