Evolution of Quantum Waves: Numerical and Physical Aspects

of and beyond BEC

Dr. Mark Edwards (Physics), Dr. Cheng Zhang (ME) and Dr. Shi-Jun Zheng (Math)

sponsored by the COSM Interdisciplinary Pilot Project



Goal: To develop PDE solver and library on the network


Scope: The Scientific Computation & Numerical PDE group conducts research that highlights the connections between physics, engineering and mathematical modeling. Our group promotes collaborations between a variety of closely-related areas whose methods are based on analytical theory and computational algorithms, with the support of simulations as well as experiments.



Our group organizes a series of joint seminars, colloquiums, lectures, and workshops, and publishes research in high-quality peer-reviewed journals. We also offer graduate courses, direct theses, and expose graduate and undergraduate students to the frontier research in this challenging field.

Ring BEC and GP equation with electromagnetic fields

Quantum mechanics is the foundation of modern physics in the last century as classical mechanics for Newton's physics in the nineteen's century.  The suspicion on QM has been vanishing as  more  and more  evidence  to verify the  solid and rigor of the theory.   This project is to further make quantum more accessible and available to practical applications, meanwhile to make the abstract theory more valid and true so it Exists in daily people's life.


Analytical solution and design for the local and global existence and uniqueness as well as scattering

Time-splitting pseudo-spectral algorithm to reconstruct standing waves, solition using NLS with trapping potential and nonlinear perturbation.


Installation OpenFOAM

An open-source C++ library package, OpenFOAM (Open Field Operation and Manipulation) is being used to develop solver for the
non-linear Schrödinger and Navier-Stokes. OpenFOAM provides an open development platform for solving PDEs in parallel mode using finite volume method.
OpenFOAM has a large user base across many areas of engineering and science, from both academic and industrial organizations.

ComSol.multiphysics also has information on Femlab solving PDEs (based on Matlab 2d/3d)

Rediscover further properties including quantized superfluidity, vortex, plasmer and superconductivity

Simulation of certain dumbell-potential influenced system that resembles a routine circuit with R, C and L

Scientific Computing and Computational Mathematics

Electromagnetic Non-Linear Schrödinger equations

\(iu_t=\left(-\frac{1}{2}\sum_{j=1}^n (\partial_j -iA_j)^2+V \right)u+ g(|u|^2)u\)


Standard NLS \( i \partial_t u + \Delta u - \mu |u|^{p-1} u = 0.  \)

 Gross-Pitaevskii hierarch:  The mean field limit of \(N\)-body quantum dynamics is generated by the Hamiltonian

\[ H_N:=-\sum_{j=1}^N \Delta_j+\frac{1}{N}\sum^N_{i<j}V(x_j-x_j) \]

where \(i \partial_t\psi_N= H_N \psi_N \), which leads to the Hartree equation describing the condensate wave function evolution

\[iu_t = -\Delta u+ (V*|u|^2) u\]

FD, FEM and FV LAB

Wave-matter in ultra-cold dilute gaseous state with wave link


This project is based on the the NLS that governs statistical quantum mechanics motivated by current computational and experimental physics, which has been conducted in GSU and NIST labs.  It requires both mathematical tools and techniques as well as computational physics.

Bose-Einstein condensation has developed into an tremendously important branch of modern physics with progress and results that has been challenged by mathematical conjecture and physical modeling.  Our group aims to use state of the art method to treat the NLS system in various context, pertubative, variational, geometric and probabilistic. Analytically our approach and theory rely on the dispersive and concentration compactness nature of the equation and therefore Strichartz estimate, Morawetz inequalities are established in order to obtain local and global wellposedness and scattering. 

  Dr. Edwards recently executed a couple of projects with members in the NIST; e.g. using Lagrange variational method to compute efficiently the ring-shaped BEC.  Following that Edwards proposed dumbell-shaped potential to simulate the analogue experiment fulfiled in NIST.  The atoms are 23^Na in a setting that involves femion and boson atoms.  The ring-BEC can be used to make sensoring navigation system, the dumbell potential to make  atom circuit. They all immitate the classical system but have profound far-reaching applications in e.g. information technology, green energy and so on, which are New that are non-existing before. 


Dr. Zhang uses parallel computing to further study and compare these numerial results via different schemes.

Based on a rigorously derivation of N-body GP equation by Grillakis-Machedon, Chen et al, also the study of Carles et al in the semi-classical case.  Dr. Zheng obtained LWP and GWP for the NLS with low regularity for the initial data.
 The analysis has natural physical intuition which relies on the conservation of number of particals, conservation of energy and momentum and virial identitiy in the magnetic case. He also studies the profile decompositon of the focusing NLS with quadratic magnetic potentials, which can describe exactly how the shape of the initial data affects the finite time blowup of the solution.

Broader Impact:   Our study of the ring BEC model provides qualitative and quantitative understanding of the special wave matter phenomena occuring in quantum nature. The proposed project combines the genuine knowledge and strength from math and physics, which help promote the interdisciplinary working communicatoins between mathematicians and physicists and therefore might provide new insights and tools as well as motivation for two kinds of scientists who are able to collaborate to solve problems originated from atomic and condensed matter physics.

Interllectural Merit: The research is highlighted by the expertise of leading expert in physics. The team consists of senior and young scientists.  The acdemic results are presented in conferences and seminars.  We also invited and discuss with well-known visitors from Brown, Johns Hopkins, Universty of Illinois, UIUC and Chicago, University of Minnesota, University of Maryland.  This project is supported by the College of Science and Mathematics at Georgia Southern University. The faculty members teach and train graduate and undergraduate students that resulted in some publications in peer-reviewd professional journals and proceeding. The working experience with faculty and students 
serve as an example for inspiring and continuing teaching and learning for the young generation in order to build a more educated region in the rural of the southern east coast.

GPU source for PDE solver: