NonlinearSchrodinger: Higher-Order Algorithms and Darboux Transformations for Nonlinear Schr\"odinger Equations

TL;DR

NonlinearSchrodinger.jl is a Julia package that simplifies the numerical and analytical study of nonlinear Schrödinger equations, offering advanced algorithms and Darboux transformations with an easy-to-use interface.

Contribution

The paper introduces a Julia package that provides higher-order algorithms and Darboux transformations for solving NLSEs, combining numerical and analytical methods in a user-friendly way.

Findings

Supports 32 numerical algorithms including high-order symplectic and Runge-Kutta methods.

Enables analytical solution computation via Darboux transformations up to fifth order.

Provides simple interface for complex simulations and solutions of NLSEs.

Abstract

NonlinearSchrodinger.jl is a Julia package with a simple interface for studying solutions of nonlinear Schr\"odinger equations (NLSEs). In approximately ten lines of code, one can perform a simulation of the cubic NLSE using one of 32 algorithms, including symplectic and Runge-Kutta-Nystr\"om integrators up to eighth order. Furthermore, it is possible to compute analytical solutions via a numerical implementation of the Darboux transformation for extended NLSEs up to fifth order, with an equally simple interface. In what follows, we review the fundamentals of solving this class of equations numerically and analytically, discuss the implementation, and provide several examples.