On the symplectic integration of the discrete nonlinear Schr\"odinger equation with disorder

TL;DR

This paper compares symplectic integration methods for solving the disordered discrete nonlinear Schrödinger equation, finding that three-part split methods are most effective for long-term simulations of large systems.

Contribution

It introduces and evaluates symplectic integrators based on two and three part operator splitting for the DDNLS equation, highlighting their efficiency and conservation properties.

Findings

Three-part split methods excel in long-term integration of large systems.

Two-part split methods are suitable for smaller lattices up to 70 sites.

Methods based on three-part splits better conserve the wave packet's norm.

Abstract

We present several methods, which utilize symplectic integration techniques based on two and three part operator splitting, for numerically solving the equations of motion of the disordered, discrete nonlinear Schr\"odinger (DDNLS) equation, and compare their efficiency. Our results suggest that the most suitable methods for the very long time integration of this one-dimensional Hamiltonian lattice model with many degrees of freedom (of the order of a few hundreds) are the ones based on three part splits of the system's Hamiltonian. Two part split techniques can be preferred for relatively small lattices having up to $N\approx\;$70 sites. An advantage of the latter methods is the better conservation of the system's second integral, i.e. the wave packet's norm.