Higher-order topological insulators (HOTIs) are recently discovered topological phases, possessing symmetry-protected corner states with fractional charges. An unexpected connection between these states and the seemingly unrelated phenomenon of bound states in the continuum (BICs) was recently unveiled. When nonlinearity is added to the HOTI system, a number of fundamentally important questions arise. For example, how does nonlinearity couple higher-order topological BICs with the rest of the system, including continuum states? In fact, thus far BICs in nonlinear HOTIs have remained unexplored. Here we unveil the interplay of nonlinearity, higher-order topology, and BICs in a photonic platform. We observe topological corner states that are also BICs in a laser-written second-order topological lattice and further demonstrate their nonlinear coupling with edge (but not bulk) modes under the proper action of both self-focusing and defocusing nonlinearities. Theoretically, we calculate the eigenvalue spectrum and analog of the Zak phase in the nonlinear regime, illustrating that a topological BIC can be actively tuned by nonlinearity in such a photonic HOTI. Our studies are applicable to other nonlinear HOTI systems, with promising applications in emerging topology-driven devices.

We report the first observation of Riemann (simple) waves, which play a crucial role for understanding the dynamics of any shock-bearing system. This was achieved by properly tailoring the phase of an ultrashort light pulse injected into a highly nonlinear fiber. Optical Riemann waves are found to evolve in excellent quantitative agreement with the remarkably simple inviscid Burgers equation, whose applicability in physical systems is often challenged by viscous or dissipative effects. Our method allows us to further demonstrate a viable novel route to efficiently control the shock formation by the proper shaping of a laser pulse phase. Our results pave the way towards the experimental study, in a convenient benchtop setup, of complex physical phenomena otherwise difficult to access.

We demonstrate theoretically and experimentally that a finite Airy beam changes its trajectory while maintaining its acceleration in nonlinear photorefractive media. During this process, the spatial spectrum reshapes dramatically, leading to negative (or positive) spectral defects on the initial spectral distribution under a self-focusing (or defocusing) nonlinearity.

We introduce the concept of spatial spectral phase gradient, and demonstrate, both theoretically and experimentally, how this concept could be employed for generating single-and multi-path self-accelerating beams. In particular, we show that the trajectories of the accelerating beams are determined a priori by different key spatial frequencies through direct spectrum-to-distance mapping. In the non-paraxial regime, our results clearly illustrate the breakup of Airy beams from a different perspective, and demonstrate how circular, elliptic or hyperbolic accelerating beams can be created by judiciously engineering the spectral phase. Furthermore, we found that the accelerating beams still follow the predicted trajectory also for vectorial wavefronts. Our approach not only generalizes the idea of Fourier-space beam engineering along arbitrary convex trajectories, but also offers new possibilities for beam/pulse manipulation not achievable through standard direct real-space approaches or by way of time-domain phase modulation. PACS number(s): 42.25.-p, 03.50.-z