Independence of one-period shift from the nonlocality parameter in strong nonlocal Thouless pumping

Establish whether, for adiabatic Thouless pumping of one‑hump solitons governed by the nonlocal nonlinear Schrödinger equation with a normalized kernel K_d(x) in a periodic lattice created by two slowly sliding commensurate sublattices, the displacement of the soliton center over one driving period becomes independent of the nonlocality parameter d in the regime where the nonlocality scale d and the soliton width both greatly exceed the lattice period.

Background

The paper analyzes Thouless pumping of solitons in a nonlocal nonlinear medium modeled by a nonlinear Schrödinger equation with an exponential kernel and a dynamically varying lattice formed by two sliding sublattices. After presenting results for this specific kernel, the authors discuss general kernels K_d(x) that satisfy a normalization condition and consider limits of weak and strong nonlocality.

In the strong nonlocality limit the kernel’s maximum scales to zero and both the nonlocality scale d and the soliton width can become much larger than the lattice period X. The authors explicitly conjecture that, in this regime, the one-period shift during pumping becomes independent of d, and note that this requires further numerical investigation beyond the scope of the paper.

References

Since d and ℓ become now much larger than the period of the lattice X: d, ℓ≫X, it can be conjectured that the one-period shift becomes independent of the nonlocality parameter d [this is what we observed in Figs. 5(a) and (b) for the nonlocal media studied in this work]. These conjectures, however require further thorough numerical study, going beyond the scope of this work.

— Thouless pumping of solitons in a nonlocal medium (2507.05840 - Ye et al., 8 Jul 2025) in Section "Some general considerations" (near the end, just before Conclusion)