Numerical inverse scattering transform for the coupled nonlinear Schrödinger equation

This paper numerically investigates the Riemann–Hilbert problem for the coupled nonlinear Schrödinger (CNLS) equation by implementing the numerical inverse scattering transform (NIST). The Riemann–Hilbert problem is constructed based on the initial conditions and the Lax pair associated with the CNLS equation. Prior to performing the NIST, we make previous preparations in two aspects. First, by introducing Chebyshev nodes and polynomials and choosing appropriate mapping functions, we compute the scattering matrix and eigenvalues with high precision in the numerical direct scattering. Second, by applying the Deift–Zhou nonlinear steepest descent method, we deform the original Riemann–Hilbert problem to mitigate the influence of oscillation terms. The numerical inverse scattering method distinguishes from the traditional numerical methods in that it allows to compute solutions at any spatial and temporal point without time stepping or spatial discretization. Starting directly from the Riemann–Hilbert problem of the CNLS equation, the NIST is effective for solving the long-term evolution of solutions.