  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.
