# Scattering of Matter Wave Solitons on Localized Potentials
## 4 Scattering of Bright Solitons
### Weak Barriers  
As mentioned in the previous section, the chemical potential of a bright soliton is given by \(\mu=-\frac{N^{2}}{8}\), and when the potential is weak compared to the chemical potential, \(N^{2}\gg V_{0}\), we expect the strong interaction potential to hold the atoms together throughout the scattering process. In Figure 1 the time evolution of the Gross-Pitaevskii wave function is shown for soliton wave packets, incident on a weak square barrier with four different incident velocities. The calculations show that the soliton keeps its sech-shape and is either reflected or transmitted by the barrier like a classical particle. In Figure 1b,c the soliton is slowed down dramatically and dwells on the potential for a rather long time, before it is eventually reflected or transmitted regaining its initial shape and speed. We define reflection and transmission coefficients as the weight of the wave function on the left and right hand side of the barrier  
\[ \begin{split} R&=\frac{1}{N}\int_{-L/2}^{0}\text {d}x\ |\psi|^{2}\\ T&=\frac{1}{N}\int_{0}^{L/2}\!\!\text{d}x\ |\psi|^{2}, \end{split}  \tag{29}\]  
where \(L\) is the length of the box used for our calculations, and the barrier is centred at \(x=0\). In Figure 2 transmission and reflection coefficients are shown for two different barrier widths. The figure confirms that the solitons are almost exclusively reflected or transmitted by the wide barrier, while the narrow barrier causes a minor splitting of the mean field wave packet. To get a better understanding of the observed phenomena, we assume that the soliton keeps its sech-shape during the scattering process, a fair assumption as seen from the numerical calculations in Figure 1,  
\[\psi_{x_{0}}(x)=B\;\text{sech}[B(x-x_{0})]\text{e}^{i\dot{x}_{0}(x-x_{0})+i\dot{ \varphi}}, \tag{30}\]  
and we use the Lagrange approach presented in [17; 18; 42] to identify equations of motion for the time dependent parameters \(B\), \(x_{0}\) and \(\phi\). The effective Lagrangian of the system is  
\[\mathcal{L}=\frac{1}{2}N\dot{x}_{0}^{2}-N\dot{\phi}+\frac{1}{6}NB(N-B)-V_{\text {eff}}(x_{0}), \tag{31}\]  
where \(V_{\text{eff}}(x_{0})\) is  
\[V_{\text{eff}}(x_{0})=\int_{-\infty}^{\infty}\!\!\!\text{d}x\;V(x)|\psi_{x_{0} }(x)|^{2}. \tag{32}\]  
Figure 1: (Color online) Time evolution of the condensate density. The external potential, indicated by the dashed lines, is a square barrier with height \(V_{0}=0.04\) and width \(b=10\) and the incoming soliton has \(N=1\). The threshold velocity, Equation (37), is \(v_{t}\approx 0.289\). In **(a)** the incoming soliton velocity is \(v=0.24\), which is well below the threshold velocity, in **(b)** the velocity is \(v=0.29\), which is just about the threshold velocity and in **(c,d)** the velocities are \(v=0.3\) and \(v=0.4\) respectively, which are above the threshold velocity. Here and in all the following figures all physical quantities are presented in the dimensionless units defined by Equations (3)–(6). For a square barrier, i.e., an external potential of the form  
\[V(x)= \begin{cases}0&\text{for }|x|>\frac{b}{2}\\ V_{0}&\text{for }|x|<\frac{b}{2},\end{cases}  \tag{33}\]  
the effective potential becomes  
\[V_{\text{eff}}(x_{0})= \begin{cases}0&\text{for }|x|>\frac{b}{2}\\ \frac{V_{0}N}{2}\bigl{(}\tanh\Bigl{[}B\left(\frac{b}{2}-x_{0}\right)\Bigr{]} \\ +\tanh\Bigl{[}B\left(\frac{b}{2}+x_{0}\right)\Bigr{]}\bigr{)}&\text{for }|x|<\frac{b}{2}.\end{cases}  \tag{34}\]  
The Hamiltonian of the motion is found from the Lagrangian:  
\[H=N\dot{x}_{0}^{2}-\mathcal{L}=\frac{1}{2}N\dot{x}_{0}^{2}+\frac{1}{6}NB(B-N) +V_{\text{eff}}(x_{0}), \tag{35}\]  
and from this we find the classical equation of motion for the soliton center \(x_{0}\)  
\[\ddot{x}_{0}=-\frac{1}{N}\frac{\partial V_{\text{eff}}(x_{0})}{\partial x_{0}}. \tag{36}\]  
This is the equation of motion of a classical particle with mass \(N\) and center of mass \(x_{0}\) moving in the effective potential, \(V_{\text{eff}}\), defined by Equation (32).