This paper builds a numerical toolkit to solve the coupled modified Korteweg–de Vries (mKdV) equation using the inverse scattering transform. The coupled system models two interacting wave fields and leads to a 3×3 matrix version of a Riemann–Hilbert problem. A Riemann–Hilbert problem is a way of encoding the wave data and reconstructing the solution by solving a matrix-valued boundary problem in the complex plane. The authors’ numerical inverse scattering transform (NIST) computes the solution directly at chosen space and time points, without stepping forward in time.

To make this work they first perform the “direct scattering” step. Starting from the Lax pair (a pair of linear equations associated with the nonlinear system), they form a 3×3 scattering matrix and compute its entries, including vector-valued reflection coefficients and any discrete eigenvalues. These data are computed numerically by solving the matrix spectral problem with a Chebyshev collocation method and suitable mappings. Chebyshev collocation is a high-accuracy spectral method that represents functions by special polynomials and enforces the equations at selected points.

For the inverse step they reformulate the problem as an oscillatory Riemann–Hilbert problem and then use the Deift–Zhou nonlinear steepest descent method to deform contours and tame the oscillations. The phase function in this problem has two stationary points that are symmetric about zero, and that structure leads the authors to split the (x,t) plane into three regions. Each region needs a different contour deformation so that the oscillatory terms become exponentially small and the numerical inverse problem is well behaved.

Why this matters: traditional numerical methods usually march forward in time, which can be costly and inaccurate for very long times. The NIST computes the solution at any given (x,t) directly, which the authors show in numerical experiments can capture the main long-time, asymptotic features of the coupled mKdV solutions. Their implementation also includes analysis of the stability and convergence of the direct scattering computation.