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arxiv: 2311.09139 · v1 · pith:DYGGNZNBnew · submitted 2023-11-15 · 🌊 nlin.PS

Skin modes in a nonlinear Hatano-Nelson model

Bertin Many Manda , Ricardo Carretero-Gonz\'alez , Panayotis G. Kevrekidis , Vassos Achilleos This is my paper

Pith reviewed 2026-05-25 08:11 UTC · model grok-4.3

classification 🌊 nlin.PS
keywords non-Hermitian skin effectHatano-Nelson modelKerr nonlinearitynonlinear skin modesperturbation theorystability analysisdiscrete solitonsopen boundary conditions
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The pith

Nonlinear skin modes emerge from linear ones in the Hatano-Nelson model at any non-reciprocal strength via perturbation theory.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines how the non-Hermitian skin effect in the Hatano-Nelson lattice changes when Kerr nonlinearity is added. Perturbation theory is used to continue the linear edge-localized modes into nonlinear stationary solutions that remain localized at the favored boundary for any non-reciprocal coupling strength. Focusing nonlinearity makes these modes more localized and connects them to discrete solitons as couplings approach zero, while defocusing makes the modes more extended than in the linear case. Linear stability analysis is performed on the full family of solutions together with selected dynamical simulations.

Core claim

Based on perturbation theory, families of nonlinear skin modes emerge from the linear ones at any non-reciprocal strength. In the case of focusing nonlinearity, these families of nonlinear skin modes tend to exhibit enhanced localization, bridging the gap between weakly nonlinear modes and the highly nonlinear states (discrete solitons) when approaching the anti-continuum limit with vanishing couplings. Conversely, for defocusing nonlinearity, these nonlinear skin modes tend to become more extended than their linear counterpart.

What carries the argument

Perturbation theory that continues linear skin modes of the open-boundary Hatano-Nelson chain into the nonlinear regime with Kerr nonlinearity.

If this is right

  • Families of nonlinear skin modes exist for any non-reciprocal strength.
  • Focusing nonlinearity produces enhanced localization relative to the linear modes.
  • Defocusing nonlinearity produces more extended profiles than the linear modes.
  • Focusing nonlinear skin modes connect continuously to discrete solitons in the anti-continuum limit.
  • Linear stability analysis applies uniformly across the obtained nonlinear mode families.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The continuation result implies that non-Hermitian skin localization can be preserved under weak nonlinear perturbations in open chains.
  • The same perturbative construction could be tested on other non-reciprocal lattices or with different nonlinearities such as saturable or nonlocal terms.
  • In driven-dissipative photonic or acoustic platforms the intensity dependence of localization could be used to steer wave propagation toward one boundary.
  • Stability of the nonlinear skin modes opens the possibility of observing them as attractors in long-time evolution under small perturbations.

Load-bearing premise

The nonlinearity remains a small perturbation around the linear skin modes so that the perturbative expansion remains valid across the parameter range studied.

What would settle it

Numerical computation of exact nonlinear stationary solutions at moderate nonlinearity strength that deviate in profile or existence from the first-order perturbative predictions.

Figures

Figures reproduced from arXiv: 2311.09139 by Bertin Many Manda, Panayotis G. Kevrekidis, Ricardo Carretero-Gonz\'alez, Vassos Achilleos.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) The proposed nonlinear HN model where closed [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) The total intensity, [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Intensity [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Total intensity, [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (b) sorted by increasing value of the participation FIG. 5. Eigenvalues λ of the linear stability problem for the stationary solutions with (a) C = 0.501 and (b) C = 0.527 of the family of NLSMs emerging from the second skin linear mode at constant energy, red curve in [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Nonlinear HN lattice dynamics. The initial conditions [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: shows, in the focusing case (σ = +1), the results of the comparison of the families of NLSMs bi￾furcating from the first linear modes for various t values obtained using numerical (bold colored curves) and RPT (black curves). The plot displays the dependence of the total intensity as a function of the coupling strengths for these families and shows a good agreement between the two results for small amplitu… view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Same as in Fig [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
read the original abstract

Non-Hermitian lattices with non-reciprocal couplings under open boundary conditions are known to possess linear modes exponentially localized on one edge of the chain. This phenomenon, dubbed non-Hermitian skin effect, induces all input waves in the linearized limit of the system to unidirectionally propagate toward the system's preferred boundary. Here we investigate the fate of the non-Hermitian skin effect in the presence of Kerr-type nonlinearity within the well-established Hatano-Nelson lattice model. Our method is to probe the presence of nonlinear stationary modes which are localized at the favored edge, when the Hatano-Nelson model deviates from the linear regime. Based on perturbation theory, we show that families of nonlinear skin modes emerge from the linear ones at any non-reciprocal strength. Our findings reveal that, in the case of focusing nonlinearity, these families of nonlinear skin modes tend to exhibit enhanced localization, bridging the gap between weakly nonlinear modes and the highly nonlinear states (discrete solitons) when approaching the anti-continuum limit with vanishing couplings. Conversely, for defocusing nonlinearity, these nonlinear skin modes tend to become more extended than their linear counterpart. To assess the stability of these solutions, we conduct a linear stability analysis across the entire spectrum of obtained nonlinear modes and also explore representative examples of their evolution dynamics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript studies the nonlinear Hatano-Nelson chain with Kerr nonlinearity under open boundaries. Using perturbation theory around the known linear skin modes, it constructs families of nonlinear stationary skin modes that exist for any non-reciprocal hopping asymmetry. Focusing nonlinearity is shown to enhance localization (approaching discrete solitons in the anti-continuum limit), while defocusing nonlinearity delocalizes the modes relative to the linear case. Linear stability analysis is performed over the obtained families, supplemented by representative dynamical simulations.

