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arXiv · 2106.13713

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

Xin Yang·Bernard Deconinck·Thomas Trogdon
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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.

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BibTeXRIS

Xin Yang, Bernard Deconinck, Thomas Trogdon. 2021-06-11. The Numerical Unified Transform Method for the Nonlinear Schr\"odinger equation on the half-line. https://arxiv.org/abs/2106.13713

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