Skin-Mode-Assisted Solitons in Non-Hermitian Lattices

• SMASs are nonlinear localized states defined by the interplay of NHSE-induced skin mode accumulation and Kerr-type self-trapping, resulting in boundary-pinned solitons.
• SMASs manifest in diverse settings from 1D Hatano-Nelson chains to 2D lattices and temporal photonic feedforward arrays, highlighting boundaries’ role in lowering soliton formation thresholds.
• The phenomenon distinguishes SMASs from nonreciprocity-dressed solitons by requiring open boundary conditions and showcasing stability reentrance along with asymmetric localization profiles.

Skin-Mode-Assisted Solitons (SMASs) are nonlinear localized states in non-Hermitian lattices whose formation is fundamentally tied to skin-mode localization induced by the non-Hermitian skin effect (NHSE). In the formulation that explicitly separates mechanisms, SMASs are a distinct soliton phase rooted in the skin effect rather than in band nonreciprocity alone; they are localized at the sites or boundaries where skin modes accumulate under open boundary conditions and arise through the interplay of asymmetric hopping and Kerr-type self-trapping (Li et al., 4 Aug 2025). Across one-dimensional Hatano-Nelson chains, nonlinear optical waveguide arrays, two-dimensional lattices, and temporal photonic feedforward lattices, the term denotes edge- or corner-pinned nonlinear states whose existence, thresholds, spatial asymmetry, and stability are controlled by both non-reciprocal transport and nonlinearity (Hou et al., 1 Jul 2026).

1. Definition and conceptual scope

In non-Hermitian lattices with non-reciprocal couplings, the NHSE causes eigenmodes under open boundary conditions to accumulate exponentially at a preferred boundary. SMASs are the nonlinear localized states that inherit this boundary preference from skin modes while acquiring self-trapping from nonlinearity. A precise characterization states that SMASs are “a distinct soliton phase in non-Hermitian lattices whose formation is fundamentally tied to the presence of skin-mode localization,” and that they are “always localized at the sites/boundaries where skin modes accumulate under open boundary conditions” (Li et al., 4 Aug 2025).

This definition is narrower than the broader category of nonlinear skin modes. In the nonlinear Hatano-Nelson model, families of nonlinear stationary modes can be continued from linear skin modes at any non-reciprocal strength, and in the focusing case these families become more localized and connect to highly nonlinear discrete solitons in the anti-continuum limit (Manda et al., 2023). Within that lineage, SMASs denote the regime in which skin-mode localization is not merely a perturbative remnant of the linear problem but an active ingredient lowering the formation threshold and shaping the final soliton profile (Li et al., 4 Aug 2025).

A recurrent point in the literature is that SMASs require boundaries in the sense relevant to the NHSE. One formulation states that SMASs only exist under open boundary conditions, whereas nonreciprocity-dressed solitons (NRDSs) persist under periodic boundary conditions and are governed by bulk band nonreciprocity rather than by the NHSE (Li et al., 4 Aug 2025). This distinction is central to current usage of the term.

2. Canonical models and mathematical formulation

A standard one-dimensional setting is the nonlinear Hatano-Nelson lattice with Kerr nonlinearity. In optical waveguide arrays, the governing equation is

$$i \frac{d}{dz} U_n = C_L U_{n+1} + C_R U_{n-1} + \sigma |U_n|^2 U_n,$$

where $U_n$ is the complex field amplitude at site $n$, $z$ is the propagation distance, $C_L$ and $C_R$ are nonreciprocal coupling coefficients, and $\sigma$ is the Kerr nonlinearity parameter. The degree of non-Hermiticity is quantified by

$h = C_R - C_L.$

Open boundary conditions are used to observe the NHSE (Hou et al., 1 Jul 2026).

A related one-dimensional formulation uses

$$i\frac{d\psi_n}{d\tau} = C(\psi_{n+1} + t\psi_{n-1}) + \sigma |\psi_n|^2\psi_n,$$

with stationary solutions

$E u_n = C(u_{n+1} + t u_{n-1}) + \sigma|u_n|^2u_n,$

under $U_n$0. In this representation, $U_n$1 encodes the non-reciprocal parameter and $U_n$2 distinguishes focusing and defocusing Kerr response (Manda et al., 2023).

