NRDS: Nonreciprocity-Dressed Solitons

• Nonreciprocity-Dressed Solitons (NRDSs) are nonlinear solitary excitations whose defining characteristic is that their propagation and stability are governed by intrinsic nonreciprocal mechanisms rather than boundary-induced effects.
• In non-Hermitian lattices, NRDSs emerge from bulk band nonreciprocity, persisting under periodic boundary conditions and exhibiting asymmetric dispersion distinct from skin-mode-assisted solitons.
• Active metamaterial systems demonstrate NRDS behavior via local nonreciprocal drive and power-dependent dynamics, leading to topological, breathing, and self-induced soliton states with tunable stability.

Searching arXiv for the cited NRDS-related papers to ground the article in current literature. arXiv search query: (Li et al., 4 Aug 2025) arXiv search query: (Veenstra et al., 2023) Nonreciprocity-dressed solitons (NRDSs) are solitary nonlinear excitations whose existence, transport, localization profile, or internal dynamics are modified in an essential way by nonreciprocity. In current arXiv literature, the label is used in more than one closely related sense. In non-Hermitian lattices, NRDSs denote solitons that originate from bulk band nonreciprocity rather than from skin-mode accumulation at a boundary (Li et al., 4 Aug 2025). In active mechanical metamaterials, the same label is used more broadly for topological or breathing solitons whose motion is “dressed” by a local non-reciprocal drive, so that nonreciprocity selects propagation direction, terminal velocity, and stability properties (Veenstra et al., 2023, Brandenbourger et al., 2024). A neighboring development studies solitons with self-induced nonreciprocal dynamics in a discrete nonlinear Schrödinger setting, where the effect is power dependent and tied to point-gap topology (Castro et al., 2024). Taken together, these works suggest that NRDSs are best understood not as a single canonical solution family, but as a class of nonlinear localized states in which reciprocity breaking is constitutive rather than perturbative.

1. Terminology and distinguishing criteria

A central distinction in the recent literature is between solitons assisted by the non-Hermitian skin effect (NHSE) and solitons governed by band nonreciprocity. The stacked Su–Schrieffer–Heeger-like study explicitly separates three nonlinear regimes in the $y$-$g$ plane: the nonlinear perturbative skin mode (NPSM), the skin-mode-assisted soliton (SMAS), and the nonreciprocity-dressed soliton (NRDS) (Li et al., 4 Aug 2025). In that usage, SMASs originate from skin-mode localization, whereas NRDSs stem from bulk asymmetric dispersion and remain present even when the NHSE is removed by imposing periodic boundary conditions.

This distinction is important because NRDSs are often misidentified with any localized nonlinear state in a non-Hermitian lattice. The phase-diagram analysis instead treats SMASs and NRDSs as physically distinct objects with different formation mechanisms, spatial profiles, and boundary-condition dependence.

Class	Dominant mechanism	Behavior under PBCs
NPSM	Weakly nonlinear continuation of a skin mode	Vanishes as a distinct skin-localized state
SMAS	Skin-mode localization lowers the soliton threshold	Disappears
NRDS	Bulk band nonreciprocity and asymmetric dispersion	Persists

A second terminological extension appears in active metamaterials. There, NRDS refers to topological solitons or breathing solitons whose effective dynamics are altered by odd, local, non-reciprocal feedback. In that setting, the “dressing” is dynamical: nonreciprocity does not merely shift a pre-existing soliton solution, but pumps and stabilizes a one-way nonlinear excitation (Veenstra et al., 2023, Brandenbourger et al., 2024). This broader usage preserves the same conceptual core: nonreciprocity is embedded in the soliton’s governing mechanism.

2. Bulk band-nonreciprocity NRDSs in stacked SSH-like lattices

The prototype lattice for the band-nonreciprocity formulation is a two-dimensional stacked SSH-like model with reciprocal intracell hopping $t_x$ along $x$, reciprocal intercell hopping $t_y$ along $y$, and nonreciprocal intercell hopping along $x$,

$K_R = K(1-y), \qquad K_L = K(1+y),$

where $y \in [0,1]$ is the nonreciprocity parameter (Li et al., 4 Aug 2025). The nonlinear eigenproblem is

$E u_n = H_{nm} u_m + g |u_n|^2 u_n,$

with on-site Kerr nonlinearity $g$0, and $g$1 corresponding to the focusing regime. The representative parameter set reported for this model is

$g$2

The defining claim of this work is that NRDSs are not boundary-induced objects. Under open boundary conditions, both NHSE and band nonreciprocity are present, so SMASs and NRDSs can both occur. Under periodic boundary conditions, the NHSE disappears, SMASs disappear, but NRDSs remain. This persistence is the operational criterion by which the paper attributes NRDS formation to bulk band nonreciprocity rather than to edge accumulation.

