V-HONLS: Viscous Damping in Higher-Order NLS

• V-HONLS is a weakly viscous extension of the higher-order nonlinear Schrödinger equation that models modulational instability, rogue-wave formation, and spectral downshifting in deep-water wave trains.
• The model incorporates a leading-order linear damping term (iΓu) and a higher-order correction (ε·2Γ uₓ), which produce broader, less localized rogue waves and robust spectral downshifting.
• Numerical simulations using Floquet spectral diagnostics reveal disordered spectral evolution and a decoupling between rogue-wave events and permanent carrier downshift, distinguishing V-HONLS from other damping models.

Viscous Damping Model (V-HONLS) denotes the weakly viscous extension of the higher-order nonlinear Schrödinger equation used to study modulational instability, rogue-wave formation, and spectral downshifting in deep-water wave trains. In the formulation analyzed in "Soliton-like Rogue Wave Dynamics in Dissipative Higher-Order NLS Models: A Floquet Spectral Perspective" (Schober et al., 27 Jul 2025), V-HONLS is obtained from a common higher-order NLS skeleton by setting the nonlinear mean-flow damping parameter to zero and retaining a small viscous parameter $\Gamma$, so that dissipation enters through a leading-order linear damping term and a higher-order correction acting on the whole envelope. Within that study, V-HONLS is contrasted with both conservative HONLS and the nonlinear mean-flow damping model (NLD-HONLS), and is characterized by disordered Floquet spectral evolution, broader and less localized rogue waves, and robust but gradually realized spectral downshifting (Schober et al., 27 Jul 2025).

1. Governing equation and model identity

The common dissipative higher-order NLS framework considered for HONLS, NLD-HONLS, and V-HONLS is

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

where $\epsilon>0$ is the higher-order parameter, $0<\Gamma,\beta\ll 1$, $u(x,t)$ is the complex envelope, periodic in $x$ with period $L$, and $\mathscr H$ is the Hilbert transform. The conservative HONLS is recovered by setting $\Gamma=\beta=0$, while NLD-HONLS is obtained by $\Gamma=0$ and $$i u_t + u_{xx} + 2u|u|^2 + i \Gamma u + \epsilon\left[ 2u (1 + i\beta)\mathscr{H} \left(|u|^2\right)_x - 8i|u|^2 u_x + \frac{i}{2} u_{xxx} + 2\Gamma u_x \right] = 0,$$0 (Schober et al., 27 Jul 2025).

V-HONLS is the specialization

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

with $$i u_t + u_{xx} + 2u|u|^2 + i \Gamma u + \epsilon\left[ 2u (1 + i\beta)\mathscr{H} \left(|u|^2\right)_x - 8i|u|^2 u_x + \frac{i}{2} u_{xxx} + 2\Gamma u_x \right] = 0,$$2. Its viscous contributions are therefore the leading-order linear damping term $$i u_t + u_{xx} + 2u|u|^2 + i \Gamma u + \epsilon\left[ 2u (1 + i\beta)\mathscr{H} \left(|u|^2\right)_x - 8i|u|^2 u_x + \frac{i}{2} u_{xxx} + 2\Gamma u_x \right] = 0,$$3 and the higher-order correction $$i u_t + u_{xx} + 2u|u|^2 + i \Gamma u + \epsilon\left[ 2u (1 + i\beta)\mathscr{H} \left(|u|^2\right)_x - 8i|u|^2 u_x + \frac{i}{2} u_{xxx} + 2\Gamma u_x \right] = 0,$$4. The study fixes $$i u_t + u_{xx} + 2u|u|^2 + i \Gamma u + \epsilon\left[ 2u (1 + i\beta)\mathscr{H} \left(|u|^2\right)_x - 8i|u|^2 u_x + \frac{i}{2} u_{xxx} + 2\Gamma u_x \right] = 0,$$5 and varies $$i u_t + u_{xx} + 2u|u|^2 + i \Gamma u + \epsilon\left[ 2u (1 + i\beta)\mathscr{H} \left(|u|^2\right)_x - 8i|u|^2 u_x + \frac{i}{2} u_{xxx} + 2\Gamma u_x \right] = 0,$$6 over

$$i u_t + u_{xx} + 2u|u|^2 + i \Gamma u + \epsilon\left[ 2u (1 + i\beta)\mathscr{H} \left(|u|^2\right)_x - 8i|u|^2 u_x + \frac{i}{2} u_{xxx} + 2\Gamma u_x \right] = 0,$$7

The same source states that an additional ad hoc term, $$i u_t + u_{xx} + 2u|u|^2 + i \Gamma u + \epsilon\left[ 2u (1 + i\beta)\mathscr{H} \left(|u|^2\right)_x - 8i|u|^2 u_x + \frac{i}{2} u_{xxx} + 2\Gamma u_x \right] = 0,$$8, used in some viscous Dysthe or HONLS variants, is not included because it has negligible impact on the timescales considered. In this sense, the published V-HONLS is a specific weakly viscous closure rather than an all-inclusive dissipative HONLS family.