Two-dimensional generalization is provided by the nonlinear Hatano-Nelson lattice on an $U_n$3 square lattice,

$U_n$4

with open boundary conditions $U_n$5. Here $U_n$6 and $U_n$7 quantify the coupling asymmetry in the two directions, and asymmetric hopping localizes eigenmodes at a favored corner when asymmetry exists in both directions (Kokkinakis et al., 18 May 2025).

Another experimentally realized platform is the temporal photonic feedforward lattice, where the amplitudes $U_n$8 and $U_n$9 in two coupled fiber loops obey

$n$0

$n$1

where $n$2 is the non-Hermiticity parameter, $n$3 is the coupling angle, and $n$4 is the effective Kerr nonlinear coefficient produced by optoelectronic feedforward (Wang et al., 2024).

These models share a common structure: non-reciprocal hopping creates skin localization under open boundaries, and local nonlinearity supplies the self-action required for stationary or dynamically sustained localization. This suggests that SMASs are best regarded as a mechanism class rather than a single lattice-specific object.

3. Formation mechanisms: NHSE, self-trapping, and threshold reduction

The basic mechanism is the interplay between NHSE-driven transport and nonlinear self-trapping. In one-dimensional waveguide arrays, a single-channel excitation creates stable solitons supported by the interplay of the Kerr nonlinearity and NHSE, while in the Hermitian case the corresponding solitons are self-trapped only by the balance between Kerr nonlinearity and diffraction (Hou et al., 1 Jul 2026). In the focusing Hatano-Nelson model, nonlinearity and non-reciprocal hopping act synergistically: nonlinear skin modes become narrower and more strongly localized at the skin edge as amplitude increases, eventually connecting to discrete solitons localized at the favored boundary (Manda et al., 2023).

A key advance is the separation of two distinct nonlinear localization mechanisms. In a stacked Su-Schrieffer-Heeger-like model, SMASs and NRDSs occupy different regions of parameter space and have different physical origins: “SMASs originate from skin effect, while NRDSs stem from band nonreciprocity” (Li et al., 4 Aug 2025). The same work states that for SMASs, skin-mode localization reduces wave broadening at the localization sites, thereby lowering the formation threshold. In the associated Wannier-function-based nonlinear Hamiltonian, the threshold for SMASs satisfies

$n$5

where $n$6 is the hopping amplitude opposite to the skin-mode localization direction; as nonreciprocity increases and $n$7 decreases, $n$8 decreases (Li et al., 4 Aug 2025).

Threshold behavior also appears in direct dynamical studies. In nonlinear optical waveguide arrays, the symbolic-regression analysis yields the soliton existence boundary

$n$9

where $z$0 is the effective nonlinear coefficient. In that formulation, increasing $z$1 suppresses the formation of nonlinear localized states, so higher nonlinearity or input intensity is required to compensate (Hou et al., 1 Jul 2026). In two dimensions, there is a threshold power above which a localized soliton can form, and the threshold depends strongly on the excitation position and the value of non-Hermiticity $z$2; near the favored skin corner the threshold is lower, whereas in the bulk or unfavored corners it increases substantially with $z$3, and for sufficiently strong $z$4 self-trapping in the bulk can become impossible (Kokkinakis et al., 18 May 2025).

These results are sometimes read as contradictory because one set emphasizes threshold lowering and another threshold increase. The distinction lies in what is being compared. In the mechanism-specific phase-diagram treatment, skin localization lowers the threshold at the skin localization site relative to broader-dispersion alternatives such as NRDS formation (Li et al., 4 Aug 2025). In spatially resolved waveguide and two-dimensional Hatano-Nelson settings, stronger non-Hermiticity can make self-trapping away from the favored boundary more difficult, even though localization near the preferred boundary is enhanced (Hou et al., 1 Jul 2026).

