--- title: Nonlinear Hatano–Nelson Model url: https://www.emergentmind.com/topics/nonlinear-hatano-nelson-model type: topic --- # Nonlinear Hatano–Nelson Model Searching arXiv for recent and foundational papers on nonlinear and interacting Hatano–Nelson models. The nonlinear Hatano–Nelson model denotes a family of non-Hermitian extensions of the Hatano–Nelson chain in which asymmetric hopping is combined either with explicit amplitude-dependent nonlinearities, most commonly onsite Kerr or saturable terms, or with genuine many-body interactions in second-quantized lattice Hamiltonians. In the strict dynamical-systems sense, the term refers most directly to discrete nonlinear Schrödinger–type or continuum nonlinear wave equations built on Hatano–Nelson nonreciprocity, such as Kerr-nonlinear lattices and saturating nonlinear drift models [2311.09139; 2112.06241; 2604.02263; 2602.17439; 2501.01226]. In a broader many-body sense, it is also used for interacting fermionic Hatano–Nelson models, where the evolution remains linear in Hilbert space but interactions generate nontrivial collective effects, Luttinger-liquid descriptions, modified skin physics, and non-Hermitian quantum phase transitions [2208.12017; 2201.12653; 2304.09688; 2408.07122]. A central distinction across the literature is therefore between nonlinear state equations and interacting many-body generalizations: both extend Hatano–Nelson physics beyond the linear single-particle chain, but they do so through different mechanisms and support different observables, stability structures, and boundary phenomena. ## 1. Definitions and model classes The standard Hatano–Nelson model is a one-dimensional non-Hermitian lattice with asymmetric hopping. In nonlinear generalizations, the asymmetry is retained while nonlinear feedback is added either at the level of the wave amplitude or through many-body interactions. The resulting literature separates naturally into four classes. First, Kerr-type discrete nonlinear Schrödinger lattices add an onsite cubic term to the Hatano–Nelson chain. One representative model is \[ i\frac{d\psi_n}{d\tau} = C\left(\psi_{n+1}+t\psi_{n-1}\right) +\sigma |\psi_n|^2\psi_n, \] with open boundary conditions \(\psi_0=\psi_{N+1}=0\), coupling \(C\), non-reciprocity parameter \(t\), and Kerr coefficient \(\sigma=\pm1\), where \(\sigma=+1\) is focusing and \(\sigma=-1\) is defocusing [2311.09139]. A related nonreciprocal discrete nonlinear Schrödinger form is \[ i\frac{d\psi_n}{dt} +\kappa_{\mathrm{R}}\psi_{n+1} +\kappa_{\mathrm{L}}\psi_{n-1} -\left(\kappa_{\mathrm{R}}+\kappa_{\mathrm{L}}\right)\psi_n +\xi |\psi_n|^2 \psi_n =0, \] with \(\kappa_{\mathrm{R}}=\kappa(1+\lambda)\) and \(\kappa_{\mathrm{L}}=\kappa(1-\lambda)\), used to study nonlinear dynamical skin formation and self-trapping [2112.06241]. A further formulation emphasizes amplified bulk wave-packet transport in the discrete equation \[ -i\frac{dy_n}{dt} = gy_{n-1} + y_{n+1} + \alpha |y_n|^2 y_n, \] where \(g>1\) is nonreciprocal and \(\alpha\) is the onsite cubic nonlinearity [2604.02263]. Under periodic boundary conditions, another Kerr-nonlinear Hatano–Nelson model is written as \[ i \frac{d \psi_n}{dt}= \kappa \exp(h) \psi_{n+1}+\kappa \exp(-h) \psi_{n-1}+ \chi | \psi_n|^2 \psi_n , \] with imaginary gauge field \(h\) and Kerr strength \(\chi\) [2501.01226]. Second, continuum nonlinear Hatano–Nelson models can be formulated as stationary nonlinear eigenvalue problems with amplitude-dependent nonreciprocity. One example is \[ \hat H(\psi)\,\psi = E\,\psi , \qquad \hat H(\psi)=\frac{\hat k^2}{2}+ i\,F(|\psi|^2)\,\hat k,\qquad \hat k \coloneqq -i\partial_x , \] with \[ F(z)=\gamma + a z - b z^2. \] Here \(\gamma\) is linear nonreciprocity, \(a>0\) a cubic nonlinear enhancement, and \(b>0\) a quintic saturating suppression [2602.17439]. Third, interacting