2. Dissipation mechanism and physical interpretation

In the cited formulation, V-HONLS is derived from a weakly viscous extension of the Euler equations associated with Carter & Govan (2016) and Dias–Dyachenko–Zakharov (2008), and is described as accounting for energy loss due to background fluid viscosity. The parameter $$i u_t + u_{xx} + 2u|u|^2 + i \Gamma u + \epsilon\left[ 2u (1 + i\beta)\mathscr{H} \left(|u|^2\right)_x - 8i|u|^2 u_x + \frac{i}{2} u_{xxx} + 2\Gamma u_x \right] = 0,$$9 therefore represents bulk-viscosity effects in deep-water wave trains, rather than boundary-layer-specific losses (Schober et al., 27 Jul 2025).

A central distinction between V-HONLS and NLD-HONLS is the manner in which damping is distributed. In V-HONLS, damping is linear in $\epsilon>0$0 and acts directly on the entire wave envelope through $\epsilon>0$1 and $\epsilon>0$2. In NLD-HONLS, by contrast, dissipation enters only through the imaginary part of the nonlocal mean-flow feedback term,

$\epsilon>0$3

and is therefore strongly steepness-dependent and localized near steep envelope crests. The data explicitly describe the NLD-HONLS damping as negligible on the background and significant mainly near the crest of steep, strongly modulated envelopes, whereas V-HONLS damps high- and low-amplitude regions in a more uniform way, modulo amplitude (Schober et al., 27 Jul 2025).

This distinction organizes the phenomenology. For weak viscosity, early modulational-instability growth and the first rogue-wave events remain close to the conservative HONLS dynamics, but longer-time evolution, phase coherence, and downshifting are substantially altered. V-HONLS is thus neither a small perturbation in every dynamical respect nor a selective coherence-preserving mechanism; it is a weak but global dissipative perturbation of the HONLS envelope dynamics.

3. Floquet spectral framework

The principal diagnostic used to analyze V-HONLS is the instantaneous Floquet spectrum associated with the Zakharov–Shabat spectral problem of integrable NLS. The $\epsilon>0$4- and $\epsilon>0$5-Lax operators are written as

$\epsilon>0$6

and

$\epsilon>0$7

Given a fundamental solution matrix $\epsilon>0$8, the Floquet discriminant is

$\epsilon>0$9

and the Floquet spectrum is

$0<\Gamma,\beta\ll 1$0

The analysis tracks periodic or antiperiodic points, simple periodic points, critical points, and double points. Each pair of simple points defines a nonlinear mode; complex double points signal exponential instabilities; complex critical points identify weaker but dynamically important band crossings (Schober et al., 27 Jul 2025).

For dissipative evolutions such as V-HONLS, the Floquet spectrum is not invariant. The procedure therefore recomputes $0<\Gamma,\beta\ll 1$1 at discrete times and treats it as an instantaneous diagnostic of nonlinear mode content. Rogue waves are classified using two simultaneous conditions. First, the strength

$0<\Gamma,\beta\ll 1$2

must satisfy $0<\Gamma,\beta\ll 1$3. Second, a one- or two-mode soliton-like rogue wave (SRW) requires at least one band length

$0<\Gamma,\beta\ll 1$4

to satisfy

$0<\Gamma,\beta\ll 1$5

Within this diagnostic, V-HONLS exhibits transient soliton-like states at early times but generally evolves toward broader, multimode, non-SRW events (Schober et al., 27 Jul 2025).

4. Rogue-wave regimes and phase coherence

The reported V-HONLS dynamics depend strongly on initial steepness. For steep initial data given by an exact two-mode spatially periodic breather (SPB), and for $0<\Gamma,\beta\ll 1$6, $0<\Gamma,\beta\ll 1$7, the evolution initially follows HONLS closely. At the first rogue wave, around $0<\Gamma,\beta\ll 1$8, the two upper spectral bands $0<\Gamma,\beta\ll 1$9 and $u(x,t)$0 are both short and satisfy the soliton threshold $u(x,t)$1, so the event is classified as a two-mode SRW (Schober et al., 27 Jul 2025).

Later SPB dynamics are qualitatively different. The number of rogue waves is reduced relative to HONLS, and later events occur in more disordered spectral states. The paper identifies an example of a one-mode SRW at $u(x,t)$2, where one band is short, and a generic rogue wave at $u(x,t)$3, where the bands are not short. The broad trend is that early one- and two-mode SRWs remain possible, but long-lived soliton-like regimes are not sustained under viscous damping. The background instead undergoes repeated complex critical-point crossings and favors a more disordered multimode state (Schober et al., 27 Jul 2025).

