Skin Solitons in Non-Hermitian Media

• Skin solitons are nonlinear localized excitations that arise in non-Hermitian media, characterized by asymmetric boundary localization and the influence of nonreciprocal dynamics.
• They form when the NHSE-induced piling of linear eigenmodes interacts with nonlinear Kerr self-trapping, resulting in sharp localization or tilted propagation profiles.
• Realizations in platforms such as temporal photonic lattices, stacked SSH-like models, and mechanical metamaterials demonstrate their diverse applications and critical threshold behaviors.

Skin solitons are nonlinear localized excitations in non-Hermitian or nonreciprocal media whose existence, profile, or dynamics are governed by the non-Hermitian skin effect (NHSE), by band nonreciprocity, or by nonreciprocal driving. In lattice systems they arise when asymmetric transport, which under open boundary conditions piles linear eigenmodes at a preferred edge or corner, interacts with Kerr self-trapping or other nonlinear mechanisms. Recent work has established several concrete realizations: nonlinear continuations of linear skin modes in the one-dimensional Hatano–Nelson model (Manda et al., 2023), corner-directed skin solitons in two-dimensional nonlinear Hatano–Nelson lattices (Kokkinakis et al., 18 May 2025), the distinction between skin-mode-assisted solitons and nonreciprocity-dressed solitons in stacked SSH-like lattices (Li et al., 4 Aug 2025), experimentally demonstrated skin solitons in temporal photonic feedforward lattices (Wang et al., 2024), and one-way topological solitons obtained by quenching non-Hermitian skin waves in active mechanical metamaterials (Veenstra et al., 2023).

1. Conceptual scope and defining features

In the narrowest usage, a skin soliton is a nonlinear stationary or propagating state that inherits its spatial asymmetry from the NHSE. In the linear limit of a nonreciprocal lattice under open boundary conditions, all eigenmodes can become exponentially localized at a preferred boundary. Nonlinearity then either sharpens that localization, inhibits the boundary-directed transport, or reorganizes it into a stable self-localized state. This mechanism is explicit in Hatano–Nelson-type lattices and in temporal photonic feedforward systems (Manda et al., 2023).

The recent literature also shows that the term cannot be used as a synonym for any asymmetric soliton in a non-Hermitian lattice. In the stacked SSH-like model, two distinct nonlinear phases are identified: skin-mode-assisted solitons (SMASs), which originate from the skin effect under open boundary conditions, and nonreciprocity-dressed solitons (NRDSs), which originate from bulk band nonreciprocity and persist under periodic boundary conditions (Li et al., 4 Aug 2025). In active mechanical metamaterials, the relevant nonlinear objects are one-way topological solitons produced when non-Hermitian skin waves, unstable in the linear regime, are stabilized by a bistable substrate (Veenstra et al., 2023).

A plausible implication is that “skin soliton” is best treated as a mechanism-sensitive term. Boundary localization, bulk dispersion asymmetry, and nonreciprocal driving can all generate nonlinear asymmetric solitary states, but they are not equivalent mechanisms.

2. Canonical theoretical frameworks

The canonical one-dimensional setting is the nonlinear Hatano–Nelson chain. Its dynamics are governed by

$$i\,\frac{d\psi_n}{dt} =-J_R\,\psi_{n+1}-J_L\,\psi_{n-1} +g\,|\psi_n|^2\,\psi_n, \qquad \psi_0=\psi_{N+1}=0,$$

with nonreciprocal hoppings $J_R\neq J_L$ and Kerr coefficient $g$. Under the stationary ansatz $\psi_n(t)=\phi_n e^{-i\mu t}$, one obtains

$$-J_R\,\phi_{n+1}-J_L\,\phi_{n-1} +g\,|\phi_n|^2\,\phi_n =\mu\,\phi_n.$$

In the linear limit, the open-boundary eigenmodes are skin modes exponentially localized at one edge; nonlinear stationary modes continue from these states (Manda et al., 2023).