many-body Hatano–Nelson models introduce density-density or Hubbard interactions but preserve linear Schrödinger evolution in Fock space. For spinless fermions with nearest-neighbor interaction and open boundaries, a standard microscopic form is \[ H=\sum_{n=1}^{N-1} \frac J2 e^{ah}c_n^\dagger c_{n+1}+\frac J2 e^{-ah} c_{n+1}^\dagger c_n +U\, c_n^\dagger c_{n}c_{n+1}^\dagger c_{n+1}, \] at half filling [2208.12017; 2304.09688]. Under periodic closure, a related interacting model is \[ \hat{H}=\sum_{\ell}[(t+\gamma)\hat{c}_{\ell}^{\dagger}\hat{c}_{\ell+1}+(t-\gamma)\hat{c}_{\ell+1}^{\dagger}\hat{c}_{\ell}+U\hat{n}_{\ell}\hat{n}_{\ell+1}], \] used to study symmetry breaking and spectral topology [2201.12653]. A spinful two-leg extension with onsite Hubbard interaction is \[ \hat{\cal H}=\hat{\cal K}+\hat{\cal U}, \] with asymmetric intraleg hopping, Hermitian rung coupling \(V_0\), and onsite repulsion \[ \hat{\cal U}=U\sum_{j,\alpha} \hat{n}_{j,\alpha \uparrow}\hat{n}_{j,\alpha \downarrow}, \] in a balanced ladder geometry \(\delta_A=-\delta_B\equiv \delta\) [2604.14533]. Fourth, spinful linear extensions with spin-dependent gauge fields enrich Hatano–Nelson phenomenology without adding nonlinearity. These are not nonlinear models, but they clarify which NHSE mechanisms can survive or fail when internal channels are added [2505.05458]. This distinction matters because several papers explicitly warn that “interacting” does not imply “nonlinear” in the Gross–Pitaevskii or DNLS sense [2304.09688; 2408.07122; 2208.12017; 2201.12653]. ## 2. Nonlinear stationary states and skin-mode continuation A central question in explicitly nonlinear Hatano–Nelson systems is whether the non-Hermitian skin effect survives once cubic self-action is added. In the Kerr lattice with stationary ansatz \(\psi_n(\tau)=u_n e^{iE\tau}\), the nonlinear algebraic problem is \[ E u_n = C\left(u_{n+1}+t u_{n-1}\right)+\sigma |u_n|^2u_n. \] The total intensity used to parameterize nonlinear branches is \[ S=\sum_n |u_n|^2, \] while the conserved weighted norm is \[ S_D=\sum_n t^n |u_n|^2 \] on a finite chain [2311.09139]. This conservation law is specific to the nonreciprocal model and distinguishes it from Hermitian DNLS dynamics. In the linear limit \(\sigma=0\), the open-chain Hatano–Nelson spectrum is real, \[ E_q = 2\sqrt{t}\cos\!\left(\frac{q\pi}{N+1}\right), \] with right eigenmodes \[ u^{(0)}_{q,n} = \sqrt{\frac{2}{N+1}\, t^{n/2}\, \sin\!\left(\frac{n q\pi}{N+1}\right), \] which are exponentially skin-localized toward the left edge for \(t<1\) and toward the right edge for \(t>1\) [2311.09139]. The nonlinear problem admits perturbative continuation from every linear skin mode. Fixing \(E=E_q\), the branch is expanded as \[ C = 1 + C_1 \epsilon^2 + \mathcal O(\epsilon^4), \qquad u_n = \epsilon u_n^{(0)} + \epsilon^2 u_n^{(1)} + \epsilon^3 u_n^{(2)} + \mathcal O(\epsilon^4), \] and the solvability condition yields \[ C_1 =-\frac{\sigma}{E_q}\left(\frac{2}{N+1}\right)^2 \sum_{n=1}^N t^{n}\sin^4\!\left(\frac{nq\pi}{N+1}\right). \] This establishes that families of nonlinear skin modes emerge from linear ones for arbitrary non-reciprocity [2311.09139]. The sign of the Kerr nonlinearity controls localization. For focusing nonlinearity, these nonlinear skin-mode families become more localized and interpolate continuously toward anti-continuum discrete-soliton states as \(C\to0\). For defocusing nonlinearity, the branches broaden, and the \(q=1\) family can become almost extended over the finite lattice [2311.09139]. The paper reports threshold behavior in the anti-continuum endpoint: for the branch from \(q=1\), the endpoint reaches the leftmost single site when \(t1\), branches connect to consecutive left-edge excited sites when \(t0\) each excited site satisfies \(|u_n|^2=E\) [2311.09139]. This continuity