Recent explorations of quantized soliton transport in optical waveguides have thrust nonlinear topological pumping into the spotlight. In this work, we introduce a unified topological invariant applicable across both weakly and strongly nonlinear regimes. In the weak nonlinearity regime, where the nonlinear bands are well separated, the invariant reduces to the Abelian Chern number of the occupied nonlinear band. Consequently, the pumped charge is quantized to an integer value. As the nonlinearity increases, the nonlinear bands start to intertwine, leading to a situation where the invariant is expressed as the non-Abelian Chern number divided by the number of interacting bands. This could result in a fractional quantization of the pumped charge. Our unified-topological-invariant approach not only advances the understanding of the soliton dynamics but also provides implications for the future design of nonlinear topological systems.

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• Symmetry protected topological states
• Topological effects in photonic systems
• Topological phases of matter
• Solitons