4. Dynamical regimes and propagation scenarios

Single-site excitation, broad-pulse excitation, and stationary nonlinear eigenmodes realize different aspects of SMAS physics. In one-dimensional waveguide arrays, a single-channel input $z$5 produces robust edge-localized skin solitons when Kerr nonlinearity is present. In the linear limit, the corresponding solution under single-site excitation is

$z$6

where $z$7 is the Bessel function of order $z$8 (Hou et al., 1 Jul 2026).

For broad excitations, the discrete dynamics can be approximated by a nonlinear Schrödinger equation with drift,

$z$9

which after rescaling becomes

$C_L$0

with $C_L$1. In the non-Hermitian case $C_L$2, broad solitons experience self-acceleration toward the edge, acquiring positional drift and amplitude growth; the reported conclusion is that NHSE accelerates the propagation of the broad soliton toward the boundary and ultimately causes tight localization at the edge, described as a hallmark of the NHSE in the continuum limit (Hou et al., 1 Jul 2026).

In temporal photonic feedforward lattices, the dynamical picture is especially explicit. At the preferred edge, NHSE and Kerr self-trapping cooperate, yielding highly confined edge skin solitons. Away from the preferred boundary, strong enough Kerr nonlinearity can arrest NHSE-induced transport and form stable localized states in the bulk or at the less favored edge (Wang et al., 2024). This establishes that nonlinear control can either reinforce or inhibit the transport tendency associated with linear skin accumulation.

A plausible implication is that “SMAS” names a family of nonlinear transport outcomes rather than only a single stationary branch: some realizations emphasize adiabatic continuation from a linear skin mode, while others emphasize nonlinear arrest of boundary-directed transport.

5. Stationary states, continuation from linear modes, and stability

The stationary-state viewpoint is central to the formal identification of SMASs. In the nonlinear Hatano-Nelson model with Kerr nonlinearity, perturbation theory shows that each linear skin mode seeds a family of nonlinear skin modes. For focusing nonlinearity, these families bridge the gap between weakly nonlinear states and highly nonlinear discrete solitons as the anti-continuum limit is approached; for defocusing nonlinearity, the nonlinear skin modes become more extended than their linear counterparts (Manda et al., 2023).

In non-Hermitian waveguide arrays, stationary eigenmode analysis separates nonlinear bulk modes in the Hermitian regime from near-edge skin solitons in the non-Hermitian regime. The nonlinear bulk modes are compressed toward the edge by non-reciprocity, and this is described as the nonlinear extension of the NHSE. The same study reports a one-to-one mapping between linear skin modes and nonlinear skin solitons, demonstrated via Newton-Raphson continuation from linear eigenmodes, and states that introducing nonlinearity before or after non-Hermiticity yields identical final edge-localized solitons, indicating no path dependence or hysteresis (Hou et al., 1 Jul 2026).

Stability is similarly mechanism-dependent. The stacked SSH-like analysis states that the transition from nonlinear perturbative skin modes (NPSMs) to SMASs is accompanied by a “stability reentrance,” whereas NRDSs are dynamically stable across their domain (Li et al., 4 Aug 2025). In the nonlinear Hatano-Nelson study, linear stability analysis shows that for focusing nonlinearity and the lowest branch, nonlinear skin modes are stable near both the linear and anti-continuum limits but can exhibit unstable intervals at intermediate intensities; for defocusing nonlinearity, the lowest mode is always stable (Manda et al., 2023). In two-dimensional lattices, solitons near the favored corner remain highly stable for all tested $C_L$3, whereas stability diminishes away from that corner as $C_L$4 increases (Kokkinakis et al., 18 May 2025).

A broader fixed-point perspective extends the taxonomy of nonlinear skin localization. In a generalized one-dimensional tight-binding lattice with asymmetric couplings and several nonlinearities, the zero fixed point governs skin modes under semi-infinite boundary conditions, while open boundaries generate power-dependent spectra, degeneracy, and power-energy discontinuity. That work also identifies localized modes that are neither skin nor scale-free localized modes and shows that an impurity can induce discrete dark and anti-dark solitons (Yuce, 2024). This suggests that SMASs belong to a wider nonlinear non-Hermitian landscape in which boundary conditions, continuation structure, and nonlinear fixed points interact nontrivially.

