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