# Scattering of Matter Wave Solitons on Localized Potentials
## 5 Scattering of Dark-Bright Solitons  
Dark and dark-bright solitons retain their localized nature when they collide or propagate in slowly varying potentials [4; 5; 6; 41; 45; 46; 47; 48; 4], and their scattering on delta-function potentials has been studied in [49]. Here, we shall study the collisions of dark-bright vector  
Figure 5: (Color online) Transmission and reflection coefficients of a soliton with \(N=1\) on a strong barrier for various incoming soliton velocities. The coefficients have been found by solving the GPE numerically (symbols), by Equation (38) with \(E=\frac{1}{2}\varphi^{2}\) (dashed-dotted lines), and from Equation (38) with \(E=\frac{1}{2}\varphi^{2}-\frac{N^{2}}{24}\) (dashed lines), see the text. In (**a**) the barrier has strength \(V_{0}=0.6\) and width \(b=2\), in (**b**) the barrier has strength \(V_{0}=0.2\) and strength \(b=2\), and in (**c**) the barrier has \(V_{0}=0.8\) and \(b=0.5\), and thus the same area as in (**b**). All barriers are higher than the chemical potential, \(|\mu|=\frac{1}{8}\). In (**d**) the barrier is a delta barrier of strength \(\alpha=0.4\) which means that the barrier has the same area as the barriers in (**b**,**c**). For the delta potential the transmission probability reads \(T=\Theta(E)\cdot\frac{1}{1+a^{2}},a=\frac{\alpha}{\sqrt{2}E}\). solitons on a rapidly varying square potential, which we choose to only directly affect the bright component such that the coupled GP equations are given by  
\[i\frac{\partial\psi_{\text{D}}(x,t)}{\partial t} =\left(-\frac{1}{2}\frac{\partial^{2}}{\partial x^{2}}+\left|\psi_ {\text{D}}(x,t)\right|^{2}\right. \tag{40}\] \[+\left|\psi_{\text{B}}(x,t)\right|^{2}\right)\psi_{\text{D}}(x,t),\] \[i\frac{\partial\psi_{\text{B}}(x,t)}{\partial t} =\left(-\frac{1}{2}\frac{\partial^{2}}{\partial x^{2}}+\left| \psi_{\text{B}}(x,t)\right|^{2}\right.\] \[+\left|\psi_{\text{D}}(x,t)\right|^{2}+V(x)\left.\right)\psi_{ \text{B}}(x,t).\]  
Because of the rapid spatial variation at the edge of the square potential, we expect the vector soliton to break up into a reflected and a transmitted part as we observed for the single bright soliton. However due to the interaction with the dark component, this system is more complicated, and the linear approximation successfully applied to the bright soliton is not valid here. The results, shown in Figure 6, are obtained by numerical simulations of the system (in all numerical calculations \(\mu=1\)). It is seen that even though the external potential only affects the bright component directly, the dark component feels the external potential through its interaction with the bright component. Both components of the dark-bright soliton are partly reflected and partly transmitted through the external barrier. After impact on the external potential the reflected(transmitted) bright and dark components are still bound in joint center of mass motion. The transmission and reflection coefficients, \(T_{\text{B}}\) and \(R_{\text{B}}\), are defined in the same way as for the single bright soliton in Equation (29). For the dark component, we define the reflection and transmission coefficients as the number of atoms missing with respect to the homogeneous background condensate, on the reflection and transmission side of the  
Figure 6: (Color online) Time evolution of the condensate density when the dark-bright soliton scatters on a barrier. (**a**) shows the bright component and (**b**) the dark component of the condensate. The incoming soliton has \(N_{\text{B}}=3\) and \(\sin\alpha=0.5\), and the barrier, which only directly affects the bright component, has height \(V_{0}=0.2\) and width \(b=2\), its location is indicated by the dashed lines. barrier, divided by the missing number of atoms \(N_{D}\) in the incident dark component of the vector soliton,  
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