NLD-HONLS: Nonlinear Mean-Flow Damping Model

• The NLD-HONLS model is a higher-order nonlinear Schrödinger formulation that integrates a complex mean-flow feedback term to introduce nonlocal damping.
• It embeds dissipation in the carrier–sideband interaction, yielding phase corrections that sustain organized Floquet-band dynamics and recurrent focusing.
• Numerical studies reveal that NLD-HONLS supports soliton-like rogue events and permanent spectral downshifting distinct from linear viscous decay.

The nonlinear mean-flow damping model, usually denoted NLD-HONLS, is a dissipative higher-order nonlinear Schrödinger formulation in which the Dysthe-type nonlocal mean-flow feedback is assigned a complex coefficient, so that dissipation enters through the same carrier–sideband interaction channels as the conservative mean-flow term rather than through uniform modewise decay. In the periodic deep-water setting studied in recent work, NLD-HONLS is the $\Gamma=0,\beta>0$ specialization of a unified damped HONLS family, and its distinctive signature is the appearance of interaction-dependent dissipative phase corrections that preserve organized Floquet-band dynamics during recurrent focusing while also producing permanent downshifting and strongly localized damping near steep crests (Schober, 26 May 2026, Schober et al., 27 Jul 2025).

1. Governing equation and model class

In the formulation used for spatially periodic complex envelopes $u(x,t)$ of period $L$, with $\mu=2\pi/L$, the damped higher-order NLS family is

$$i u_t + u_{xx} + 2 |u|^2 u + \epsilon \left[ 2 u (1 + i \beta) \mathscr{H}\!\left((|u|^2)_x\right) - 8 |u|^2 u_x + \frac{1}{2} u_{xxx} \right] + i \Gamma u + 2 i \epsilon \Gamma u_x = 0,$$

where $\epsilon>0$ is a small higher-order parameter, $\Gamma\ge 0$ is the viscous-damping coefficient, $\beta\ge 0$ is the nonlinear mean-flow damping coefficient, and $\mathscr H$ denotes the Hilbert transform (Schober, 26 May 2026). The leading $u_{xx}$ and $u(x,t)$0 terms are the focusing NLS dispersion and cubic nonlinearity; $u(x,t)$1 is the third-order dispersive correction; $u(x,t)$2 is the self-steepening correction; and the nonlocal mean-flow feedback is carried by the Hilbert-transform term.

Three closely related regimes are distinguished in the literature.

Regime	Parameters	Dominant damping mechanism
HONLS	$u(x,t)$3	None
NLD-HONLS	$u(x,t)$4	Nonlocal interaction-dependent damping
V-HONLS	$u(x,t)$5	Linear modewise viscous decay

The defining feature of NLD-HONLS is the factor $u(x,t)$6 multiplying the conservative mean-flow interaction. Its real part yields the usual HONLS mean-flow feedback, while the imaginary part introduces nonlinear dissipation through the same nonlocal structure. In the physical interpretation adopted in later Floquet studies, this damping is steepness-selective and localized near strongly modulated crests, and it is associated with dissipative mean-flow response induced by viscosity, turbulence, or micro-breaking (Schober et al., 27 Jul 2025). By contrast, the viscous terms $u(x,t)$7 are diagonal in Fourier space and primarily produce amplitude decay without directly entering the leading interaction-phase equations (Schober, 26 May 2026).

The energy–momentum budgets make this distinction explicit. For the unified model,

$u(x,t)$8

with $u(x,t)$9 defined by spatial averages of $L$0, $L$1, and mean-flow couplings. In the NLD-HONLS regime $L$2, one has $L$3 and $L$4, showing that energy decay and momentum evolution are governed by the nonlocal damping rather than by uniform linear attenuation (Schober et al., 27 Jul 2025).

2. Five-mode reduction and carrier–sideband structure

To isolate the dominant coherent exchange between the carrier and its first few sidebands, the 2026 interaction-phase study adopts a closed five-mode truncation,

$L$5

retaining the carrier $L$6, first sidebands $L$7, and second sidebands $L$8 (Schober, 26 May 2026). Projecting the PDE onto these Fourier modes yields a finite-dimensional nonlinear system in which linear dispersion, cubic interactions, mean-flow couplings, self-steepening, and dissipation remain explicitly separated.

