This paper introduces the concept of a nonlinear Chern number to characterize topological phases in nonlinear systems, extending the notion of topology beyond linear systems. The authors demonstrate a nonlinear extension of the Chern number based on nonlinear eigenvalue problems, proving the existence of a bulk-boundary correspondence even in strongly nonlinear regimes. They reveal nonlinearity-induced topological phase transitions where the presence of topological edge modes depends on the amplitude of oscillatory modes. A minimal model of a nonlinear Chern insulator is analyzed, showcasing the amplitude dependence of the nonlinear Chern number and confirming the bulk-boundary correspondence. This work reveals genuinely nonlinear topological phases adiabatically disconnected from the linear regime, opening avenues for exploring nonlinear topological materials.
