  Although topological phenomena attract growing interest not only in linear
systems but also in nonlinear systems, the bulk-edge correspondence under the
nonlinearity of eigenvalues has not been established so far. We address this
issue by introducing auxiliary eigenvalues. We reveal that the topological edge
states of auxiliary eigenstates are topologically inherited as physical edge
states when the nonlinearity is weak but finite (i.e., auxiliary eigenvalues
are monotonic as for the physical one). This result leads to the bulk-edge
correspondence with the nonlinearity of eigenvalues.