between linear skin modes and discrete solitons is one of the most explicit nonlinear continuations of NHSE in the literature. ## 3. Dynamical nonlinear skin formation, trapping, and wave-packet transport Beyond stationary branches, several works study the real-time dynamics of nonlinear Hatano–Nelson systems. In the quench-dynamics setting with a single-site pulse, \[ \psi_n(t)=\delta_{n,m_0}\qquad \text{at } t=0, \] the interplay between asymmetric hopping and onsite nonlinearity yields distinct nonlinear localization patterns [2112.06241]. The time-averaged diagnostic \[ S_n \equiv \frac{1}{T/2}\int_{T/2}^{T} |\psi_n(t)|\,dt \] is used to classify long-time states [2112.06241]. In one dimension, four phases are reported: skin, trap-skin, shifted-trap-skin, and embedded-trap-skin [2112.06241]. The skin phase corresponds to ordinary NHSE-driven accumulation at the preferred boundary. The trap-skin phase combines boundary skin accumulation with persistent trapping at the initial site, signaled by \(S_{m_0}=1\). The shifted-trap-skin phase displays a trapped peak shifted from the initial site, typically to \(m_0+1\). The embedded-trap-skin phase retains \(S_{m_0}=1\) but with the initial-site memory embedded within a broader skin profile [2112.06241]. The phase boundaries are determined by gaps in the dynamical indicators \(S_{m_0}\) and \(S_{m_0+1}\), rather than by a linear spectral gap [2112.06241]. In two dimensions, analogous nonlinear higher-order corner-skin behavior is found, but the shifted-trap-skin phase is absent [2112.06241]. A complementary dynamical perspective follows a broad Gaussian bulk packet in the nonlinear equation \[ -i\frac{dy_n}{dt} = gy_{n-1} + y_{n+1} + \alpha |y_n|^2 y_n, \] with initial condition \[ y_n(0)=\exp\left(\frac{-(n-n_0)^2}{4\sigma_0^2}\right)e^{ik_0 n}, \] and \(k_0=0\) [2604.02263]. The paper shows that amplification makes even weak nonlinearities grow dynamically important, organizing the motion into three regimes: nonlinear-skin, wave-mixing, and self-trapping [2604.02263]. In the continuum approximation \[ -i \frac{\partial y}{\partial t} = -b \frac{\partial y}{\partial x} + D \frac{\partial^2 y}{\partial x^2} + \alpha |y|^2 y, \qquad b=g-1,\quad D=\frac{g+1}{2}, \] a Gaussian variational ansatz yields collective-coordinate equations for the width \(\sigma(t)\) and center of mass \(X(t)\) [2604.02263]. Eliminating auxiliary parameters gives \[ \ddot{\sigma} = \frac{D^2}{\sigma^3} - \frac{\alpha D S}{2\sqrt{\pi}\sigma^2} \] and \[ \ddot{X} = \frac{2bD}{\sigma^2} + \frac{2b}{D}\dot{\sigma}^2 - \frac{\alpha b S}{2\sqrt{\pi}\sigma}. \] These formulas show that nonlinearity couples amplification, dispersion, and nonreciprocity [2604.02263]. Focusing nonlinearities \(\alpha>0\) suppress the acceleration and make it decrease in time, whereas defocusing nonlinearities \(\alpha<0\) enhance it and make it increase [2604.02263]. The same paper identifies spectral criteria for the three regimes using the real-frequency width \[ \Delta = 2(g+1) \] and average spacing \[ d=\frac{\Delta}{N}. \] The nonlinear-skin regime obeys \(\nu(t)\Delta\), where \(\nu(t)\) is the amplification-enhanced nonlinear frequency shift [2604.02263]. The crossover times are \[ T_{\mathrm{mix}}^2 = \frac{2\sigma_0^2}{(g-1)^2} \log\left( \frac{2\sqrt{2}\sigma_0(g+1)}{\alpha N} \right), \] \[ T_{\mathrm{trap}}^2 = \frac{2\sigma_0^2}{(g-1)^2} \log\left( \frac{2\sqrt{2}\sigma_0(g+1)}{\alpha} \right). \] A notable conclusion is that nonlinear interactions typically destroy coherent evolution before the linear non-Hermitian jump can occur [2604.02263]. Under periodic boundary conditions, the nonlinear dynamics can differ qualitatively. In the Kerr-nonlinear ring \[ i \frac{d \psi_n}{dt}= \kappa \exp(h) \psi_{n+1}+\kappa \exp(-h) \psi_{n-1}+ \chi | \psi_n|^2 \psi_n , \] exact nonlinear