Koopman Nonlinear Non-Hermitian Skin Effect

TL;DR

This paper introduces a Koopman-based method to characterize nonlinear non-Hermitian skin effects, defining localization through Koopman eigenfunctions in a lifted observable space, revealing unique dynamical signatures.

Contribution

It proposes a novel Koopman framework for identifying nonlinear skin effects, extending the concept beyond linear eigenstates to Koopman eigenfunctions in a lifted space.

Findings

Koopman eigenfunctions localize sharply on higher-order observables.

Localization in the lifted space influences sensitivity to boundary perturbations.

The framework distinguishes nonlinear skin effects from linear counterparts.

Abstract

Non-Hermitian skin effects are conventionally manifested as boundary localization of eigenstates in linear systems. In nonlinear settings, however, where eigenstates are no longer well defined, it becomes unclear how skin effects should be faithfully characterized. Here, we propose a Koopman-based characterization of nonlinear skin effects, in which localization is defined in terms of Koopman eigenfunctions in a lifted observable space, rather than physical states. Using a minimal nonlinear extension of the Hatano-Nelson model, we show that dominant Koopman eigenfunctions localize sharply on higher-order observables, in stark contrast to linear skin effects confined to linear observables. This lifted-space localization governs the sensitivity to boundary amplitude perturbations, providing a distinct dynamical signature of the nonlinear skin effect. Our results establish the Koopman framework as a natural setting in which skin effects unique to nonlinear non-Hermitian systems can be identified.