  We study the nonlinear perturbation of a high-order exceptional point (EP) of
the order equal to the system site number $L$ in a Hatano-Nelson model with
unidirectional hopping and Kerr nonlinearity. Notably, We find a class of
discrete breathers that aggregate to one boundary, here named as skin discrete
breathers (SDBs). The nonlinear spectrum of these SDBs shows a hierarchical
power-law scaling near the EP. Specifically, the response of nonlinear energy
to the perturbation is given by $E_m\propto \varGamma^{\alpha_{m}}$, where
$\alpha_m=3^{m-1}$ is the power with $m=1,\cdots,L$ labeling the nonlinear
energy bands. This is in sharp contrast to the $L$-th root of a linear
perturbation in general. These SDBs decay in a double-exponential manner,
unlike the edge states or skin modes in linear systems, which decay
exponentially. Furthermore, these SDBs can survive over the full range of
nonlinearity strength and are continuously connected to the self-trapped states
in the limit of large nonlinearity. They are also stable, as confirmed by a
defined nonlinear fidelity of an adiabatic evolution from the stability
analysis. As nonreciprocal nonlinear models may be experimentally realized in
various platforms, such as the classical platform of optical waveguides, where
Kerr nonlinearity is naturally present, and the quantum platform of optical
lattices with Bose-Einstein condensates, our analytical results may inspire
further exploration of the interplay between nonlinearity and non-Hermiticity,
particularly on high-order EPs, and benchmark the relevant simulations.
