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

Abstract

In this paper, a direct method is proposed to calculate the eigenvalue of the Zakharov-Shabat system. The main tools of our method are Chebyshev polynomials and the QR algorithm. After introducing the hyperbolic tangent mapping, the eigenfunctions and potential function defined in the real field can be represented by Chebyshev polynomials. Using Chebyshev nodes, the Zakharov-Shabat eigenvalue problem is transformed into a matrix eigenvalue problem. The matrix eigenvalue problem is solved by the QR algorithm. Our method is used to calculate eigenvalues of the Zakharov-Shabat equation with three potentials, the rationality of our method is verified by comparison with analytical results