To interpret this, the authors formulate a Wannier-function-based nonlinear Hamiltonian by expanding

$g$3

and, under the approximation that the nonlinear interaction is dominated by valence-band Wannier functions within the same unit cell, derive

$g$4

with

$g$5

The real-space dispersion kernel is obtained from the Bloch bands through

$g$6

This formulation is used to show that soliton formation depends on how skin-mode localization and band nonreciprocity respectively suppress or enhance wave dispersion. For SMASs, the relevant suppression occurs at the edge where the skin mode is already concentrated. For NRDSs, the crucial effect is asymmetric bulk dispersion.

3. Phase boundaries, localization profiles, and stability

The stacked SSH-like analysis identifies three thresholds: $g$7 for SMAS formation, $g$8 for NRDS formation, and $g$9 for a type transition between NRDS-II and NRDS-I. For the representative case $t_x$0, the reported values are

$t_x$1

(Li et al., 4 Aug 2025). The threshold trends extracted from the two-leading-Wannier-function approximation are that $t_x$2 decreases as $t_x$3 increases, $t_x$4 increases with $t_x$5, and $t_x$6 is almost independent of $t_x$7, remaining close to the molecular-limit prediction

$t_x$8

The spatial profiles of the phases are correspondingly different. Linear skin modes are boundary localized under open boundary conditions. NPSMs remain close to those linear skin modes under weak nonlinearity. SMASs form at the same boundary region as the linear skin mode, are strongly confined, often within one unit cell, and decay exponentially into the bulk. NRDSs, by contrast, appear in the bulk, away from the skin-localized region, and decay exponentially on both sides with asymmetric decay lengths that reflect the nonreciprocal hopping asymmetry.

The dynamical-stability analysis perturbs a nonlinear eigenstate as

$t_x$9

and diagnoses stability through the eigenvalues of the stability matrix $x$0, using

$x$1

as the indicator. The reported stability structure is selective rather than generic: a stability gap appears only near the NPSM-to-SMAS transition at $x$2, while the other transitions, including those involving NRDSs, are dynamically stable. This produces a stability reentrance sequence—stable linear skin mode, unstable intermediate NPSM, stable SMAS—whereas the NRDS regime does not exhibit the same instability corridor.

A common misconception is that asymmetric localization in a non-Hermitian lattice must be a skin-effect signature. The persistence of NRDSs under periodic boundary conditions shows that asymmetry in the soliton profile can instead be a bulk-dispersion effect.

4. Non-reciprocal topological solitons in active metamaterials

In the active-mechanical formulation, NRDSs are topological solitons or antisolitons whose motion is modified by local non-reciprocal feedback. The experimental platform is a chain of 50 mechanical oscillators elastically coupled by links of stiffness $x$3, equipped with motors, sensors, and feedback control, with active torque

$x$4

(Veenstra et al., 2023). To convert unstable skin-wave-like excitations into propagating topological domain walls, the system is endowed with a bistable on-site potential, realized by magnets attached to the rotor arms and a periodic array of magnets on the substrate.

The corresponding discrete non-reciprocal Frenkel–Kontorova model is

$x$5

where $x$6, $x$7 is dissipation, and $x$8 is the bistable potential strength. In the continuum limit this becomes the odd sine-Gordon equation,

$x$9

The soliton velocity obeys the perturbative law

$t_y$0

For $t_y$1, this gives a stable fixed point at

$t_y$2

The perturbative stability analysis yields

$t_y$3

with instability onset at

$t_y$4

and the highest wavenumbers becoming unstable first. In the overdamped limit,

$t_y$5

so $t_y$6 always.

A defining property of this class is directional degeneracy between kinks and antikinks. Under a reciprocal constant drive, solitons and antisolitons move in opposite directions; under the local non-reciprocal term $t_y$7, both propagate in the same direction. The paper interprets this as coupling directly to the translational Goldstone mode of the domain wall.

