Nonlinear topological pumping of edge solitons

We study how nonlinear strength affects topological pumping of edge solitons by using the Gross--Pitaevskii equation. For weak nonlinear strength, the introduction of nonlinearity breaks the symmetry of the energy spectrum, which makes the topological pumping from the left edge to the right edge differ from the inverse process. For moderate nonlinear strength, self-crossing structures appear in the spectrum, and the right-to-left adiabatic pumping channel is destroyed, only left-to-right topological pumping can be achieved under slow modulation. As the nonlinear strength further increases, although left-to-right topological pumping in one pumping cycle also breaks down, we find that a thin soliton, located at a single left edge, can be mixed with the bulk soliton, and hybridized topological pumping of edge and bulk solitons can be realized after multiple pumping cycles. For stronger nonlinear strength, edge solitons are self-trapped and all topological pumping channels are shut down. Our work could trigger further studies of the interplay between nonlinearity and topology.