Significance. If the perturbative construction is uniformly valid, the work supplies a systematic route from linear skin modes to their nonlinear counterparts in a canonical non-Hermitian lattice, together with a stability survey. This bridges the weakly nonlinear regime to the anti-continuum limit and clarifies how non-reciprocity and nonlinearity interact to control localization.

major comments (1)
  1. [Perturbative construction (around the linear eigenmodes)] The central claim that nonlinear skin-mode families exist “at any non-reciprocal strength” rests on the perturbative expansion remaining controlled for arbitrary asymmetry. Because the linear eigenmodes localize exponentially with increasing non-reciprocity, the overlap integrals that set the first-order frequency shift and higher-order corrections can grow; the manuscript provides no explicit bound or radius-of-convergence estimate that is independent of the asymmetry parameter. This uniformity assumption is load-bearing for the stated result.
minor comments (2)
  1. [Abstract] The abstract states that stability is assessed “across the entire spectrum of obtained nonlinear modes”; a brief statement of the range of nonlinearity coefficients and system sizes actually examined would improve clarity.
  2. [Figures] Figure captions would benefit from explicit listing of the non-reciprocal parameter values and nonlinearity strengths used in each panel.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed reading and the constructive comment on the perturbative construction. We address the point below and indicate planned revisions.

read point-by-point responses

  1. Referee: [Perturbative construction (around the linear eigenmodes)] The central claim that nonlinear skin-mode families exist “at any non-reciprocal strength” rests on the perturbative expansion remaining controlled for arbitrary asymmetry. Because the linear eigenmodes localize exponentially with increasing non-reciprocity, the overlap integrals that set the first-order frequency shift and higher-order corrections can grow; the manuscript provides no explicit bound or radius-of-convergence estimate that is independent of the asymmetry parameter. This uniformity assumption is load-bearing for the stated result.

    Authors: We agree that the manuscript does not furnish an explicit radius-of-convergence bound that is uniform with respect to the non-reciprocal asymmetry. Our construction proceeds for each fixed value of the asymmetry parameter, with the Kerr nonlinearity strength serving as the small parameter. For any fixed asymmetry the linear skin mode is a well-defined, exponentially localized eigenfunction of the linear operator; standard perturbative arguments (Lyapunov–Schmidt reduction or the implicit-function theorem in suitable function spaces) then guarantee the local continuation of a nonlinear branch for sufficiently small nonlinearity amplitude. The size of the admissible nonlinearity interval may indeed depend on the asymmetry through the growth of nonlinear overlap integrals, but this dependence does not invalidate the existence statement “at any non-reciprocal strength,” which refers to each fixed strength separately. We will revise the manuscript to clarify the scope of the perturbation (fixed asymmetry, small nonlinearity) and to add a short remark acknowledging the possible shrinkage of the convergence radius with increasing asymmetry. revision: yes

Circularity Check

0 steps flagged

No circularity: standard perturbative construction from independently known linear modes

full rationale

The paper starts from the established linear eigenmodes of the Hatano-Nelson model (known from prior non-self-referential literature) and applies standard perturbation theory to generate nonlinear corrections. No equation reduces to a fitted parameter renamed as a prediction, no self-definitional loop exists between the claimed nonlinear families and their linear seeds, and no load-bearing uniqueness theorem or ansatz is imported via self-citation. The derivation chain remains self-contained against external benchmarks (linear spectrum and perturbation theory), yielding an independent result rather than a tautology.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The work rests on the standard discrete nonlinear Schrödinger equation with asymmetric nearest-neighbor couplings and open boundaries; no new entities are introduced.

free parameters (2)
  • nonlinearity coefficient
    Strength of the Kerr term; treated as a tunable parameter rather than fitted to external data.
  • non-reciprocal coupling strength
    Asymmetric hopping parameter varied across the reported families.
axioms (2)
  • domain assumption The underlying lattice is described by a discrete nonlinear Schrödinger equation with non-reciprocal nearest-neighbor couplings under open boundary conditions.
    Standard modeling choice for Hatano-Nelson-type systems.
  • domain assumption Perturbation theory around the linear eigenmodes remains valid for the range of nonlinearity strengths considered.
    Central technical assumption enabling the construction of nonlinear families.

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