6. Dimensional extensions, experimental realizations, and applications

The one-dimensional optical-waveguide setting provides a direct realization of strongly localized skin solitons through non-reciprocal couplings and Kerr nonlinearity (Hou et al., 1 Jul 2026). Two-dimensional nonlinear Hatano-Nelson lattices generalize the phenomenon from edge localization to corner localization: when asymmetry exists in both directions, all bulk eigenmodes localize at a single favored corner in the linear regime, and two-dimensional skin soliton solutions inherit strong spatial asymmetry with tails extending predominantly toward the skin-corner direction (Kokkinakis et al., 18 May 2025).

The temporal photonic feedforward lattice supplies an experimental demonstration of the nonlinear NHSE in an effective Kerr nonlinear temporal photonic lattice. In that platform, artificial nonlinearity is created by optoelectronic feedforward, which overcomes the high-power requirements and lack of tunability of material Kerr media. The reported experimental results include edge skin solitons tightly confined within $C_L$5 sites, with average inverse participation ratio in the linear case of approximately $C_L$6, and strong robustness of nonlinear skin solitons against random phase disorder (Wang et al., 2024). The same work uses nonlinearity-controlled NHSE to design an optical router with a flexibly tuned output port, where the center of mass of the output distribution shifts continuously as $C_L$7 is tuned (Wang et al., 2024).

Applications mentioned across the literature remain primarily photonic and wave-based. The experimental feedforward study associates nonlinear NHSE and skin solitons with robust signal transmission, routing, and processing (Wang et al., 2024). The phase-diagram framework for SMASs and NRDSs states that the theory spans experimentally realizable non-Hermitian systems “from optics to mechanics” and offers predictive tools for engineering solitons in such settings (Li et al., 4 Aug 2025). The waveguide-array study further points to integrated nonlinear and non-Hermitian photonics, including robust edge-state engineering and topological lasers, though these are framed as implications rather than as demonstrated devices (Hou et al., 1 Jul 2026).

7. Distinctions, misconceptions, and open interpretive issues

One common misconception is that every nonlinear localized mode in a non-reciprocal lattice is a skin-effect soliton. The most explicit correction is the distinction between SMASs and NRDSs: SMASs are boundary-localized and require open boundaries because they are rooted in the NHSE, whereas NRDSs are bulk-localized, arise from band nonreciprocity, and persist under periodic boundary conditions (Li et al., 4 Aug 2025).

A second misconception is that the NHSE is purely a linear phenomenon. Several studies describe a nonlinear extension of the NHSE, in which nonlinearity enhances edge localization, compresses nonlinear bulk modes toward the boundary, or suppresses NHSE-induced transport depending on launch conditions and position (Hou et al., 1 Jul 2026). In the experimental temporal lattice, Kerr self-trapping can improve localization at the preferred boundary, but away from that boundary it can inhibit boundary-directed transport and produce stable localized states elsewhere (Wang et al., 2024). The fixed-point study strengthens this point by showing that nonlinearity leaves the spectral region unchanged under semi-infinite boundary conditions while dramatically modifying it under open boundary conditions, so nonlinear boundary physics is not a trivial perturbation of the linear case (Yuce, 2024).

A third interpretive issue concerns the role of nonlinearity sign. In the focusing Kerr regime, nonlinearity typically enhances localization and supports continuation toward discrete solitons; in the defocusing regime, it tends to broaden nonlinear skin modes relative to linear skin modes (Manda et al., 2023). This indicates that “skin-mode-assisted” should not be read as synonymous with stronger localization under all nonlinear responses.

Taken together, the literature presents SMASs as a boundary-conditioned nonlinear phase of non-Hermitian matter or wave transport: they are generated by the cooperation or competition of asymmetric hopping and nonlinear self-action, they admit multiple realizations from one-dimensional edges to two-dimensional corners, and their most precise modern definition depends on separating NHSE-assisted soliton formation from nonlinear localization driven solely by bulk nonreciprocity (Li et al., 4 Aug 2025).