具有四阶精度的新型数值算法用于直接Zakharov-Shabat问题

Exponential Fourth Order Schemes for Direct Zakharov-Shabat problem

We propose two finite-difference algorithms of fourth order of accuracy for solving the initial problem of the Zakharov-Shabat system. Both schemes have the exponential form and conserve quadratic invariant of Zakharov-Shabat system. The second scheme contains the spectral parameter in exponent only and allows to apply the fast computational algorithm.

Fast sixth-order algorithm based on the generalized Cayley transform for the Zakharov-Shabat system in optical applications

Based on the generalized Cayley transform, a family of conservative one-step schemes of the sixth order of accuracy for the Zakharov-Shabat system is constructed. The exponential integrator is a special case. Schemes based on rational approximation allow the use of fast algorithms to solve the initial problem for a large number of values of the spectral parameter.

正散射问题的散射数据计算

Scattering data computation for the Zakharov-Shabat system

本文提出了一种数值方法，用于求解与非线性薛定谔方程初值问题相关的正散射问题，通过求解具有结构核的维尔斯特拉积分系统并识别多项式-指数和中的系数和参数，验证了方法的有效性。

Introducing phase jump tracking -- a fast method for eigenvalue evaluation of the direct Zakharov-Shabat problem

We propose a new method for finding discrete eigenvalues for the direct Zakharov-Shabat problem, based on moving in the complex plane along the argument jumps of the function $a(ζ)$, the localization of which does not require great accuracy. It allows to find all discrete eigenvalues taking into account their multiplicity faster than matrix methods and contour integrals. The method shows significant advantage over other methods when calculating a large discrete spectrum, both in speed and accuracy.

Efficient method for calculating the eigenvalue of the Zakharov-Shabat system

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 g…

基于切比雪夫多项式的快速八阶帕德格式，用于直接扎哈罗夫-沙巴特问题

Fast eight-order Pade schemes based on Chebyshev polynomials for direct Zakharov-Shabat problem

该研究提出基于切比雪夫多项式的快速八阶帕德格式，用于直接扎哈罗夫-沙巴特问题，提升了连续非线性谱计算的精度与效率。