Dynamics of Solitons in the One-Dimensional Nonlinear Schr\"odinger Equation

TL;DR

This paper develops a variational approach using coupled Gaussian functions to efficiently model the dynamics of bright solitons in the one-dimensional nonlinear Schrödinger equation, including stationary states and collisions.

Contribution

It introduces a computationally efficient variational method that surpasses analytical solutions and matches numerical simulations for soliton dynamics.

Findings

The variational approach accurately reproduces stationary soliton states.

It effectively models breathing oscillations of excited solitons.

High-energy collision dynamics show excellent agreement with exact simulations.

Abstract

We investigate bright solitons in the one-dimensional Schr\"odinger equation in the framework of an extended variational approach. We apply the latter to the stationary ground state of the system as well as to coherent collisions between two or more solitons. Using coupled Gaussian trial wave functions, we demonstrate that the variational approach is a powerful method to calculate the soliton dynamics. This method has the advantage that it is computationally faster compared to numerically exact grid calculations. In addition, it goes far beyond the capability of analytical ground state solutions, because the variational approach provides the ability to treat excited solitons as well as dynamical interactions between different wave packets. To demonstrate the power of the variational approach, we calculate the stationary ground state of the soliton and compare it with the analytical solution showing the convergence to the exact solution. Furthermore, we extend our calculations to nonstationary solitons by investigating the breathing oscillations of excited solitons and the coherent collisions of several wave packets in both the low- and high-energy regime. Comparisons of the variational approach with numerically exact simulations on grids reveal excellent agreement in the high-energy regime while deviations can be observed for low energies.