We study soliton Thouless pumping in an extended diagonal Aubry-André-Harper model with on-site nonlinearities and intersite nonlinearities. We show that the intersite nonlinearities can make solitons acquire anomalous transport distances far beyond the ones predicted by the linear bands, and the quantized displacements can be engineered well. We uncover that nonlinear instabilities require lower limits on sweeping rates for soliton pumping, challenging the common notion that slower modulation enables a more favorable realization of topological transport. The nonlinear interactions between solitons make multisoliton pumping generally lack the robustness characteristic of Thouless pumping as linear systems. Our results provide many possibilities to engineer topological pumping by nonlinearities and are also a step forward for realizing applications of soliton pumping.

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• Geometric & topological phases
• Quantum transport
• Transport phenomena
• Solitons