Nonlinear skin modes and fixed points

We investigate a one-dimensional tight-binding lattice with asymmetrical couplings and various type of nonlinearities to study the nonlinear non-Hermitian skin effect. Our focus is on the exploration of nonlinear skin modes through a fixed-point perspective. The nonlinear interactions are shown to have no impact on the spectral region in a semi-infinite system; however, they induce considerable changes when boundaries are present. The spectrum under open boundary conditions is found not to be a subset of the corresponding spectrum under the semi-infinite boundary conditions. We identify distinctive features of nonlinear skin modes, such as degeneracy and power-energy discontinuity. Furthermore, we demonstrate that a family of localized modes that are neither skin nor scale-free localized modes is formed with the introduction of a coupling impurity. Additionally, we show that an impurity can induce discrete dark and antidark solitons.