Recently, interest has surged in eigenvalue nonlinearity-based topological transport governed by the equation of auxiliary eigenvalues $H\mathrm{\Psi }=\omega S\left(\omega \right)\mathrm{\Psi }$ [Isobe et al., Phys. Rev. Lett. 132, 126601 (2024); Bai and Liang, Phys. Rev. A 111, 042201 (2025); Bai and Liang, Phys. Rev. A 112, 052207 (2025)] rather than the conventional Schrödinger equation $H\mathrm{\Psi }=E\mathrm{\Psi }$ in conservative settings, bit non-Hermitian generalizations remain uncharted. In this work, we are motivated to investigate the nonlinear Thouless pumping in a non-Hermitian and nonlinear Rice-Mele model. In particular, we uncover how non-Hermiticity 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 framework of the equation of auxiliary eigenvalues, directly linking nonlinear spectral characteristics to the bulk-boundary correspondence. Our findings reveal emergent phenomena arising from the interplay between nonlinearity and non-Hermiticity, providing key insights for the design of topological insulators and the controlled manipulation of quantum edge states in the real world.

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• Dynamics of nonlinear optical systems
• Geometric & topological phases
• Non-reciprocal propagation
• Nonlinear waves
• Schroedinger equation