# Scattering of Matter Wave Solitons on Localized Potentials
## 6 Conclusions  
We have investigated scattering of solitons in Bose-Einstein condensates on localized potentials. The homogeneous Gross-Pitaevskii equation supports matter wave solitons, and we have focused on two types of solitons: Bright solitons, which are wave packets without dispersion due to the attractive, non-linear interactions between the atoms in the soliton,  
Figure 7: Transmission and reflection coefficients of the dark soliton and the bright soliton as a function of the barrier height \(V_{0}\). In (**a**) \(N_{\text{B}}=3\) and in (**b**) \(N_{\text{B}}=1\). In both cases \(\sin\alpha=0.5\) and the barrier width is that of a dark-bright soliton with \(\sin\alpha=0\) (\(\kappa^{-1}\), Equation (22)), for \(N_{\text{B}}=3\) that is \(b=2\) and for \(N_{\text{B}}=1\) it is \(b\approx 1.3\).  
and dark-bright solitons in two separate condensate components. In the latter the bright soliton is stabilized by the presence of the dark soliton, a dip in the condensate density.  
The scattering properties of a dark-bright soliton on a barrier only affecting the bright component could be expected to resemble that of a single bright soliton scattering on a barrier. But we find that the scattering of a dark-bright soliton is in fact very different. The differences are due to the interaction between the two components of the dark-bright condensate, which enables an indirect interaction between the dark soliton and the external potential. This interaction is strongest when the bright component is large compared to the dark one.  
We have obtained numerical results by mean field theory and provided their interpretation by classical particle and single particle wave function arguments. The splitting of solitons by scattering potentials has also been proposed as a promising candidate for observation of effects beyond Gross-Pitaevskii mean field theory, thus exploring the atomic equivalent to non-classical states in quantum optics. Scattering with partial reflection on a potential barrier may thus produce Schrodinger cat like, mesoscopic superposition states with all atoms in the reflected and in the transmitted wavepacket mode [22; 29]. Such states as well as states with other types of occupation number correlations have promising applications in metrology [30; 53; 54; 56], but they are fragile against disintegration of the state and spreading of the atoms over many single particle states. A proper analysis of such effects thus necessitate multi-atom, multi-mode theory like the multi-orbital theory [55], positive-P simulations [28; 37; 56] or truncated Wigner approaches [28; 57]. The work presented here may guide efforts and assess relevant parameter regimes prior to more advanced research efforts beyond mean field theory and single and few mode approximations. We note that experimental criteria have been proposed for when the atomic system may be restricted to populate only few scattering modes [58]. Experimental realization of solitons in 1D-condensates is well developed [59], and dark-bright solitons have previously been studied in slowly varying potentials. Thus the prospects are promising for experimental studies along the lines of our calculations.  
Conceptualization, S.D.H., N.N. and K.M.; formal analysis, S.D.H., N.N. and K.M.; writing--original draft preparation, S.D.H., N.N. and K.M.; supervision, N.N. and K.M.; All authors have read and agreed to the published version of the manuscript.  
APC was funded by Villum Fonden: QUSCOPE.  
Not applicable.  
The authors declare no conflict of interest.