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Higher-order topological insulators (HOTIs) are recently discoveredtopological phases, possessing symmetry-protected corner states with fractionalcharges. An unexpected connection between these states and the seeminglyunrelated phenomenon of bound states in the continuum (BICs) was recentlyunveiled. When nonlinearity is added to a HOTI system, a number offundamentally important questions arise. For example, how does nonlinearitycouple higher-order topological BICs with the rest of the system, includingcontinuum states? In fact, thus far BICs in nonlinear HOTIs have remainedunexplored. Here, we demonstrate the interplay of nonlinearity, higher-ordertopology, and BICs in a photonic platform. We observe topological corner stateswhich, serendipitously, are also BICs in a laser-written second-ordertopological lattice. We further demonstrate nonlinear coupling with edge statesat a low nonlinearity, transitioning to solitons at a high nonlinearity.Theoretically, we calculate the analog of the Zak phase in the nonlinearregime, illustrating that a topological BIC can be actively tuned by bothfocusing and defocusing nonlinearities. Our studies are applicable to othernonlinear HOTI systems, with promising applications in emerging topology-drivendevices.

Nonlinear control of photonic higher-order topological bound states in the continuum