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Non-Hermitian lattices with non-reciprocal couplings under open boundaryconditions are known to possess linear modes exponentially localized on oneedge of the chain. This phenomenon, dubbed non-Hermitian skin effect, inducesall input waves in the linearized limit of the system to unidirectionallypropagate toward the system's preferred boundary. Here we investigate the fateof the non-Hermitian skin effect in the presence of Kerr-type nonlinearitywithin the well-established Hatano-Nelson lattice model. Our method is to probethe presence of nonlinear stationary modes which are localized at the favorededge, when the Hatano-Nelson model deviates from the linear regime. Based onperturbation theory, we show that families of nonlinear skin modes emerge fromthe linear ones at any non-reciprocal strength. Our findings reveal that, inthe case of focusing nonlinearity, these families of nonlinear skin modes tendto exhibit enhanced localization, bridging the gap between weakly nonlinearmodes and the highly nonlinear states (discrete solitons) when approaching theanti-continuum limit with vanishing couplings. Conversely, for defocusingnonlinearity, these nonlinear skin modes tend to become more extended thantheir linear counterpart. To assess the stability of these solutions, weconduct a linear stability analysis across the entire spectrum of obtainednonlinear modes and also explore representative examples of their evolutiondynamics.

Skin modes in a nonlinear Hatano-Nelson model