The linear dispersive coefficient is

$L$9

while the projected quadratic spectral field $\mu=2\pi/L$0 determines the cubic and mean-flow interaction coefficients $\mu=2\pi/L$1, $\mu=2\pi/L$2, and $\mu=2\pi/L$3. The resulting modal equations are

$\mu=2\pi/L$4

This decomposition makes the structural difference between the two dissipative mechanisms transparent. In V-HONLS, dissipation stays diagonal through $\mu=2\pi/L$5, so the nonlinear interaction tensors $\mu=2\pi/L$6, $\mu=2\pi/L$7, and $\mu=2\pi/L$8 remain unchanged. In NLD-HONLS, the dissipative contribution is $\mu=2\pi/L$9, so the damping is injected directly into the same projected mean-flow interaction operator that governs conservative carrier–sideband exchange (Schober, 26 May 2026).

Writing $$i u_t + u_{xx} + 2 |u|^2 u + \epsilon \left[ 2 u (1 + i \beta) \mathscr{H}\!\left((|u|^2)_x\right) - 8 |u|^2 u_x + \frac{1}{2} u_{xxx} \right] + i \Gamma u + 2 i \epsilon \Gamma u_x = 0,$$0 yields amplitude and phase equations. The amplitude dynamics contain both conservative and dissipative projections of $$i u_t + u_{xx} + 2 |u|^2 u + \epsilon \left[ 2 u (1 + i \beta) \mathscr{H}\!\left((|u|^2)_x\right) - 8 |u|^2 u_x + \frac{1}{2} u_{xxx} \right] + i \Gamma u + 2 i \epsilon \Gamma u_x = 0,$$1, $$i u_t + u_{xx} + 2 |u|^2 u + \epsilon \left[ 2 u (1 + i \beta) \mathscr{H}\!\left((|u|^2)_x\right) - 8 |u|^2 u_x + \frac{1}{2} u_{xxx} \right] + i \Gamma u + 2 i \epsilon \Gamma u_x = 0,$$2, and $$i u_t + u_{xx} + 2 |u|^2 u + \epsilon \left[ 2 u (1 + i \beta) \mathscr{H}\!\left((|u|^2)_x\right) - 8 |u|^2 u_x + \frac{1}{2} u_{xxx} \right] + i \Gamma u + 2 i \epsilon \Gamma u_x = 0,$$3, together with the viscous decay terms. The phase dynamics contain the modal detuning $$i u_t + u_{xx} + 2 |u|^2 u + \epsilon \left[ 2 u (1 + i \beta) \mathscr{H}\!\left((|u|^2)_x\right) - 8 |u|^2 u_x + \frac{1}{2} u_{xxx} \right] + i \Gamma u + 2 i \epsilon \Gamma u_x = 0,$$4 and the real or imaginary projections of the same interaction coefficients divided by $$i u_t + u_{xx} + 2 |u|^2 u + \epsilon \left[ 2 u (1 + i \beta) \mathscr{H}\!\left((|u|^2)_x\right) - 8 |u|^2 u_x + \frac{1}{2} u_{xxx} \right] + i \Gamma u + 2 i \epsilon \Gamma u_x = 0,$$5. This representation is central because the carrier–sideband dynamics are governed less by the individual modal phases $$i u_t + u_{xx} + 2 |u|^2 u + \epsilon \left[ 2 u (1 + i \beta) \mathscr{H}\!\left((|u|^2)_x\right) - 8 |u|^2 u_x + \frac{1}{2} u_{xxx} \right] + i \Gamma u + 2 i \epsilon \Gamma u_x = 0,$$6 than by specific four-wave phase combinations.

