# Scattering of Matter Wave Solitons on Localized Potentials
## 2 Mean Field Theory of Interacting Atoms  
We treat the Bose-Einstein condensate of atoms at zero temperature in a Hartree mean field theory, which neglects particles outside the condensate and treats short range interactions in the gas by an interaction potential which is proportional to the mean atomic density. We assume either a guided condensate with a high transversal excitation frequency [40] or the situation of a wide condensate with approximate translational invariance in the transversal directions. This yields the 1D timedependent Gross-Pitaevskii equation (GPE)  
\[i\hbar\frac{\partial\psi(x,t)}{\partial t} =\big{(}-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial x^{2}}+V (x,t) \tag{1}\] \[+g|\psi(x,t)|^{2}\big{)}\psi(x,t),\]  
where the order parameter, the macroscopic wave function, is normalized to the total number of atoms  
\[\int dx|\psi(x,t)|^{2}=N \tag{2}\]  
so that \(|\psi(x,t)|^{2}\) represents the particle density. The 1D interaction strength \(g\) is a known function of the 3D low energy scattering properties and the transverse confinement of the atoms [4].  
In an attractive BEC with \(g<0\), (meta-)stable bright soliton solutions with a finite number of particles exist, while in a repulsive BEC with \(g>0\), dark solitons may exist in the form of holes in the background density.  
For numerical purposes it is advantageous to rewrite the GPE in a dimensionless form, and for a homogeneous condensate with repulsive interactions, the linear atomic density, \(n_{0}\) and the healing length \(\xi=\sqrt{\frac{\hbar^{2}}{m\xi n_{0}}}\) provide natural scale factors. We thus introduce dimensionless quantities for length, energy and time, labelled by a tilde  
\[x=\tilde{x}\xi,\qquad E=\tilde{E}\frac{\hbar^{2}}{m\xi^{2}}=\tilde{E}n_{0}g, \qquad t=\tilde{t}\frac{m\xi^{2}}{\hbar}. \tag{3}\]  
and obtain the equation  
\[i\frac{\partial\tilde{\psi}}{\partial\tilde{t}}=-\frac{1}{2}\frac{2^{2} \tilde{\psi}}{\partial\tilde{x}^{2}}+|\tilde{\psi}|^{2}\tilde{\psi}+\tilde{V} \tilde{\psi}. \tag{4}\]  
where the dimensionless wave function  
\[\tilde{\psi}=\frac{\psi}{\sqrt{n_{0}}}, \tag{5}\]  
is normalized to unit bulk density \(|\tilde{\psi}|^{2}=1\) in the homogeneous part of the condensate.  
A condensate with attractive interactions has no natural length scale like the healing length, but for convenience in the following, we introduce an arbitrary length unit \(l\), which may be chosen to fit a relevant experimental length scale, and we set  
\[x=\bar{x}l,\qquad E=\bar{E}\frac{\hbar^{2}}{ml^{2}},\qquad t=\bar{t}\frac{ml^{2} }{\hbar}. \tag{6}\]  
Inserting this in the GPE, Equation (1), and introducing the rescaled wave function, \(\left|\tilde{\psi}\right|^{2}=|g|\frac{ml^{2}}{\hbar^{2}}|\psi|^{2}\), leads to the dimensionless GPE,  
\[i\frac{\partial\tilde{\psi}}{\partial\bar{t}}=-\frac{1}{2}\frac{\partial^{2} \tilde{\psi}}{\partial\bar{x}^{2}}-|\tilde{\psi}|^{2}\tilde{\psi}+\tilde{V} \tilde{\psi}. \tag{7}\]  
The rescaled number of particles in the condensate is  
\[\tilde{N}=\int\mathrm{d}\bar{x}\left|\tilde{\psi}\right|^{2}=N\frac{ml|g|}{ \hbar^{2}}. \tag{8}\]  
Thus for given \(N\), the choice of \(\tilde{N}\) determines the length scale \(l\) and sets the dimensions of the problem.  
Due to the presence of the non-linear term it is in general necessary to solve the Gross-Pitaevskii equation by numerical means. The numerical results presented in the following are based on propagation of the one-dimensional, dimensionless and non-homogeneous GPE using the Crank-Nicolson method. For brevity we will omit the tilde above dimensionless quantities throughout.