Fractional Thouless pumping of solitons: A unique manifestation of bulk-edge correspondence of nonlinear eigenvalue problems

Recent foundational studies have established the bulk-edge correspondence for nonlinear eigenvalue problems using auxiliary eigenvalues $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}\mathrm{\ensuremath{\Psi}}=\ensuremath{\omega}S(\ensuremath{\omega})\mathrm{\ensuremath{\Psi}}$, spanning both linear [T. Isobe et al., Phys. Rev. Lett. 132, 126601 (2024)] and nonlinear [C. Bai and Z. Liang, Phys. Rev. A 111, 042201 (2025)] Hamiltionians. This progress prompts a fundamental question: Can eigenvalue nonlinearity generate observable physical phenomena absent in conventional approaches ($\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}\mathrm{\ensuremath{\Psi}}=E\mathrm{\ensuremath{\Psi}}$)? In this work, we address this question by demonstrating a uniquely nonlinear manifestation of the bulk-edge correspondence: fractional Thouless pumping of solitons. Through systematic investigation of nonlinear Thouless pumping in an extended Rice-Mele model incorporating next-nearest-neighbor (NNN) couplings, we uncover that NNN interaction parameters can induce fractional topological phases---even in the presence of quantized topological invariants as predicted by conventional linear approaches. Crucially, these fractional phases are naturally explained within the auxiliary eigenvalue framework, directly linking nonlinear spectral characteristics to the bulk-boundary correspondence. Our findings reveal emergent phenomena arising from the interplay between nonlinearity and NNN couplings, providing key insights for the design of topological insulators and the controlled manipulation of quantum edge states in nonlinear regimes.