Numerical inverse scattering transform for the focusing and defocusing Kundu–Eckhaus equations

In this paper, we develop the numerical inverse scattering transform (NIST) for the focusing and defocusing Kundu–Eckhaus (KE) equations. The NIST consists of numerical direct scattering and numerical inverse scattering. In numerical direct scattering, tanh mapping is introduced to improve the Chebyshev collocation method, which can help us obtain high-precision scattering data. Benefitting from the Deift–Zhou nonlinear steepest descent method and Olver's numerical method, we can effectively complete the calculation for numerical inverse scattering. Different from traditional methods, the NIST does not require advance time during the calculation, and it will be more effective in a long time. Thus the NIST is of great significance in studying the long-time evolution for solutions.