Numerical inverse scattering for the sine-Gordon equation

TL;DR

This paper develops a spectrally accurate numerical method for solving the sine-Gordon equation using inverse scattering, enabling solution computation at any point without discretization or time-stepping.

Contribution

It implements a fully spectral numerical inverse scattering transform for the sine-Gordon equation, advancing computational techniques for integrable PDEs.

Findings

The method achieves uniform accuracy across the domain.

It allows solution computation at arbitrary points without discretization.

The implementation is based on the nonlinear steepest descent method.

Abstract

We implement the numerical inverse scattering transform (NIST) for the sine-Gordon equation in laboratory coordinates on the real line using the method developed by Trogdon, Olver and Deconinck. The NIST allows one to compute the solution at any x and t without having spatial discretization or time-stepping. The numerical implementation is fully spectrally accurate. With the help of the method of nonlinear steepest descent, the NIST is demonstrated to be uniformly accurate.