Computer-Aided Derivation of Multi-scale Models: A Rewriting Framework

We introduce a framework for computer-aided derivation of multi-scale models. It relies on a combination of an asymptotic method used in the field of partial differential equations with term rewriting techniques coming from computer science. In our approach, a multi-scale model derivation is characterized by the features taken into account in the asymptotic analysis. Its formulation consists in a derivation of a reference model associated to an elementary nominal model, and in a set of transformations to apply to this proof until it takes into account the wanted features. In addition to the reference model proof, the framework includes first order rewriting principles designed for asymptotic model derivations, and second order rewriting principles dedicated to transformations of model derivations. We apply the method to generate a family of homogenized models for second order elliptic equations with periodic coefficients that could be posed in multi-dimensional domains, with possibly multi-domains and/or thin domains.Comment: 26 page

Josephson Oscillation and Transition to Self-Trapping for Bose-Einstein-Condensates in a Triple-Well Trap

We investigate the tunnelling dynamics of Bose-Einstein-Condensates(BECs) in a symmetric as well as in a tilted triple-well trap within the framework of mean-field treatment. The eigenenergies as the functions of the zero-point energy difference between the tilted wells show a striking entangled star structure when the atomic interaction is large. We then achieve insight into the oscillation solutions around the corresponding eigenstates and observe several new types of Josephson oscillations. With increasing the atomic interaction, the Josephson-type oscillation is blocked and the self-trapping solution emerges. The condensates are self-trapped either in one well or in two wells but no scaling-law is observed near transition points. In particular, we find that the transition from the Josephson-type oscillation to the self-trapping is accompanied with some irregular regime where tunnelling dynamics is dominated by chaos. The above analysis is facilitated with the help of the Poicar\'{e} section method that visualizes the motions of BECs in a reduced phase plane.Comment: 10 pages, 11 figure