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Nonlinear Thouless pumps for bosons exhibit quantized pumping via solitonmotion, despite the lack of a meaningful notion of filled bands. However, thetheoretical underpinning of this quantization, as well as its relationship tothe Chern number, has thus far been lacking. Here we show that for low powersolitons, transport is dictated by the Chern number of the band from which thesoliton bifurcates. We do this by expanding the discrete nonlinearSchr\"odinger equation (equivalently, the Gross-Pitaevskii equation) in thebasis of Wannier states, showing that the soliton's position is dictated bythat of the Wannier state throughout the pump cycle. Furthermore, we describesoliton pumping in two dimensions.

The Chern Number Governs Soliton Motion in Nonlinear Thouless Pumps