Although the topology of a non-interacting linear system can be characterized by the Chern number that is computed from its eigenvectors, the topology of a nonlinear system is classified by the nonlinear Chern number, which utilizes the nonlinear extension of the eigenequation. In weakly nonlinear regions (that is, small amplitude), the nonlinear Chern number predicts the existence of edge modes corresponding to those in linear systems. Specifically, when nonlinear systems exhibit edge-localized steady states, both nonlinear and linear Chern numbers are non-zero (top). If we inject higher energy into the system and consider the eigenmodes with large amplitudes, the nonlinear band structure can become gapless. At such a gapless point, a nonlinearity-induced topological phase transition can occur, where topological boundary modes appear with the non-zero nonlinear Chern number (bottom). The nonlinearity-induced topological phases exhibit boundary modes that cannot be predicted from the linear Chern number. Therefore, such topological phases are genuinely unique to nonlinear systems.
