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.

翻译：本文提出一种基于Chebyshev多项式的数值方法，用于计算Zakharov-Shabat系统的特征值。针对Zakharov-Shabat特征值问题中势函数的渐近性质，构造了形如tanh(ax)的映射。该映射能根据势函数的梯度合理分布Chebyshev节点。通过结合Chebyshev多项式、tanh(ax)映射及Chebyshev节点，将Zakharov-Shabat特征值问题转化为矩阵特征值问题，并采用QR算法求解。该方法对Satsuma-Yajima势具有良好收敛性，且收敛速度快于Fourier配点法。该方法不仅适用于简单势函数，对复杂Y型势同样能快速收敛，并可进一步推广求解其他线性特征值问题。