Improved numerical approach for time-independent Gross-Pitaevskii nonlinear Schroedinger equation

TL;DR

This paper presents an improved numerical method for solving the time-independent Gross-Pitaevskii nonlinear Schrödinger equation, utilizing scaling, shooting, Runge-Kutta, and secant methods for efficient and accurate solutions.

Contribution

The work introduces a novel combination of scaling, shooting, Runge-Kutta, and secant methods to enhance the numerical solution of the Gross-Pitaevskii equation.

Findings

Enhanced convergence speed of the numerical method

Accurate wave-function normalization after solution

Effective handling of boundary conditions at infinity

Abstract

In the present work, we improve a numerical method, developed to solve the Gross-Pitaevkii nonlinear Schroedinger equation. A particular scaling is used in the equation, which permits to evaluate the wave-function normalization after the numerical solution. We have a two point boundary value problem, where the second point is taken at infinity. The differential equation is solved using the shooting method and Runge-Kutta integration method, requiring that the asymptotic constants, for the function and its derivative, are equal for large distances. In order to obtain fast convergence, the secant method is used.