Nonlinear periodic lattices occur in a large variety of systems, such as biological molecules, nonlinear optical waveguides, solid-state systems and Bose-Einstein condensates. The underlying dynamics in these systems is dominated by the interplay between tunnelling between adjacent potential wells and nonlinearity. A balance between these two effects can result in a self-localized state: a lattice or 'discrete' soliton. Direct observation of lattice solitons has so far been limited to one-dimensional systems, namely in arrays of nonlinear optical waveguides. However, many fundamental features are expected to occur in higher dimensions, such as vortex lattice solitons, bright lattice solitons that carry angular momentum, and three-dimensional collisions between lattice solitons. Here, we report the experimental observation of two-dimensional (2D) lattice solitons. We use optical induction, the interference of two or more plane waves in a photosensitive material, to create a 2D photonic lattice in which the solitons form. Our results pave the way for the realization of a variety of nonlinear localization phenomena in photonic lattices and crystals. Finally, our observation directly relates to the proposed lattice solitons in Bose-Einstein condensates, which can be observed in optically induced periodic potentials.

We demonstrate that optical discrete solitons are possible in appropriately oriented biased photorefractive crystals. This can be accomplished in optically induced periodic waveguide lattices that are created via plane-wave interference. Our method paves the way towards the observation of entirely new families of discrete solitons. These include, for example, discrete solitons in two-dimensional self-focusing and defocusing lattices of different group symmetries, incoherently coupled vector discrete solitons, discrete soliton states in optical diatomic chains, as well as their associated collision properties and interactions. We also present results concerning transport anomalies of discrete solitons that depend on their initial momentum within the Brillouin zone.

In most classical fluids, shock waves are strongly dissipative, their energy being quickly lost through viscous damping. But in systems such as cold plasmas, superfluids and Bose-Einstein condensates, where viscosity is negligible or non-existent, a fundamentally different type of shock wave can emerge whose behaviour is dominated by dispersion rather than dissipation.Dispersive shock waves are difficult to study experimentally, and analytical solutions to the equations that govern them have only been found in one dimension (1D). By exploiting a wellknown, but little appreciated, correspondence between the behaviour of superfluids and nonlinear optical materials, we demonstrate an all-optical experimental platform for studying the dynamics of dispersive shock waves. This enables us to observe the propagation and nonlinear response of dispersive shock waves, including the interaction of colliding shock waves, in 1D and 2D. Our system offers a versatile and more accessible means for exploring superfluid-like and related dispersive phenomena.Unlike dissipative shock waves in ordinary gases/fluids, which have a well-defined shock front due to viscosity, dispersive superfluid-like shock waves have an oscillatory front. These oscillations result from two basic, and related, properties of the superfluid state: nonlinearity and coherence. Coherence results from cooling the fluid, so that the constituent particles of the condensate are perfectly correlated, while nonlinearity refers to the inter-particle interactions which make this correlation possible. For different reasons, these two properties also appear in nonlinear optics. While the relationship is well known in condensate community [e.g. nonlinear "atom optics" studies in Bose-Einstein condensates (BEC) 1-3 ], the relationship has been underappreciated from the opposite perspective. Here, we build on previous theoretical 4, 5 and experimental 6, 7 work on superfluid behavior in BEC to examine the optical equivalent of condensate shock waves. We demonstrate basic dispersive, dissipationless shock waves in one and two transverse dimensions, characterize their nonlinear properties, and reveal the nontrivial interactions when two such shocks collide.While dispersive shock waves in optics have been studied previously for temporal pulses in fibers 8-16 , they have not yet been considered in the spatial domain. In this case, the extra dimensional freedom allows consideration of wavefront geometry, which is shown to significantly affect shock propagation and interaction. The particular system considered here is a spatial one in which a continuous optical wave propagates in a nonlinear Kerr-like medium, mainly along the z-axis. To an excellent approximation, the slowly-varying amplitude ψ of such a field can be described by the nonlinear Schrödinger equation: