Efficient algorithms for solving the spectral scattering problems\\ for the Manakov system of nonlinear Schroedinger equations

TL;DR

This paper introduces efficient vectorial algorithms for solving spectral scattering problems related to the Manakov system, extending scalar methods to handle wave polarization in nonlinear Schrödinger equations.

Contribution

It develops a new algebraic framework using 4-block matrices to generalize scalar spectral algorithms to the vector case for the Manakov system.

Findings

Algorithms successfully recover known analytical solutions.

Numerical efficiency confirmed through testing.

Applicable to both focusing and defocusing cases.

Abstract

``Vectorial'' numerical algorithms are proposed for solving the inverse and direct spectral scattering problems for the nonlinear vector Schroedinger equation, taking into account wave polarization, known as the Manakov system. It is shown that a new algebraic group of 4-block matrices with off-diagonal blocks consisting of special vector-like matrices makes possible the generalization of numerical algorithms of the scalar problem to the vector case, both for the focusing and defocusing Manakov systems. As in the scalar case, the solution of the inverse scattering problem consists of inversion of matrices of the discretized system of Gelfand-Levitan-Marchenko integral equations using the Toeplitz Inner Bordering algorithm of Levinson's type. Also similar to the scalar case, the algorithm for solving the direct scattering problem obtained by inversion of steps of the algorithm for the inverse scattering problem. Testing of the vector algorithms performed by comparing the results of the calculations with the known exact analytical solution (the Manakov vector soliton) confirmed the numerical efficiency of the vector algorithms.