A growing number of dynamical situations involve the coupling of particles or singularities with physical waves. In principle these situations are very far from the wave particle duality at quantum scale where the wave is probabilistic by nature. Yet some dual characteristics were observed in a system where a macroscopic droplet is guided by a pilot wave it generates. Here we investigate the behaviour of these entities when confined in a two-dimensional harmonic potential well. A discrete set of stable orbits is observed, in the shape of successive generalized Cassinian-like curves (circles, ovals, lemniscates, trefoils and so on). Along these specific trajectories, the droplet motion is characterized by a double quantization of the orbit spatial extent and of the angular momentum. We show that these trajectories are intertwined with the dynamical build-up of central wave-field modes. These dual self-organized modes form a basis of eigenstates on which more complex motions are naturally decomposed.

Wave control is usually performed by spatially engineering the properties of a medium. Because time and space play similar roles in wave propagation, manipulating time boundaries provides a complementary approach. Here, we experimentally demonstrate the relevance of this concept by introducing instantaneous time mirrors. We show with water waves that a sudden change of the effective gravity generates time-reversed waves that refocus at the source. We generalize this concept for all kinds of waves introducing a universal framework which explains the effect of any time disruption on wave propagation. We show that sudden changes of the medium properties generate instant wave sources that emerge instantaneously from the entire space at the time disruption. The time-reversed waves originate from these "Cauchy sources" which are the counterpart of Huygens virtual sources on a time boundary. It allows us to revisit the holographic method and introduce a new approach for wave control.Holographic methods are based on the time-reversal invariance of wave equations. They rely on the fact that any wave field can be completely determined within a volume by knowing the field (and its normal derivative) on any enclosing surface 1,2 . Hence, information reaching the 2D surface is sufficient to recover all information inside the whole volume. Based on these properties, Denis Gabor introduced the Holographic method, which provides an elegant way to back-propagate a monochromatic wave field and obtain 3D images. More recently, time-reversal mirrors exploited the same principles extended to a broadband spectrum to create time-reversed waves. This latter approach has been implemented with acoustic 3 , elastic 4 , electromagnetic 5 and water waves 6,7 . It requires the use of emitter-receptor antennas positioned on an arbitrary enclosing surface. The wave is recorded, digitized, stored, time-reversed and rebroadcasted by the antenna array. If the array intercepts the entire forward wave with a good spatial sampling, it generates a perfect backwardpropagating copy. Note that this process is difficult to implement in optics 8,9 , and the standard solution is to work with monochromatic light and use nonlinear regimes such as three-wave or four-wave mixing 10,11 .Here, within the general concept of spacetime transformations [12][13][14][15][16] , we completely revisit the holographic method and introduce a new way to create wideband time-reversed wave fields in 2D or 3D by manipulating time boundaries. Time boundaries have recently received much attention because they have been shown to play a major role in several phenomena such as time refraction, dynamic Casimir effect, Hawking radiation, photon acceleration and self-phase modulation [17][18][19][20][21][22][23][24][25] . In addition, different suggestions to process wideband time-reversal have been proposed in optics to associate both time and spatial modulation of the medium refractive index. These suggestions, mainly for 1D propagation, rely on a dynamic tuning of photonic crystals ...