# Scattering of Matter Wave Solitons on Localized Potentials
## 4 Scattering of Bright Solitons
### Strong Barriers  
Scattering of bright solitons on strong barriers with \(V_{0}\gg\mu\) has been studied in the limit of high soliton velocities [18; 19; 23; 24]. In this regime the soliton wave function may coherently split into reflected and transmitted components, and the asymptotic behavior of the transmission and reflection as the soliton velocity increases is the same as for a single particle, described by the linear Schrodinger equation. In the limit of low soliton velocities the soliton is also split into a reflected and a transmitted component. Here, we shall consider scattering of bright solitons on strong potentials with special attention to the behavior at low soliton velocities, where the single particle description fails. Figure 3 shows the time evolution of a bright soliton incident on a rectangular potential barrier. We observe both a reflected and a transmitted component in the calculation. When the potential barrier is high enough to split the soliton, i.e., \(V_{0}\gg\mu\), and at the same time wide compared to the soliton width, a part of the soliton is seen to be transiently trapped inside the barrier, see Figure 4. The trapped component oscillates and is gradually emitted out of the barrier region. Numerical simulations as well as analytical calculations show that this trapping is due to the wave nature of the soliton, and not the interactions. This behavior is not seen when the barrier is narrow compared to the soliton width. Figure 3: (Color online) Time evolution of the condensate density. The external potential is a square barrier placed at \(x=0\). The potential has height \(V_{0}=0.8\) and width \(b=0.5\), and the incoming soliton has \(v=0.5\) and \(N=1\). We have solved the GPE numerically for different cases, and Figure 5 summarizes our results (symbols) as functions of the incident velocity. When the external potential is strong compared to the chemical potential of the soliton, we expect the interaction term in the GPE to be less important. The results in Figure 5 may thus be approximated by the usual expression for the transmission probability for a single particle with energy \(E\) incident on a square potential barrier,  
\[ \begin{split} T=&\Theta(E)\times\\ &\begin{cases}\Big{[}1+\frac{V_{0}^{2}}{4E(V_{0}-E)}\sinh^{2} \Big{(}b\sqrt{2(V_{0}-E)}\Big{)}\Big{]}^{-1}\\ \text{for }E<V_{0}\\ \Big{[}1+\frac{V_{0}^{2}}{4E(E-V_{0})}\sinh^{2}\Big{(}b\sqrt{2(E-V_{0})} \Big{)}\Big{]}^{-1}\\ \text{for }E>V_{0},\end{cases}\end{split}  \tag{38}\]  
where the Heaviside step function, \(\Theta(E)\), ensures that the energy of the particle scattered is positive, which is evident in the single particle case but not in the interacting particle case. The non-linear interaction term cannot be omitted from the GPE in regions of space away from the strong potential barrier. In these regions, however, the soliton behavior is well known, and in particular we know the average energy per particle \(\epsilon\):  
\[\epsilon=\frac{E_{\text{B}}}{N}=\frac{1}{2}\
u^{2}-\frac{N^{2}}{24}. \tag{39}\]  
Because of the attractive interactions between the particles, \(\epsilon\) becomes negative when \(v<\frac{N}{\sqrt{12}}\) (we have chosen \(N=1\) in our calculations which gives \(\frac{N}{\sqrt{12}}\approx 0.29\) in our dimensionless units). To describe scattering of the soliton in the linear approximation, it makes good sense to replace the energy \(E\) in Equation (38) by the average energy per particle in the soliton, \(\epsilon\) in Equation (39), and in the velocity regime \(v\leq\frac{N}{\sqrt{12}}\) the Heaviside stepfunction in Equation (38) suppresses the transmission and predicts full reflection of the soliton. The prediction based on this description is shown in Figure 5 and is in very good agreement with the solution of the GPE. The small transmission probability observed below the velocity threshold may be due, in parts, to our use of wave packets with finite spread, and hence with a momentum distribution exceeding the threshold value.