3. Interaction-phase dynamics and the $$i u_t + u_{xx} + 2 |u|^2 u + \epsilon \left[ 2 u (1 + i \beta) \mathscr{H}\!\left((|u|^2)_x\right) - 8 |u|^2 u_x + \frac{1}{2} u_{xxx} \right] + i \Gamma u + 2 i \epsilon \Gamma u_x = 0,$$7 mechanism

The dominant carrier–sideband interaction phases are

$$i u_t + u_{xx} + 2 |u|^2 u + \epsilon \left[ 2 u (1 + i \beta) \mathscr{H}\!\left((|u|^2)_x\right) - 8 |u|^2 u_x + \frac{1}{2} u_{xxx} \right] + i \Gamma u + 2 i \epsilon \Gamma u_x = 0,$$8

the arguments of the principal four-wave products $$i u_t + u_{xx} + 2 |u|^2 u + \epsilon \left[ 2 u (1 + i \beta) \mathscr{H}\!\left((|u|^2)_x\right) - 8 |u|^2 u_x + \frac{1}{2} u_{xxx} \right] + i \Gamma u + 2 i \epsilon \Gamma u_x = 0,$$9 and $\epsilon>0$0 (Schober, 26 May 2026). Differentiation yields exact phase-balance equations of the form

$\epsilon>0$1

where $\epsilon>0$2 is the linear phase mismatch and $\epsilon>0$3 are the projected cubic, mean-flow, and self-steepening contributions.

The decisive simplification occurs in the carrier–sideband regime

$\epsilon>0$4

In that regime, the dissipative part of the mean-flow projection contributes leading phase-dependent terms proportional to $\epsilon>0$5, and the interaction-phase equations acquire the explicit corrections

$\epsilon>0$6

with

$\epsilon>0$7

Equivalently,

$\epsilon>0$8

$\epsilon>0$9

These terms are interaction-dependent dissipative corrections produced by the imaginary part of the mean-flow projection, not by any diagonal damping channel (Schober, 26 May 2026).

The interpretation given in the 2026 analysis is precise. The $\Gamma\ge 0$0 terms provide restoring-type feedback within the dominant carrier–sideband interaction dynamics. They do not imply rigid phase locking, but they oppose sustained monotone drift away from phase configurations favorable to recurrent focusing. This explains why NLD-HONLS can exhibit substantial interaction-phase restructuring without losing spectral organization. A common misconception is that strong phase evolution necessarily signals diffuse or disordered modulation; the finite-gap benchmarks used in the same study show that localized restructuring and cumulative drift in one interaction phase can coexist with organized quasiperiodic Floquet structure (Schober, 26 May 2026).

By contrast, viscous damping does not produce any direct $\Gamma\ge 0$1 contribution at leading order. It alters $\Gamma\ge 0$2 only indirectly through its effect on the amplitudes $\Gamma\ge 0$3. This distinction is central to the observed contrast between recurrent carrier–sideband focusing under NLD-HONLS and progressively diffuse multimode evolution under V-HONLS (Schober, 26 May 2026).

4. Floquet spectrum, spectral organization, and finite-gap interpretation

The Floquet spectral framework used throughout this literature is inherited from the focusing NLS Zakharov–Shabat problem. For an $\Gamma\ge 0$4-periodic potential $\Gamma\ge 0$5, the spatial operator is

$\Gamma\ge 0$6

with spectral parameter $\Gamma\ge 0$7. The monodromy over one spatial period defines the Floquet discriminant

$\Gamma\ge 0$8

and the Floquet spectrum is

$\Gamma\ge 0$9

Simple periodic or antiperiodic points are zeros of $\beta\ge 0$0 with nonzero derivative; double points satisfy $\beta\ge 0$1 as well; and critical points satisfy $\beta\ge 0$2 without necessarily obeying $\beta\ge 0$3 (Schober, 26 May 2026).

The 2026 comparison between nonlinear mean-flow damping and viscous damping distinguishes two types of nonintegrable spectral evolution. In an “organized” evolution, dominant bands remain separated and localized in the upper half-plane, band lengths vary smoothly, and no critical-point crossings occur. In a “reconnection” regime, bands repeatedly cross through critical points, connectivity changes, band identities swap, and the spectrum becomes diffuse and multimode (Schober, 26 May 2026). The localization criterion used both in the 2026 interaction-phase paper and in the earlier rogue-wave studies is

$\beta\ge 0$4

so that a band of length below $\beta\ge 0$5 is treated as localized or “soliton-like” (Schober, 26 May 2026, Schober et al., 2022).