plane waves exist, but all are modulationally unstable for \(h\neq 0\), independent of wave number and independent of the sign of \(\chi\) [2501.01226]. The instability generates irregular intensity landscapes that act as self-induced disorder through effective onsite potentials \[ V_n(t) \equiv \chi | \phi_n(t)|^2, \qquad W_n(t) \equiv \chi \phi_n^2(t), \] around an irregular solution \(\phi_n(t)\) [2501.01226]. The reported consequence is “growth blockade”: the total intensity \[ P(t)=\sum_{n=1}^N |\psi_n(t)|^2 \] initially amplifies but then abruptly ceases secular growth and fluctuates around a nearly constant value [2501.01226]. The interpretation is that modulational instability dynamically generates an irregular effective potential that arrests the convective motion responsible for linear Hatano–Nelson growth [2501.01226]. ## 4. Stability theory, bifurcations, and basin geometry The stability analysis of nonlinear Hatano–Nelson modes combines non-Hermitian linearization with nonlinear bifurcation theory. For stationary Kerr modes, a Bogoliubov–de Gennes–type perturbation \[ w_n(\tau)=a_n e^{i\lambda\tau}+b_n^* e^{-i\lambda^*\tau} \] leads to a \(2N\times 2N\) eigenvalue problem [2311.09139]. Although the linearization is non-Hermitian for \(t\neq1\), a similarity transformation renders the relevant operator symmetric, implying the quartet symmetry \[ \lambda,\ \lambda^*,\ -\lambda,\ -\lambda^* . \] A nonlinear skin mode is linearly stable when all \(\lambda\) are real and unstable when a complex quartet appears [2311.09139]. For focusing nonlinearity, branches are stable near both the linear limit \(C\approx1\) and the anti-continuum limit \(C\approx0\), with instability bubbles at intermediate intensity; for defocusing nonlinearity, the \(q=1\) branch is reported to be always stable [2311.09139]. A different stability language emerges in the continuum model with saturating nonlinear nonreciprocity, \[ \partial_x^2\psi -2\Bigl(\gamma+a\psi^2-b\psi^4\Bigr)\,\partial_x\psi +2E\,\psi =0, \] treated as a spatial dynamical system in \((\psi,v)\) with \(v=\partial_x\psi\) [2602.17439]. The flow is \[ \partial_x \psi = v,\qquad \partial_x v = 2\bigl(\gamma+a\psi^2-b\psi^4\bigr)\,v - 2E\,\psi. \] Here skin modes correspond to trajectories attracted to the origin, while extended states correspond to trajectories attracted to a stable limit cycle [2602.17439]. This replaces the usual spectral classification by attractor-basin geometry. The paper identifies a subcritical Hopf bifurcation at \(\gamma=0\) and a saddle-node of limit cycles at \(\gamma=\gamma_c<0\), producing three regimes: skin-only for \(\gamma<\gamma_c\), coexistence for \(\gamma_c<\gamma<0\), and extended-only for \(\gamma>0\) [2602.17439]. Under the slow-amplitude approximation, averaging yields the radial equation \[ \partial_x r = h(r),\qquad h(r)\equiv r\left(\gamma+\frac{a}{4}r^2-\frac{b}{8}r^4\right), \] with nontrivial cycle amplitudes \[ A_{\rm in}(\gamma)=\sqrt{\frac{a}{b}\left(1-\sqrt{1+\frac{8b\gamma}{a^2}}\right)}, \qquad A_{\rm out}(\gamma)=\sqrt{\frac{a}{b}\left(1+\sqrt{1+\frac{8b\gamma}{a^2}}\right)}. \] The saddle-node threshold predicted by averaging is \[ \gamma_c^{\rm (th)} = -\frac{a^2}{8b}. \] For the representative choice \(a=\frac12\), \(b=\frac1{32}\), \(E=8\), this gives \(\gamma_c^{\rm(th)}=-1\) [2602.17439]. Within the coexistence window, skin and extended stationary states are both stable at the same fixed energy \(E\), separated by a nonlinear basin separatrix rather than by a spectral mobility edge [2602.17439]. The paper introduces a basin-fraction order parameter \[ p_{\mathrm{skin}}(\gamma;\mu)=\Pr_{s\sim\mu}\!