We investigate the dynamics of a Bose-Einstein condensate in a progressively bended three dimensional cigar shaped potential. The interplay between geometry and nonlinearity is considered. At high curvature, topological localization occurs and becomes frustrated by the generation of curved dispersive shock-waves when the strength of nonlinearity is increased. The analysis is supported by four-dimensional parallel simulations.The effect of geometry on wave propagation has been considered since early developments of the theory of sound, for example, in the vibrations of membranes with different shapes [1]. Geometry largely affects energy velocity, wave transformations upon propagation, and various classical and quantum undulatory phenomena, either in constrained geometries, or at an astrophysical scale; renowned examples are the Einstein lensing effect [2], the Hawking radiation [3],or the Unruh effect [4] Albeit this old history, the link between geometry and waves is continuously fascinating many researchers, and is currently driving important applications as transformation optics [5,6], curved and twisted waveguides, [5,7] and analog of gravity in Bose-Einstein condensation (BEC) [8,9] or nonlinear optics [10][11][12]. It is also known that nonlinear waves may be affected by geometrical constraints, resulting in interesting phenomena, as solitons in curved manifolds [13,14].A striking effect of curvature is the topologically induced localization, originally considered in [15]: when the wave is constrained on extremely deformed surfaces, or lines, propagation may be inhibited and energy is trapped in the regions with the highest curvature. The way this kind of geometrical localization competes with nonlinearity, and the kind of dynamic effects resulting from a nonlinear response sufficiently strong to overcome topological bounds is largely unconsidered.If geometrical localization occurs in a reduced dimensionality (e.g., a curved surface in a three dimensional space), when nonlinearity is very effective the whole three-dimensional (3D) space becomes involved, and any treatment based on nonlinear wave equations with reduced dimensionality may be questioned. This is a key difficulty in this problem; any theoretical prediction must be numerically tested by using 3D simulations.In addition, for high nonlinearity, shock waves (SW) are generated [16] . In recent years, SW have been largely considered in optics and BEC [17][18][19][20][21]; SW originate from singular solutions of the hydrodynamic reduction of the nonlinear Schroedinger equation (NLS) regularized by oscillating wave fronts, named undular bores, or dispersive SW (D-SW). To the best of our knowledge, the effect of a curved space on DSW is un-explored.In this Letter, we investigate the way a geometrical localization is frustrated by a defocusing nonlinearity, we study this effect by one-dimensional (1D) theoretical analysis and 3D+1 simulations of the Gross-Pitaevskii (GP) equation, and report on DSW in curved potentials.The model -The adimensionalized GP ...