The two-dimensional nonlinear Hatano–Nelson lattice generalizes this picture to a square $N\times N$ lattice with open boundaries:

$$i\frac{d\psi_{n_x,n_y}}{dz} +J_x^+\,\psi_{n_x+1,n_y} +J_x^-\,\psi_{n_x-1,n_y} +J_y^+\,\psi_{n_x,n_y+1} +J_y^-\,\psi_{n_x,n_y-1} +g\,|\psi_{n_x,n_y}|^2\psi_{n_x,n_y}=0,$$

with $J_x^\pm=e^{\mp h_x}$ and $J_y^\pm=e^{\mp h_y}$. For $h_x>0$ and $J_R\neq J_L$0, linear eigenmodes pile up at the corner $J_R\neq J_L$1, and nonlinear stationary solutions become two-dimensional skin solitons tilted toward that preferred corner (Kokkinakis et al., 18 May 2025).

A different framework is the stacked SSH-like model with nonreciprocal intercell hopping $J_R\neq J_L$2, $J_R\neq J_L$3, reciprocal $J_R\neq J_L$4 and $J_R\neq J_L$5, and Kerr nonlinearity $J_R\neq J_L$6. Its nonlinear eigenproblem is

$J_R\neq J_L$7

A Wannier-function-based effective Hamiltonian separates the role of real-space dispersive hoppings $J_R\neq J_L$8 from local nonlinear overlaps, making it possible to distinguish NHSE-driven suppression of edge dispersion from nonreciprocity-driven enhancement of bulk dispersion (Li et al., 4 Aug 2025).

In temporal photonics, skin solitons are realized in an effective Floquet lattice. The one-step map for the two-component field $J_R\neq J_L$9 combines asymmetric gain/loss $g$0, coupler mixing angle $g$1, and an effective Kerr coefficient $g$2 produced by optoelectronic feedback. A Floquet ansatz $g$3 yields a nonlinear eigenvalue problem in which the Floquet phase $g$4 must lie in a gap of the linear band structure (Wang et al., 2024).

The nonreciprocal topological-soliton setting differs structurally but remains closely related. Its continuum limit is a non-reciprocal sine–Gordon equation,

$g$5

where $g$6 is a local non-reciprocal odd drive, $g$7 is damping, and $g$8 is an optional uniform drive (Veenstra et al., 2023).

3. Formation mechanisms and threshold structure

In the two-dimensional nonlinear Hatano–Nelson lattice, the central dynamical competition is explicit: linear NHSE drives an excitation toward the preferred corner, whereas Kerr nonlinearity tends to arrest diffraction and lock amplitude at the launch site. In the Hermitian limit, the self-trapping threshold satisfies roughly

$g$9

and with asymmetry one obtains

$\psi_n(t)=\phi_n e^{-i\mu t}$0

Increasing asymmetry raises $\psi_n(t)=\phi_n e^{-i\mu t}$1. Numerically, excitations launched near the preferred corner require amplitudes close to this estimate, whereas central or opposite-corner launches must also overcome the NHSE drift; for sufficiently large $\psi_n(t)=\phi_n e^{-i\mu t}$2, the bulk can never self-trap, numerically up to $\psi_n(t)=\phi_n e^{-i\mu t}$3 (Kokkinakis et al., 18 May 2025).

In the stacked SSH-like model, the threshold structure separates into phases. Under open boundary conditions there are nonlinear perturbative skin modes for $\psi_n(t)=\phi_n e^{-i\mu t}$4, SMAS for $\psi_n(t)=\phi_n e^{-i\mu t}$5 but $\psi_n(t)=\phi_n e^{-i\mu t}$6, and NRDS for $\psi_n(t)=\phi_n e^{-i\mu t}$7. In the focusing regime $\psi_n(t)=\phi_n e^{-i\mu t}$8,

$\psi_n(t)=\phi_n e^{-i\mu t}$9

while

$$-J_R\,\phi_{n+1}-J_L\,\phi_{n-1} +g\,|\phi_n|^2\,\phi_n =\mu\,\phi_n.$$0

marks a transition between two molecular-limit soliton types. As $$-J_R\,\phi_{n+1}-J_L\,\phi_{n-1} +g\,|\phi_n|^2\,\phi_n =\mu\,\phi_n.$$1 increases, $$-J_R\,\phi_{n+1}-J_L\,\phi_{n-1} +g\,|\phi_n|^2\,\phi_n =\mu\,\phi_n.$$2 decreases because skin-mode localization suppresses edge dispersion, whereas $$-J_R\,\phi_{n+1}-J_L\,\phi_{n-1} +g\,|\phi_n|^2\,\phi_n =\mu\,\phi_n.$$3 increases because band nonreciprocity enhances the most dispersive bulk channel (Li et al., 4 Aug 2025).

