  Higher-order topological insulators are unusual materials that can support
topologically protected states, whose dimensionality is lower than the
dimensionality of the structure at least by 2. Among the most intriguing
examples of such states are zero-dimensional corner modes existing in
two-dimensional higher-order insulators. In contrast to corner states, recently
discovered disclination states also belong to the class of higher-order
topological states, but are bound to the boundary of the disclination defect of
the higher-order topological insulator and can be predicted using the
bulk-disclination correspondence principle. Here, we present the first example
of the nonlinear photonic disclination state bifurcating from its linear
counterpart in the disclination lattice with a pentagonal or heptagonal core.
We show that nonlinearity allows to tune location of the disclination states in
the bandgap and notably affects their shapes. The structure of the disclination
lattice is crucial for stability of these nonlinear topological states: for
example, disclination states are stable in the heptagonal lattice and are
unstable nearly in the entire gap of the pentagonal lattice. Nonlinear
disclination states reported here are thresholdless and can be excited even at
low powers. Nonlinear zero-energy states coexisting in these structures with
disclination states are also studied. Our results suggest that disclination
lattices can be used in the design of various nonlinear topological functional
devices, while disclination states supported by them may play an important role
in applications, where strong field confinement together with topological
protection are important, such as the design of topological lasers and
enhancement of generation of high harmonics.
