论文标题
计算Zakharov-Shabat系统特征值的有效方法
Efficient method for calculating the eigenvalue of the Zakharov-Shabat system
论文作者
论文摘要
在本文中，提出了一种数值方法来计算基于Chebyshev多项式的Zakharov-Shabat系统的特征值。根据Zakharov-Shabat特征值问题的潜在功能的渐近构建Tanh（AX）的映射。考虑到潜在功能的梯度，该映射可以很好地分布Chebyshev节点。使用Chebyshev多项式，TANH（AX）映射和Chebyshev节点，Zakharov-Shabat特征值问题被转化为矩阵特征值问题，然后通过QR算法解决。该方法对于Satsuma-yajima电位具有良好的收敛性，并且收敛速度比傅立叶搭配方法快。该方法不仅适用于简单的电位功能，而且还可以快速收敛以获得复杂的Y形电位。该方法也可以进一步扩展以解决其他线性特征值问题。
In this paper, a numerical method is proposed to calculate the eigenvalues of the Zakharov-Shabat system based on Chebyshev polynomials. A mapping in the form of tanh(ax) is constructed according to the asymptotic of the potential function for the Zakharov-Shabat eigenvalue problem. The mapping could distribute Chebyshev nodes very well considering the gradient for the potential function. Using Chebyshev polynomials,tanh(ax) mapping and Chebyshev nodes, the Zakharov-Shabat eigenvalue problem is transformed into a matrix eigenvalue problem, and then solved by the QR algorithm. This method has good convergence for Satsuma-Yajima potential, and the convergence speed is faster than the fourier collocation method. This method is not only suitable for simple potential functions, but also converges quickly for complex Y-shape potential. This method can also be further extended to solve other linear eigenvalue problems.
