Algorithms for solving scattering problems for the Manakov model of nonlinear Schrödinger equations

We introduce numerical algorithms for solving the inverse and direct scattering problems for the Manakov model of vector nonlinear Schrödinger equation. We have found an algebraic group of 4-block matrices with off-diagonal blocks consisting of special vector-like matrices for generalizing the scalar problem’s efficient numerical algorithms to the vector case. The inversion of block matrices of the discretized system of Gelfand–Levitan–Marchenko integral equations solves the inverse scattering problem using the vector variant the Toeplitz Inner Bordering algorithm of Levinson’s type. The reversal of steps of the inverse problem algorithm gives the solution of the direct scattering problem. Numerical tests confirm the proposed vector algorithms’ efficiency and stability. We also present an example of the algorithms’ application to simulate the Manakov vector solitons’ collision.

The triple Wronskian solutions to the variable-coefficient Manakov model

In this paper, the variable-coefficient Manakov model whose bilinear form exists is mainly discussed. Based on the Wronskian technique, the triple Wronskian form solutions are obtained and the interactions between the two solitons are investigated.