\Bigl[\text{trajectory is attracted to the origin}\Bigr], \] where \(s=\partial_x\psi(0)\) is the boundary slope [2602.17439]. For a Cauchy-distributed \(s\), it finds a first-order-like jump at the saddle-node of limit cycles and predicts hysteresis and long-lived separatrix-induced spatial transients [2602.17439]. This suggests that in nonlinear non-Hermitian systems, global basin geometry can be as important as spectral data. ## 5. Interacting many-body Hatano–Nelson models as a distinct “nonlinear” usage Several papers treat “nonlinear Hatano–Nelson” as shorthand for interacting many-body Hatano–Nelson physics, while explicitly noting that the resulting time evolution remains linear in Hilbert space [2304.09688; 2408.07122; 2208.12017; 2201.12653]. The core spinless-fermion model with open boundaries is \[ H=\sum_{n=1}^{N-1} \frac J2 e^{ah}c_n^\dagger c_{n+1}+\frac J2 e^{-ah} c_{n+1}^\dagger c_n +U\, c_n^\dagger c_{n}c_{n+1}^\dagger c_{n+1}, \] studied at half filling [2208.12017; 2304.09688]. In bosonized form, the low-energy Hamiltonian is \[ H=\int_0^L \frac{dx}{2\pi}\, v\left[K(\pi\Pi(x)-ih)^2+\frac 1K (\partial_x\phi(x))^2\right], \] so the imaginary vector potential enters as a shift \(\pi\Pi\to \pi\Pi-ih\) [2304.09688]. A similarity transformation \[ S=\exp\left(\frac{h}{\pi}\int_0^L\phi(x')dx'\right) \] maps the bosonized Hamiltonian to an ordinary Hermitian Luttinger liquid, showing that the low-energy interacting Hatano–Nelson model is a Luttinger liquid with a non-unitary similarity twist [2304.09688]. In the ground state, many-body interactions strongly soften the visible consequences of the single-particle skin effect. The long-wavelength density tilt is \[ n_0(x)=-\frac{2Kh}{\pi^2}\ln\!\left[\tan\left(\frac{\pi x}{2L}\right)\right], \] which is only a smooth logarithmic bias rather than an exponential many-body pileup [2208.12017]. The full density retains Friedel oscillations with an \(h\)-dependent beating phase, \[ \rho(x)=\rho_0+n_0(x) +c\left(\frac{\pi\alpha}{L\sin\left(\frac{\pi x}{L}\right)}\right)^K \sin\!\left(2k_Fx+\frac{4hL}{\pi}g(x)+\delta\right), \] and the full counting statistics of particle number in a finite interval is Gaussian, with mean \[ \mu=\frac{2 h L}{\pi^2}(g(x)-g(y)) \] and variance \[ \sigma^2=\frac{1}{\pi^2}C(x,y), \] the latter being symmetric about the chain center [2208.12017]. These results indicate that many-body correlations can conceal the single-particle NHSE in local observables [2208.12017]. Quench dynamics in the interacting Luttinger-liquid regime further shows that switching the imaginary vector potential on or off launches ballistic light cones from the boundaries rather than immediate static skin accumulation [2304.09688]. The long-wavelength switch-off density and current are \[ n_0(x,t)=-\frac{Kh}{\pi^2}\sum_{\sigma=\pm}\ln\left|\tan\left(\frac{\pi (x+\sigma vt)}{2L}\right)\right|, \] \[ j_0(x,t)=\frac{vhK}{\pi^2}\sum_{\sigma=\pm}\sigma \ln\left|\tan\left(\frac{\pi (x+\sigma vt)}{2L}\right)\right|, \] while the switch-on current is exactly the negative of the switch-off current [2304.09688]. The paper emphasizes that, at least for small \(h\), ballistic boundary-emitted fronts dominate the dynamics [2304.09688]. It also shows that the long-wavelength density and current satisfy an ordinary continuity equation even under non-unitary switch-on dynamics within the bosonized sector [2304.09688]. A related finite-time ramp study considers \[ h(t)=\frac{h_0 t}{\tau} \] in the same interacting many-body model [2408.07122]. The bosonized mode amplitudes obey \[ \alpha_q(t)=g_q(\tau)\frac{1-i\omega(q)t-\exp(-i\omega(q)t)}{\omega(q)^2\tau}, \] from which the excess energy, density imbalance, and Loschmidt echo are derived [2408.07122]. The main result is a slow, oscillatory approach to adiabaticity with envelope scaling \(\tau^{-1}\) and period \(2L/v\), in contrast to the \(\tau^{-2}\) scaling of Hermitian vector-potential ramps [2408.07122]. The mean energy becomes complex despite the real instantaneous