QGLAB: A MATLAB Package for Computations on Quantum Graphs

TL;DR

QGLAB, a new MATLAB package for analyzing partial differential equations on quantum graphs is described, built on the existing, object-oriented MATLAB directed-graph class, inheriting its structure and adding additional easy-to-use features.

Abstract

We describe QGLAB, a new MATLAB package for analyzing partial differential equations on quantum graphs. The software is built on the existing, object-oriented MATLAB directed-graph class, inheriting its structure and adding additional easy-to-use features. The package allows one to construct a quantum graph and accurately compute the spectrum of elliptic operators, solutions to Poisson problems, the linear and nonlinear time evolution of a variety of PDEs, the continuation of branches of steady states (including locating and switching branches at bifurcations) and more. It overcomes the major challenge of discretizing quantum graphs -- the enforcement of vertex conditions -- using non-square differentiation matrices. It uses a unified framework to implement finite-difference and Chebyshev discretizations of differential operators on a quantum graph. For simplicity, the package overloads many built-in MATLAB functions to work on the class.

Figures (18)

• Figure 1: A directed graph with three vertices and five edges.
• Figure 1: Discretization of the interval $[0,\ell]$ using ghost points.
• Figure 1: (a) A function defined on the edges of a dumbbell graph. (b) A function defined on the edges of a tetrahedral graph.
• Figure 1: The secular determinant $\Sigma(k)$, of the Y-shaped graph discussed in the text, along with the computed values $k_j=\sqrt{-\lambda_j}$, which sit right on top of the zeros.
• Figure 1: (a) A partial bifurcation diagram for the dumbbell graph. The three blue curves are the continuations of linear eigenfunctions. The red curves were computed by continuing from branching bifurcations. The green curve was computed by computing a single large amplitude solution and then continuing it. Branching bifurcations marked with squares and folds with triangles. (b) The same diagram, plotted in different variables.