Finite-gap NLS benchmarks play an interpretive role. A one-mode finite-gap benchmark exhibits bounded oscillatory interaction phases and negligible spectral drift, while a symmetric two-mode quasiperiodic benchmark shows recurrent focusing–defocusing cycles in which one interaction phase remains relatively bounded and the other undergoes cumulative drift with localized restructuring near focusing events. In both cases the Floquet spectrum stays organized in the integrable limit, demonstrating that pronounced interaction-phase evolution does not by itself imply spectral disorder (Schober, 26 May 2026). Spatially periodic breather initial data are then used to launch damped HONLS simulations. These SPBs are relatives of the Akhmediev breather and Kuznetsov–Ma soliton and provide controlled access to modulational-instability-driven focusing cycles (Schober et al., 2022).

This spectral viewpoint also underlies the “soliton-like rogue wave” classification. Under NLD-HONLS, tiny upper-half-plane bands can pinch off, generating one- or two-soliton-like spectral states from which strongly localized rogue events emerge. In early-to-middle modulational instability, all rogue waves in the 2022 SPB study occur while the spectrum is one- or two-soliton-like, whereas near instability saturation rogue waves can also arise after the spectrum has left the soliton-like state (Schober et al., 2022).

5. Numerical framework and observed dynamics

Across the recent NLD-HONLS studies, the computational setting is a periodic domain, typically $\beta\ge 0$6, with Fourier pseudo-spectral discretization in space, $\beta\ge 0$7 modes, and ETDRK4 time stepping with $\beta\ge 0$8. Representative higher-order parameter values are $\beta\ge 0$9; NLD-HONLS simulations use $\mathscr H$0, often with $\mathscr H$1, while V-HONLS comparisons use values such as $\mathscr H$2 (Schober, 26 May 2026, Schober et al., 27 Jul 2025). Initial data include coalesced two-mode SPBs $\mathscr H$3, other SPB snapshots indexed by $\mathscr H$4, and perturbed Stokes waves of the form $\mathscr H$5 with $\mathscr H$6 and $\mathscr H$7 (Schober et al., 2022, Schober et al., 27 Jul 2025).

The diagnostics are correspondingly multi-layered. Floquet spectra are computed by solving the Zakharov–Shabat system numerically, constructing $\mathscr H$8, locating zeros of $\mathscr H$9 with Müller’s method, and tracking critical points via $u_{xx}$0 (Schober, 26 May 2026). Interaction phases are extracted from the four-wave products $u_{xx}$1, spectral drift is monitored through the dominant Fourier mode $u_{xx}$2, and focusing is quantified either by the maximum envelope amplitude $u_{xx}$3 or by the strength function $u_{xx}$4, with rogue-wave threshold $u_{xx}$5 in the SPB and SRW studies (Schober, 26 May 2026, Schober et al., 27 Jul 2025).

The numerical findings are unusually consistent across studies. In the interaction-phase analysis, NLD-HONLS exhibits persistent recurrent carrier–sideband focusing together with organized Floquet evolution, even when one interaction phase undergoes substantial restructuring. The spectrum avoids critical-point crossings, and the recurrent carrier–sideband exchange remains coherent. V-HONLS instead shows progressively diffuse modulation dynamics, repeated band reconnection through both real and complex critical points, weakening persistence of the recurrent focusing structure, and loss of clear carrier–sideband organization (Schober, 26 May 2026).

The 2025 Floquet comparison sharpens this picture. For steep SPB data at $u_{xx}$6, NLD-HONLS supports a two-mode soliton-like regime up to approximately $u_{xx}$7, then a one-mode soliton-like regime up to approximately $u_{xx}$8, with no real or complex critical-point crossings observed on $u_{xx}$9. Phase coherence remains strong, with weighted phase variance typically satisfying $u(x,t)$00. For moderately steep perturbed Stokes data, the first rogue event at $u(x,t)$01 is generic rather than soliton-like, but repeated one-mode soliton-like rogue waves then occur until approximately $u(x,t)$02, again without critical-point crossings (Schober et al., 27 Jul 2025). Increasing $u(x,t)$03 reduces the number of rogue waves and causes soliton-like rogue waves to appear earlier; for $u(x,t)$04, the first rogue waves are already soliton-like (Schober et al., 27 Jul 2025).

