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:在本文中, 提议了一种数值方法来计算Zakharov- Shabat 系统基于 Chebyshev 多元数值的精度值。 根据Zakharov- Shabat 元素值问题潜在函数的无规律性, 以 tanh( exax) 的形式构建了一种映射 。 映射可以很好地分配Chebyshev 节点值, 以潜在函数的梯度计算。 使用 Chebyshev 多元数值, tanh( ash) 映射和 Chebyshev 节点, Zakharov- Shabat 精度值问题被转化成一个矩阵的精度值问题, 然后由 QR 算法解决 。 这个方法对于 satsuma- Yajima 潜在函数来说具有良好的趋同性, 其趋同速度比 四倍合法更快。 这个方法不仅适合简单的潜在函数,, 而且对于复杂的Y shape 潜力也很快会聚合。 这个方法还可以进一步扩展到解决其他线形值问题 。</s>