The Numerical Unified Transform Method for the Nonlinear Schrödinger equation on the half-line

06/11/2021

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by   Xin Yang, et al.

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We implement the Numerical Unified Transform Method to solve the Nonlinear Schrödinger equation on the half-line. For so-called linearizable boundary conditions, the method solves the half-line problems with comparable complexity as the Numerical Inverse Scattering Transform solves whole-line problems. In particular, the method computes the solution at any x and t without spatial discretization or time stepping. Contour deformations based on the method of nonlinear steepest descent are used so that the method's computational cost does not increase for large x,t and the method is more accurate as x,t increase. Our ideas also apply to some cases where the boundary conditions are not linearizable.