The Numerical Unified Transform Method for the Nonlinear Schr\"odinger equation on the half-line

TL;DR

This paper introduces a numerical method that efficiently solves the nonlinear Schrödinger equation on a half-line, providing high accuracy and computational efficiency without traditional discretization, even for large spatial and temporal domains.

Contribution

It extends the Numerical Unified Transform Method to handle the half-line problem with linearizable boundary conditions, achieving comparable complexity to whole-line solutions and applicable to some non-linearizable cases.

Findings

The method computes solutions at any point without spatial discretization.

Contour deformation improves efficiency for large x and t.

The approach is effective for linearizable and some non-linearizable boundary conditions.

Abstract

We implement the Numerical Unified Transform Method to solve the Nonlinear Schr\"odinger 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.