spectrum, because the Hamiltonian is non-Hermitian and the evolving state is not an instantaneous eigenstate [2408.07122]. At commensurate times \(\tau=2L/v\), the entire wavefunction coincides with the adiabatic target, yielding a shortcut to adiabaticity without auxiliary controls [2408.07122]. Under periodic boundary conditions, repulsive interactions produce additional many-body effects. In the model \[ \hat{H}=\sum_{\ell}[(t+\gamma)\hat{c}_{\ell}^{\dagger}\hat{c}_{\ell+1}+(t-\gamma)\hat{c}_{\ell+1}^{\dagger}\hat{c}_{\ell}+U\hat{n}_{\ell}\hat{n}_{\ell+1}], \] two \(\mathcal{PT}\) transitions are found at half filling as \(U\) increases [2201.12653]. The first is a finite-size exceptional-point transition involving the first and second excited states, interpreted as a first-order transition into a charge-density-wave regime. Persistent currents characteristic of the Hatano–Nelson ring abruptly vanish there [2201.12653]. The second transition is a finite-size collapse of the entire spectrum onto the real axis, with critical interaction strength that scales with system size and therefore does not survive the thermodynamic limit [2201.12653]. Away from half filling and at strong interaction, the spectrum decomposes into point-gap clusters with winding numbers, such as \[ \nu_1=\operatorname{sgn}(\gamma)N, \] signaling a many-body skin effect under open boundaries [2201.12653]. A more recent multicomponent generalization is the two-orbital interacting Hatano–Nelson model, \[ \hat{\cal K} = -\sum_{j,\alpha,\sigma} \Big[ (t+\delta_\alpha)\, \hat{c}_{j+1,\alpha \sigma }^\dag \hat{c}_{j,\alpha \sigma} + (t-\delta_\alpha)\, \hat{c}^\dag_{j,\alpha \sigma} \hat{c}_{j+1,\alpha \sigma} \Big] -V_0\sum_{j,\sigma} \Big( \hat{c}^\dag_{j,A\sigma}\hat{c}_{j,B\sigma} +\hat{c}^\dag_{j,B\sigma}\hat{c}_{j,A\sigma} \Big), \] with onsite Hubbard interaction \[ \hat{\cal U}=U\sum_{j,\alpha} \hat{n}_{j,\alpha \uparrow}\hat{n}_{j,\alpha \downarrow}, \] and balanced asymmetry \(\delta_A=-\delta_B\equiv\delta\) [2604.14533]. At \(U=0\), the ladder bands are \[ E_{\pm}(k) = -2t\cos k \pm \sqrt{V_0^2-4\delta^2\sin^2 k}, \] so the spectrum is purely real under periodic boundaries when \(V_0\ge 2\delta\) [2604.14533]. In the interacting two-particle sector, the threshold for spectral reality shifts, and at strong \(U\) a detached doublon branch near \(\operatorname{Re}E\sim U\) appears [2604.14533]. In the isolated doublon regime, the point-gap winding at \(E=U\) is \[ W(E=U)=4, \] with spin-resolved decomposition \(W_\uparrow=W_\downarrow=2\) [2604.14533]. Under open boundaries, these doublon states become skin modes localized at opposite edges of the two legs [2604.14533]. This suggests that interaction-generated bound-state sectors can develop their own non-Hermitian topology, distinct from the scattering continuum. ## 6. Topology, boundaries, and common distinctions Across nonlinear and interacting variants, open boundary conditions are usually essential to skin localization. In stationary Kerr lattices, all linear eigenstates under OBC are exponentially localized, and nonlinear skin-mode families inherit that edge preference [2311.09139]. In dynamical single-pulse problems, OBC support edge or corner skin accumulation, while nonlinearity determines whether transport remains convective or is interrupted by trapping [2112.06241]. In interacting bosonized fermion chains, OBC generate the mode expansions \(\sin(qx)\) and \(\cos(qx)\), the Friedel oscillations, and the ballistic light cones [2208.12017; 2304.09688; 2408.07122]. Periodic boundary conditions serve a different role. They preserve translational invariance, permit plane-wave or Bloch analysis, and expose complex spectral loops and point-gap winding [2501.01226; 2201.12653; 2604.14533]. In the nonlinear PBC Kerr ring, exact nonlinear plane waves