The waveguide application makes this operational. The system transmits the word “ODD” in Morse code by launching soliton and antisoliton excitations from one edge. With an added constant bias $t_y$8, the terminal velocities become

$t_y$9

which enables a nonlinear filtering scheme using two chains with opposite bias $y$0 and $y$1.

5. Non-reciprocal breathing solitons

A second active-matter realization reports breathing, unidirectional, arbitrarily long-lived solitons in a non-reciprocal, non-conservative metamaterial (Brandenbourger et al., 2024). Experimentally, the platform is a ring of 20 active mechanical unit cells, each with one rotational degree of freedom $y$2, elastic couplings, a microcontroller and motor, angular encoders, and the active torque

$y$3

A programmable hammer delivers a localized kick.

The observed state is a wave packet with a fast carrier oscillation inside a compact traveling envelope. The reported propagation speed is about

$y$4

the lifetime exceeds

$y$5

the envelope amplitude remains roughly

$y$6

the width is about

$y$7

the carrier frequency is about

$y$8

and the phase velocity is

$y$9

The propagation direction reverses with the sign of $x$0.

The discrete model used to reproduce the experiments is

$x$1

For analysis, the paper uses the continuum equation

$x$2

and, in the small-amplitude regime, the envelope reduction

$x$3

which leads to the non-reciprocal dissipative nonlinear Schrödinger equation

$x$4

Inverse-scattering-based perturbation theory gives the adiabatic parameter dynamics

$x$5

together with

$x$6

The nontrivial fixed points satisfy

$x$7

provided

$x$8

and the separatrix is approximated by

$x$9

A saddle-node bifurcation occurs at

$K_R = K(1-y), \qquad K_L = K(1+y),$0

The paper emphasizes that the breathing soliton is controlled by an unstable fixed point rather than a strictly stable attractor. Long lifetimes arise when parameters and initial data place the dynamics near that fixed point and near the bifurcation. The longest-lived states occur around

$K_R = K(1-y), \qquad K_L = K(1+y),$1

matching

$K_R = K(1-y), \qquad K_L = K(1+y),$2

The initial kick must also exceed about

$K_R = K(1-y), \qquad K_L = K(1+y),$3

For the continuum kick profile

$K_R = K(1-y), \qquad K_L = K(1+y),$4

discrete spectral structure appears when

$K_R = K(1-y), \qquad K_L = K(1+y),$5

and for $K_R = K(1-y), \qquad K_L = K(1+y),$6 the breather-like regime is concentrated near

$K_R = K(1-y), \qquad K_L = K(1+y),$7

6. Self-induced topological nonreciprocity and broader interpretation

A related but distinct direction considers solitons with self-induced nonreciprocal dynamics in a discrete nonlinear Schrödinger equation (Castro et al., 2024). The abstract states that the nonreciprocal behavior is dependent on soliton power, arises from the interplay between linear and nonlinear terms in the equations of motion, is initially stable at high power, and develops nonreciprocal instabilities as power decreases, leading to unidirectional acceleration and amplification. It further attributes protection to winding numbers defined on the solitons’ mean-field Hamiltonian and stability matrix, thereby linking nonlinear dynamics to point-gap topology in non-Hermitian Hamiltonians.

The presently available figure stub for that work contains only a vertical axis $K_R = K(1-y), \qquad K_L = K(1+y),$8, a threshold $K_R = K(1-y), \qquad K_L = K(1+y),$9, and two labeled regimes above and below the threshold. Because the governing equations, stability analysis, and explicit topological construction are not available in the supplied material, the relation to NRDS terminology cannot be reconstructed in detail. Still, the abstract strongly suggests a neighboring concept in which nonreciprocal dressing is self-induced by the nonlinear state itself, rather than inherited from fixed nonreciprocal hopping or externally imposed active feedback.

Across these strands, one broad conclusion emerges. NRDSs are not defined solely by being localized waves in an asymmetric medium. Rather, the recent literature uses the term for solitary excitations whose propagation, threshold, or stability is set by one of three mechanisms: bulk band nonreciprocity in non-Hermitian lattices, local odd non-reciprocal driving in active topological media, or power-dependent self-induced nonreciprocity. This suggests that the unifying content of the term is mechanistic: nonreciprocity becomes part of the soliton’s constitutive effective dynamics.