In the temporal photonic feedforward lattice, localization is controlled by the balance between nonreciprocal amplification/attenuation and the nonlinear phase shift. For moderate $$-J_R\,\phi_{n+1}-J_L\,\phi_{n-1} +g\,|\phi_n|^2\,\phi_n =\mu\,\phi_n.$$4 and $$-J_R\,\phi_{n+1}-J_L\,\phi_{n-1} +g\,|\phi_n|^2\,\phi_n =\mu\,\phi_n.$$5, the edge skin-soliton profile is approximately exponential,

$$-J_R\,\phi_{n+1}-J_L\,\phi_{n-1} +g\,|\phi_n|^2\,\phi_n =\mu\,\phi_n.$$6

with weak-nonlinear estimate

$$-J_R\,\phi_{n+1}-J_L\,\phi_{n-1} +g\,|\phi_n|^2\,\phi_n =\mu\,\phi_n.$$7

The localization length therefore shrinks as $$-J_R\,\phi_{n+1}-J_L\,\phi_{n-1} +g\,|\phi_n|^2\,\phi_n =\mu\,\phi_n.$$8 increases, since the nonlinear phase lifts the Floquet phase deeper into the gap (Wang et al., 2024).

A complementary threshold result is obtained in non-Hermitian waveguide arrays with Kerr nonlinearity. For single-channel excitation $$-J_R\,\phi_{n+1}-J_L\,\phi_{n-1} +g\,|\phi_n|^2\,\phi_n =\mu\,\phi_n.$$9, the rescaled optical nonlinearity coefficient is $N\times N$0, and symbolic regression yields the semi-analytical soliton existence boundary

$N\times N$1

This relation separates regions in parameter space where a localized skin soliton forms from regions where the beam diffracts (Hou et al., 1 Jul 2026).

4. Stationary states, continuation, and spatial structure

In the one-dimensional Hatano–Nelson model, nonlinear skin modes emerge perturbatively from every linear skin mode at any non-reciprocal strength. For weak nonlinearity, the nonlinear eigenvalue satisfies

$N\times N$2

with $N\times N$3 the corresponding left eigenvector. Focusing nonlinearity enhances localization, while defocusing nonlinearity makes the nonlinear skin modes more extended than their linear counterparts (Manda et al., 2023).

The same model admits an anti-continuum description. When $N\times N$4, the exact edge discrete soliton is

$N\times N$5

for $N\times N$6 if $N\times N$7 and $N\times N$8 if $N\times N$9. For small but finite couplings, the solution acquires a tail $$i\frac{d\psi_{n_x,n_y}}{dz} +J_x^+\,\psi_{n_x+1,n_y} +J_x^-\,\psi_{n_x-1,n_y} +J_y^+\,\psi_{n_x,n_y+1} +J_y^-\,\psi_{n_x,n_y-1} +g\,|\psi_{n_x,n_y}|^2\psi_{n_x,n_y}=0,$$0 with

$$i\frac{d\psi_{n_x,n_y}}{dz} +J_x^+\,\psi_{n_x+1,n_y} +J_x^-\,\psi_{n_x-1,n_y} +J_y^+\,\psi_{n_x,n_y+1} +J_y^-\,\psi_{n_x,n_y-1} +g\,|\psi_{n_x,n_y}|^2\psi_{n_x,n_y}=0,$$1

This provides an explicit bridge between weakly nonlinear skin modes and strongly localized discrete solitons (Manda et al., 2023).

In two dimensions, stationary states are sought as $$i\frac{d\psi_{n_x,n_y}}{dz} +J_x^+\,\psi_{n_x+1,n_y} +J_x^-\,\psi_{n_x-1,n_y} +J_y^+\,\psi_{n_x,n_y+1} +J_y^-\,\psi_{n_x,n_y-1} +g\,|\psi_{n_x,n_y}|^2\psi_{n_x,n_y}=0,$$2, leading to

$$i\frac{d\psi_{n_x,n_y}}{dz} +J_x^+\,\psi_{n_x+1,n_y} +J_x^-\,\psi_{n_x-1,n_y} +J_y^+\,\psi_{n_x,n_y+1} +J_y^-\,\psi_{n_x,n_y-1} +g\,|\psi_{n_x,n_y}|^2\psi_{n_x,n_y}=0,$$3