exist, but their universal modulational instability destabilizes the clean translationally invariant state [2501.01226]. In interacting periodic many-body models, twisted flux insertion defines many-body winding numbers and persistent currents [2201.12653; 2604.14533]. The meaning of “topology” also changes across settings. In the single-particle or mean-field linearized picture, Hatano–Nelson topology is usually framed through point-gap winding and skin accumulation [2112.06241; 2501.01226]. In the nonlinear pulse-dynamics study, the directional skin accumulation is interpreted as topological because it remains tied to the winding-number sign even after the profile is reshaped by nonlinearity [2112.06241]. By contrast, the saturating continuum model shows that nonlinear coexistence of skin and extended states at fixed energy is controlled by attractor-basin geometry rather than a spectral mobility-edge mechanism [2602.17439]. A recurrent misconception concerns the relation between “nonlinear” and “interacting.” The interacting many-body Hatano–Nelson Hamiltonians are linear operators on Fock space and do not generate Gross–Pitaevskii- or DNLS-type evolution equations [2304.09688; 2408.07122; 2208.12017; 2201.12653]. Conversely, Kerr and saturating nonlinear lattices are genuine nonlinear state equations but do not incorporate quantum many-body correlations. The two lines of work are complementary rather than interchangeable. A second distinction concerns skin-effect robustness. In Kerr lattices, focusing nonlinearity generally reinforces localization, while defocusing broadens it [2311.09139]. In pulse-dynamics studies, skin accumulation is never eliminated but is strongly modified by self-trapping, yielding mixed trap-skin states [2112.06241]. In amplified bulk-packet dynamics, weak nonlinearity does not merely perturb the linear NHSE; amplification magnifies it until wave mixing and self-trapping dominate [2604.02263]. In interacting fermionic chains, by contrast, local observables can become much smoother than the single-particle skin intuition would suggest [2208.12017; 2304.09688]. A third distinction concerns spectral versus dynamical criteria. Stationary Kerr branches are classified by continuation, norm curves, and Bogoliubov spectra [2311.09139]. Pulse-dynamics phases are classified by time-averaged site amplitudes \(S_n\) [2112.06241]. Amplified bulk-packet regimes are classified by crossover times determined by spectral scales \(\Delta\) and \(d\) [2604.02263]. Growth blockade is diagnosed by the abrupt cessation of norm growth and by effective self-induced disorder [2501.01226]. Saturating continuum models are classified by phase portraits, limit cycles, and basin fractions [2602.17439]. Interacting many-body models use density profiles, currents, full counting statistics, spectral winding, persistent currents, Loschmidt echoes, and finite-size scaling [2208.12017; 2201.12653; 2304.09688; 2408.07122; 2604.14533]. Taken together, these results establish the nonlinear Hatano–Nelson model not as a single equation but as a research program. Its unifying structure is asymmetric hopping or imaginary-gauge drift combined with state-dependent feedback. In Kerr and saturating systems that feedback is explicit in the wave equation; in interacting many-body systems it is encoded in collective correlations and effective low-energy theories. The common outcome is that nonreciprocity ceases to be a purely linear-spectral phenomenon: it becomes entangled with bifurcation structure, modulational instability, self-trapping, attractor selection, many-body topology, and boundary-sensitive transport [2311.09139; 2112.06241; 2604.02263; 2602.17439; 2501.01226; 2208.12017; 2201.12653; 2304.09688; 2408.07122; 2604.14533]. Source: https://www.emergentmind.com/topics/nonlinear-hatano-nelson-model