High order three part split symplectic integrators: Efficient techniques for the long time simulation of the disordered discrete nonlinear Schroedinger equation

TL;DR

This paper introduces high order three-part split symplectic integrators tailored for Hamiltonian systems, demonstrating their efficiency in simulating the long-term dynamics of the disordered discrete nonlinear Schrödinger equation, a key model in wave physics.

Contribution

The paper develops and compares new high order three-part split symplectic integrators specifically designed for Hamiltonian systems with three integrable parts, applied to the DDNLS model.

Findings

Three-part split algorithms effectively simulate wave packet spreading in DDNLS.

The new integrators outperform traditional methods in efficiency for long-term simulations.

The techniques facilitate studying asymptotic behaviors in complex Hamiltonian systems.

Abstract

While symplectic integration methods based on operator splitting are well established in many branches of science, high order methods for Hamiltonian systems that split in more than two parts have not been studied in great detail. Here, we present several high order symplectic integrators for Hamiltonian systems that can be split in exactly three integrable parts. We apply these techniques, as a practical case, for the integration of the disordered, discrete nonlinear Schroedinger equation (DDNLS) and compare their efficiencies. Three part split algorithms provide effective means to numerically study the asymptotic behavior of wave packet spreading in the DDNLS - a hotly debated subject in current scientific literature.