The 2022 SPB study complements these results with a modulational-instability chronology. For coalesced two-mode SPBs, the observed spectral pathway is

$u(x,t)$05

For initialization times $u(x,t)$06, all rogue waves occur during the one- or two-soliton-like stages; near saturation, $u(x,t)$07, rogue waves may also arise after the spectrum exits the soliton-like state (Schober et al., 2022). In both the 2022 and 2025 studies, permanent spectral downshift is closely coupled to the NLD dynamics: in all reported experiments the last rogue wave precedes the permanent downshift time $u(x,t)$08, and in NLD-HONLS the delay between these events is relatively short, unlike the much later downshift seen in V-HONLS (Schober et al., 27 Jul 2025).

6. Relation to adjacent models, misconceptions, and limitations

NLD-HONLS belongs to a broader HONLS/Dysthe family and should not be conflated with either conservative mean-flow models or linearly damped HONLS formulations. In the finite-depth mean-flow theory of Gomel, Trulsen, and Slunyaev, the mean-flow contribution is explicitly inviscid and nonlocal: it reproduces the deep-water Dysthe Hilbert term at third order and the classical finite-depth local mean-flow at small $u(x,t)$09, but it does not introduce physical dissipation. That work therefore clarifies an important misconception: a mean-flow term is not inherently a damping term. The damping in NLD-HONLS arises only when the mean-flow coefficient acquires an imaginary part, as in $u(x,t)$10, whereas the finite-depth operators themselves are conservative (Gomel et al., 2023).

A second useful contrast is historical. Earlier damped HONLS studies analyzed higher-order NLS with conservative mean-flow correction and linear dissipation $u(x,t)$11, and they established a Floquet-based stabilization criterion in which stabilization corresponds to the elimination of all complex degenerate spectral elements, namely complex double points and complex critical points. Those results provide a baseline for later NLD-HONLS comparisons, but they are not themselves studies of nonlinear mean-flow damping (Schober et al., 2020). The later NLD-HONLS literature can be read as showing that when dissipation is embedded directly into the mean-flow interaction channel, the system suppresses recurrent critical-point creation more effectively than linearly damped HONLS or V-HONLS (Schober et al., 27 Jul 2025).

Two additional misconceptions are addressed directly by the recent analyses. First, strong interaction-phase drift does not necessarily imply loss of quasiperiodic spectral organization; finite-gap benchmarks and NLD-HONLS simulations show that pronounced local restructuring of $u(x,t)$12 can coexist with persistent one- or two-band Floquet organization (Schober, 26 May 2026). Second, permanent downshifting in NLD-HONLS is not merely a generic consequence of adding any weak damping. The cited studies attribute it to the amplitude-dependent, nonlocal damping structure of the mean-flow term, which breaks the balance that preserves the spectral center in linearly damped NLS/Dysthe-type models (Schober et al., 2022).

The principal limitations are also explicit. The five-mode truncation is valid only while spectral energy remains concentrated near the carrier and first few sidebands; stronger downshifting or broader spectra require additional retained modes, such as seven- or nine-mode extensions. The asymptotic setting is near-integrable, with small $u(x,t)$13, $u(x,t)$14, and $u(x,t)$15; stronger dissipation or broader-band dynamics demand refined asymptotics and more elaborate closures. The envelope description itself is narrowband and periodic. Finally, the Floquet spectral diagnostic is inherited from the integrable NLS Lax pair and is therefore not invariant under the nonintegrable HONLS, NLD-HONLS, or V-HONLS flows; it is recomputed in time as a structural diagnostic rather than an exact invariant (Schober, 26 May 2026, Schober et al., 27 Jul 2025).

Within those limits, the current consensus in the cited literature is sharp. NLD-HONLS differs from viscous damping not merely in damping strength but in damping topology: by injecting dissipation into the nonlocal mean-flow interaction itself, it generates $u(x,t)$16 feedback in the dominant carrier–sideband phases, preserves organized Floquet-band evolution over long intervals, supports persistent recurrent focusing and soliton-like rogue-wave episodes, and couples those coherent events to permanent downshifting in a way that purely modewise dissipation does not (Schober, 26 May 2026, Schober et al., 27 Jul 2025).