Merge pull request #15 from GPUE-group/3d_eqn_revision

Modified paper to use 3D GPE by default for generality

paper.md

@@ -29,11 +29,11 @@ Numerical simulations of BECs allow for new discoveries that directly mimic what
 In practice, the dynamics of BEC systems can often be found by solving the non-linear Schrödinger equation known as the Gross--Pitaevskii Equation (GPE),
 
 $$
-i\hbar \frac{\partial\Psi(x,t)}{\partial t} = \left( -\frac{\hbar^2}{2m} \frac{\partial}{\partial x^2} + V(x) + g|\Psi(x,t)|^2\right)\Psi(x,t),
+i\hbar \frac{\partial\Psi(\mathbf{r},t)}{\partial t} = \left( -\frac{\hbar^2}{2m} {\nabla^2} + V(\mathbf{r}) + g|\Psi(\mathbf{r},t)|^2\right)\Psi(\mathbf{r},t),
 $$
 
-where $\Psi(x,t)$ is the one-dimensional many-body wavefunction of the quantum system, $m$ is the atomic mass, $V(x)$ is a potential to trap the atomic system, $g = \frac{4\pi\hbar^2a_s}{m}$ is a coupling factor, and $a_s$ is the scattering length of the atomic species.
-Here, the GPE is shown in one dimension, but it can easily be extended to two or three dimensions.
+where $\Psi(\mathbf{r},t)$ is the three-dimensional many-body wavefunction of the quantum system, $\mathbf{r} = (x,y,z)$, $m$ is the atomic mass, $V(\mathbf{r})$ is an external potential, $g = \frac{4\pi\hbar^2a_s}{m}$ is a coupling factor, and $a_s$ is the scattering length of the atomic species.
+Here, the GPE is shown in three dimensions, but it can easily be modified to one or two dimensions [@PethickSmith2008].
 Though there are many methods to solve the GPE, one of the most straightforward is the split-operator method, which has previously been accelerated with GPU devices [@Ruf2009; @Bauke2011]; however, there are no generalized software packages available using this method on GPU devices that allow for user-configurable simulations and a variety of different system types.
 Even so, there are several software packages designed to simulate BECs with other methods, including GPELab [@Antoine2014] the Massively Parallel Trotter-Suzuki Solver [@Wittek2013], and XMDS [@xmds].