Inverse spectral scattering transform algorithm for the Manakov system

A numerical algorithm for solving the inverse spectral scattering problem associated with Manakov's model of the vector nonlinear Schrödinger equation is described. This model of wave processes simultaneously takes into account dispersion, nonlinear and polarization effects. It is in demand in nonlinear problems of theoretical physics and physical optics, and is especially promising for describing the propagation of optical radiation along fiber communication lines. In the presented algorithm, the solution to the inverse scattering problem is based on inverting a set of nested matrices of a discretized system of Gelfand-Levitan-Marchenko integral equations using a block version of the Toeplitz algorithm of the Levinson type. Numerical tests carried out by comparing the calculations with known exact analytical solutions confirm the stability and second-order accuracy of the proposed algorithm. An example is given of the use of the algorithm to simulate the collision of a pair of differently polarized vector Manakov solitons.