Thouless pumping of solitons in a periodic lattice yields a quantized displacement per adiabatic cycle—an integer or fractional multiple of the lattice constant, determined by the Chern number. This quantization relies on a well-defined lattice constant. Quasiperiodic lattices, however, lack this defining scale, leaving the fate of soliton transport an open question. Here, we show that solitons can nevertheless undergo Thouless pumping in quasiperiodic lattices, but in a different form: the soliton reconstructs the local lattice potential via its density distribution, spontaneously forming an effective topological structure that induces quasiquantized pumping—the displacement per cycle is not strictly quantized, yet its long-time average converges to a Chern-determined value. This breaks the existing notion that the pumped displacement per cycle is quantized. By contrast, for solitons that only weakly modify the lattice, a critical linear band topology governs the transport but cannot enforce strict quantization; residual quasiperiodic perturbations induce nonquantized drift, destroying the quantized transport.

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• Bose-Einstein condensates
• Cold gases in optical lattices
• Quantum transport
• Topological phases of matter
• 1-dimensional systems
• Solitons