The soliton power is

$$i\frac{d\psi_{n_x,n_y}}{dz} +J_x^+\,\psi_{n_x+1,n_y} +J_x^-\,\psi_{n_x-1,n_y} +J_y^+\,\psi_{n_x,n_y+1} +J_y^-\,\psi_{n_x,n_y-1} +g\,|\psi_{n_x,n_y}|^2\psi_{n_x,n_y}=0,$$4

Typical $$i\frac{d\psi_{n_x,n_y}}{dz} +J_x^+\,\psi_{n_x+1,n_y} +J_x^-\,\psi_{n_x-1,n_y} +J_y^+\,\psi_{n_x,n_y+1} +J_y^-\,\psi_{n_x,n_y-1} +g\,|\psi_{n_x,n_y}|^2\psi_{n_x,n_y}=0,$$5–$$i\frac{d\psi_{n_x,n_y}}{dz} +J_x^+\,\psi_{n_x+1,n_y} +J_x^-\,\psi_{n_x-1,n_y} +J_y^+\,\psi_{n_x,n_y+1} +J_y^-\,\psi_{n_x,n_y-1} +g\,|\psi_{n_x,n_y}|^2\psi_{n_x,n_y}=0,$$6 families have a minimum power threshold $$i\frac{d\psi_{n_x,n_y}}{dz} +J_x^+\,\psi_{n_x+1,n_y} +J_x^-\,\psi_{n_x-1,n_y} +J_y^+\,\psi_{n_x,n_y+1} +J_y^-\,\psi_{n_x,n_y-1} +g\,|\psi_{n_x,n_y}|^2\psi_{n_x,n_y}=0,$$7. For corner-centered and bulk-centered solitons, fitted thresholds are

$$i\frac{d\psi_{n_x,n_y}}{dz} +J_x^+\,\psi_{n_x+1,n_y} +J_x^-\,\psi_{n_x-1,n_y} +J_y^+\,\psi_{n_x,n_y+1} +J_y^-\,\psi_{n_x,n_y-1} +g\,|\psi_{n_x,n_y}|^2\psi_{n_x,n_y}=0,$$8

All skin solitons are strongly asymmetric, with maximal amplitude tilted toward the preferred corner (Kokkinakis et al., 18 May 2025).

In temporal photonics, nontrivial Floquet skin-soliton solutions exist only when the nonlinear phase shifts place $$i\frac{d\psi_{n_x,n_y}}{dz} +J_x^+\,\psi_{n_x+1,n_y} +J_x^-\,\psi_{n_x-1,n_y} +J_y^+\,\psi_{n_x,n_y+1} +J_y^-\,\psi_{n_x,n_y-1} +g\,|\psi_{n_x,n_y}|^2\psi_{n_x,n_y}=0,$$9 in a gap of the linear band structure. In waveguide arrays, stationary nonlinear modes satisfy

$J_x^\pm=e^{\mp h_x}$0

with Dirichlet edges. In the non-Hermitian regime, every linear skin mode continuously deforms into a nonlinear counterpart, and the peak moves toward the boundary as $J_x^\pm=e^{\mp h_x}$1 increases; the paper also reports path independence, meaning that introducing nonlinearity before non-Hermiticity or vice versa yields the same set of skin solitons (Hou et al., 1 Jul 2026).

5. Dynamics, propagation, and stability

Direct propagation in the two-dimensional nonlinear Hatano–Nelson lattice shows the antagonism between NHSE transport and self-trapping in real time. In a $J_x^\pm=e^{\mp h_x}$2 lattice with $J_x^\pm=e^{\mp h_x}$3, the linear system exhibits corner-directed drift, whereas for $J_x^\pm=e^{\mp h_x}$4 the nonlinear system first self-traps up to $J_x^\pm=e^{\mp h_x}$5 and then splits into a trapped component and a skin-driven component. For $J_x^\pm=e^{\mp h_x}$6, the mean position $J_x^\pm=e^{\mp h_x}$7 stalls and the spread $J_x^\pm=e^{\mp h_x}$8 peaks because of this two-lobe structure. Integrations with stationary initial data show that solitons near the preferred corner remain stable with $J_x^\pm=e^{\mp h_x}$9 even for $J_y^\pm=e^{\mp h_y}$0, while those farther away develop deviations $J_y^\pm=e^{\mp h_y}$1-$J_y^\pm=e^{\mp h_y}$2 (Kokkinakis et al., 18 May 2025).

