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.

中文翻译：

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求解非线性薛定谔方程的 Manakov 模型散射问题的算法

我们介绍了求解向量非线性薛定谔方程的 Manakov 模型的逆和直接散射问题的数值算法。我们发现了一个 4 块矩阵的代数群，其非对角块由特殊的类向量矩阵组成，用于将标量问题的有效数值算法推广到向量情况。Gelfand-Levitan-Marchenko 积分方程离散系统的块矩阵的求逆使用列文森类型的 Toeplitz 内边界算法的向量变体来解决逆散射问题。逆问题算法的步骤颠倒给出了直接散射问题的解决方案。数值测试证实了所提出的向量算法的效率和稳定性。