# Scattering of Matter Wave Solitons on Localized Potentials
## 1 Introduction  
Solitons are exceptionally stable wave phenomena that appear in a variety of physical systems. They propagate without spreading due to a fine balance between the dispersion and the non-linearity of the physical system. After their discovery as stable waves in shallow waters by [PERSON] in 1834, solitons have been found in very different non-linear systems. As an example, both bright optical solitons which are peaks in the intensity and dark solitons which are intensity minima have found applications in optical transmission through fibres [1, 2]. In this article we consider soliton waves in Bose-Einstein condensates of neutral atoms. Laser fields may be employed to induce waveguides, potential barriers and filters and thus enable the matter wave functionalities needed in the evolving field of atomtronics [3]. In the present work we show that in addition to their favorable propagation behavior, matter wave solitons have remarkable scattering properties that may be observed in the atomtronics architectures.  
The condensation of a cold gas of atoms permits a description of the many-body problem by a single condensate wave function order parameter, which is in turn described by the non-linear Gross-Pitaevskii wave equation [4]. The non-linearity in Bose-Einstein condensates arises from the repulsive or attractive interactions of the particles in the condensate. In equivalence with non-linear optics, dark and bright solitons are, respectively, localized dips and peaks in the Bose-Einstein condensate density. In addition to single dark and bright solitons a variety of composite solitons, for example dark-bright solitons, can exist in spinor condensates [5, 6, 7].  
The dynamics of solitons, bright and dark as well as composite dark-bright solitons, in inhomogeneous condensates with slowly varying potentials has been investigated and is adequately described by a particle-like behavior [8, 9, 10, 11, 12, 13, 14, 15]. When solitons are scattered on a localized potentials both particle- and wave-like behavior is observed [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. In this article, we shall further elaborate the analysis of matter wave solitons in 1D-condensates, investigating their scattering properties on localized potentials. We note that there is a rich literature [33, 34, 35, 36, 37] on the dynamics of non-linear superpositions of solitons, the so-called breathers and of their remarkable scattering properties [38, 39], which we shall not address in this article.  
In Section 2, we introduce the Gross-Pitaevskii equation and a convenient mapping to dimensionless coordinates that will be used throughout the paper. In Section 3, we introduce the bright, the dark, and the composite dark-bright solitons, and we briefly review their properties. In Section 4, we show that scattering of bright solitons on a low and wide potential barrier can be well understood as the motion of a classical particle governed by an effective Lagrangian, extracted from the quantum problem, while the scattering on high and narrow barriers can be understood from the single particle quantum problem. In Section 5, we present results for the scattering of a dark-bright soliton on a potential felt only by the bright soliton atoms, and we show to what extent their scattering is transferred to the dark soliton. Section 6 concludes the paper.