Numerical inverse scattering transform for the derivative nonlinear Schrodinger equation

TL;DR

This paper introduces a numerical inverse scattering transform for the derivative nonlinear Schrödinger equation, utilizing Riemann-Hilbert problem formulation and contour deformations to efficiently solve the equation without time-stepping.

Contribution

It develops a novel NIST approach for DNLS, handling complex spectra and saddle points with region-specific deformations, improving accuracy and computational efficiency.

Findings

Effective handling of complex spectra and saddle points.

Reduced computational costs through region-specific deformations.

Elimination of convergence issues common in traditional methods.

Abstract

In this paper, we develop the numerical inverse scattering transform (NIST) for solving the derivative nonlinear Schrodinger (DNLS) equation. The key technique involves formulating a Riemann-Hilbert problem (RHP) that is associated with the initial value problem and solving it numerically. Before solving the RHP, two essential operations need to be carried out. Firstly, high-precision numerical calculations are performed on the scattering data. Secondly, the RHP is deformed using the Deift-Zhou nonlinear steepest descent method. The DNLS equation has a continuous spectrum consisting of the real and imaginary axes and features three saddle points, which introduces complexity not encountered in previous NIST approaches. In our numerical inverse scattering method, we divide the $(x,t)$-plane into three regions and propose specific deformations for each region. These strategies not only help reduce computational costs but also minimize errors in the calculations. Unlike traditional numerical methods, the NIST does not rely on time-stepping to compute the solution. Instead, it directly solves the associated Riemann-Hilbert problem. This unique characteristic of the NIST eliminates convergence issues typically encountered in other numerical approaches and proves to be more effective, especially for long-time simulations.