The one-dimensional Hatano–Nelson model admits a non-Hermitian Bogoliubov–de Gennes linearization. Numerically, families of focusing and defocusing nonlinear skin modes are spectrally stable near the linear limit and near the anti-continuum limit, while intermediate-amplitude regions may show complex quartets $J_y^\pm=e^{\mp h_y}$3, signaling oscillatory instability (Manda et al., 2023).

The stacked SSH-like model exhibits a sharper phase-resolved stability structure. At the NPSM-to-SMAS transition, linear stability analysis of the Bogoliubov-type matrix shows that the maximum $J_y^\pm=e^{\mp h_y}$4 briefly opens an instability gap and then closes again after a true SMAS forms, producing a stability reentrance. Under periodic boundary conditions, SMASs disappear and only NRDSs remain above $J_y^\pm=e^{\mp h_y}$5; their transition shows no accompanying instability gap (Li et al., 4 Aug 2025).

In the non-reciprocal sine–Gordon setting, an adiabatic perturbation treatment yields the terminal velocity

$J_y^\pm=e^{\mp h_y}$6

The far-field linearized dispersion is

$J_y^\pm=e^{\mp h_y}$7

so robustness requires $J_y^\pm=e^{\mp h_y}$8 for all $J_y^\pm=e^{\mp h_y}$9, and high-$h_x>0$0 modes first go unstable when $h_x>0$1. The physical message is that linear skin waves either blow up or die out, whereas the bistable nonlinearity saturates the gain/loss and converts them into finite-velocity one-way solitons (Veenstra et al., 2023).

6. Implementations, variants, and interpretive issues

The most direct experimental realization to date is the effective Kerr nonlinear temporal photonic lattice built from coupled fiber loops and optoelectronic feedback. The reported parameters include average loop length $h_x>0$2 km, imbalance $h_x>0$3 km, coupling angle $h_x>0$4, net long-loop gain $h_x>0$5, net short-loop loss $h_x>0$6, and tunable Kerr coefficient up to $h_x>0$7. Edge skin solitons appear at $h_x>0$8, bulk solitons at $h_x>0$9, and the nonlinearity-controlled NHSE is used to implement an optical router with a flexibly tuned output port, robust against phase disorder (Wang et al., 2024).

Other proposed platforms include optical lattices with Kerr nonlinearity and engineered feed-forward coupling, including temporal synthetic dimensions; mechanical or electrical topolectrical circuits with active elements and onsite saturable nonlinearities; and active rotor chains capable of unidirectional signaling and filtering (Li et al., 4 Aug 2025). In the active waveguide demonstration of non-reciprocal topological solitons, a chain of $J_R\neq J_L$00 rotor-motors transmits information via one-way solitons at steady velocity $J_R\neq J_L$01, and opposite bias segments yield a unidirectional low-pass filter with critical interval

$J_R\neq J_L$02

for pulse collision and annihilation (Veenstra et al., 2023).

The fixed-point approach to nonlinear skin modes emphasizes that boundary conditions are not a technical detail. In a one-dimensional asymmetric lattice with local nonlinearity, the semi-infinite spectrum is independent of the nonlinearity, whereas the open-boundary spectrum is strongly modified; the spectrum under open boundary conditions is generally not a subset of the corresponding spectrum under semi-infinite boundary conditions. The same analysis identifies degeneracy, power-energy discontinuity, and a family of localized modes that are neither skin nor scale-free localized modes when a coupling impurity is introduced. In the Kerr case, a single impurity supports discrete dark and anti-dark solitons on a uniform background, governed by the cubic defect equation

$J_R\neq J_L$03

with dark solitons for $J_R\neq J_L$04 and anti-dark solitons for $J_R\neq J_L$05 (Yuce, 2024).

A common misconception is that asymmetric localization by itself establishes an NHSE-mediated soliton mechanism. The distinction between SMASs and NRDSs shows otherwise: SMASs rely on boundary-localized skin modes and disappear under periodic boundary conditions, whereas NRDSs persist under periodic boundary conditions because they originate from bulk band nonreciprocity (Li et al., 4 Aug 2025). This suggests that rigorous classification of skin solitons should specify at least three ingredients: the boundary condition, the source of asymmetry in dispersion or transport, and the nonlinear mechanism that arrests or reshapes the underlying skin dynamics.