Numerical Simulations on Nonlinear Quantum Graphs with the GraFiDi Library

TL;DR

This paper introduces the Grafidi Python library for simulating nonlinear Schr{"o}dinger equations on quantum graphs, demonstrating its effectiveness through various numerical experiments and comparison with theoretical results.

Contribution

The paper presents the first comprehensive Python library for numerical simulation of nonlinear quantum graphs, including implementation of multiple numerical schemes and validation against theoretical results.

Findings

Successful implementation of gradient flow and conjugate gradient methods for ground states.

Effective simulation of nonlinear Schr{"o}dinger dynamics on various graph types.

Validation of numerical results with existing theoretical predictions.

Abstract

Nonlinear quantum graphs are metric graphs equipped with a nonlinear Schr{\"o}dinger equation. Whereas in the last ten years they have known considerable developments on the theoretical side, their study from the numerical point of view remains in its early stages. The goal of this paper is to present the Grafidi library, a Python library which has been developed with the numerical simulation of nonlinear Schr{\"o}dinger equations on graphs in mind. We will show how, with the help of the Grafidi library, one can implement the popular normalized gradient flow and nonlinear conjugate gradient flow methods to compute ground states of a nonlinear quantum graph. We will also simulate the dynamics of the nonlinear Schr{\"o}dinger equation with a Crank-Nicolson relaxation scheme and a Strang splitting scheme. Finally, in a series of numerical experiments on various types of graphs, we will compare the outcome of our numerical calculations for ground states with the existing theoretical results, thereby illustrating the versatility and efficiency of our implementations in the framework of the Grafidi library.