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

TL;DR

This paper introduces a Chebyshev polynomial-based numerical method with tanh(ax) mapping for efficiently calculating eigenvalues of the Zakharov-Shabat system, outperforming Fourier methods especially for complex potentials.

Contribution

The paper presents a novel Chebyshev polynomial and tanh(ax) mapping approach that improves convergence speed in eigenvalue calculations for the Zakharov-Shabat system.

Findings

Faster convergence than Fourier collocation method.

Effective for both simple and complex potentials.

Applicable to other linear eigenvalue problems.

Abstract

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.