For moderately steep perturbed Stokes initial data,

$u(x,t)$4

the first V-HONLS rogue wave for $u(x,t)$5 occurs at $u(x,t)$6 and is explicitly reported as generic rather than soliton-like. For $u(x,t)$7 and $u(x,t)$8, none of the rogue waves satisfy the soliton-like band-length criterion; one-mode SRWs appear only at larger $u(x,t)$9 (Schober et al., 27 Jul 2025).

Phase coherence is quantified by the phase variance diagnostic

$x$0

with

$x$1

For steep SPB data, V-HONLS has lower mean PVD than HONLS because it suppresses the third dephased nonlinear mode that appears in HONLS. Nevertheless, its PVD remains higher and more erratic than in NLD-HONLS. The resulting rogue waves are therefore less sharply localized and less persistently soliton-like than in the nonlinear mean-flow damped model.

5. Spectral downshifting and dissipative balances

The dissipative balances in the general dissipative HONLS system are expressed through the energy and momentum densities

$x$2

For the full model,

$x$3

where

$x$4

and

$x$5

For V-HONLS, where $x$6, these reduce to

$x$7

Thus viscosity causes both energy and momentum decay, while the $x$8- and $x$9-couplings generate spectral asymmetry and net downshift (Schober et al., 27 Jul 2025).

Downshifting is tracked by the spectral mean

$L$0

and the spectral peak $L$1, defined as the Fourier mode of maximal amplitude. The adopted criterion is that frequency downshifting occurs when $L$2 and $L$3 moves to a lower mode. Permanent downshift time $L$4 is the time after which the carrier mode never again becomes dominant (Schober et al., 27 Jul 2025).

For $L$5, the paper reports that $L$6 is strictly decreasing on $L$7 for both SPB and perturbed-Stokes initial data. Permanent downshift occurs at $L$8 for SPB data and at $L$9 for Stokes data. In all V-HONLS cases studied, the last rogue wave occurs before $\mathscr H$0, often with a significant delay between the two. This is one of the paper’s main dynamical conclusions: in V-HONLS, rogue-wave occurrence and permanent downshift are largely decoupled, unlike the closer association observed in NLD-HONLS (Schober et al., 27 Jul 2025).

6. Numerical realization and diagnostics

The numerical implementation uses a Fourier spectral discretization on a periodic domain and fourth-order exponential time-differencing Runge–Kutta (ETD4RK) time stepping. Matrix exponentials are approximated by Padé schemes, and a typical time step is $\mathscr H$1. For the Stokes-wave example with two unstable modes, the simulations use $\mathscr H$2 Fourier modes, integrate to $\mathscr H$3 for rogue-wave and Floquet analysis, and to $\mathscr H$4 for downshifting studies. In the conservative case, the HONLS invariants are conserved to $\mathscr H$5 (Schober et al., 27 Jul 2025).

Floquet spectra are recomputed from the numerical solution at stored times. The computation solves the Zakharov–Shabat $\mathscr H$6-problem over one period, constructs $\mathscr H$7, locates zeros of $\mathscr H$8 with Müller’s method, and traces spectral bands and critical points. The reported spectral accuracy is $\mathscr H$9, and the output interval for spectral diagnostics is $\Gamma=\beta=0$0 (Schober et al., 27 Jul 2025).

These numerical choices matter because V-HONLS diagnostics are intrinsically spectral rather than solely waveform-based. The model’s characterization as “disordered” is not a purely visual statement about envelope profiles; it refers to repeated complex critical-point crossings, lack of sustained short-band organization, and the absence of the persistent ordered Floquet structure seen in NLD-HONLS.

7. Broader damping context

A broader theoretical context for the qualifier “viscous” is provided by generalized damping theory based on memory kernels. In the particle–bath framework of "A generalized framework for viscous and non-viscous damping models" (Ganguly et al., 11 Mar 2025), the damping force has the nonlocal form

$\Gamma=\beta=0$1

and the classical viscous model is recovered when the kernel becomes local,

$\Gamma=\beta=0$2

This suggests that the V-HONLS closure, which uses the local terms $\Gamma=\beta=0$3 and $\Gamma=\beta=0$4, may be viewed as a Markovian viscous specialization within a much larger family of dissipative models.

An additional perspective comes from "Mechanical energy and mean equivalent viscous damping for SDOF fractional oscillators" (Yuan et al., 2016), which shows that a fractional damping element can be replaced, in steady harmonic motion, by a mean equivalent viscous coefficient

$\Gamma=\beta=0$5

A plausible implication is that when memory effects are relevant, a constant viscous parameter such as $\Gamma=\beta=0$6 should be interpreted as a regime-specific local approximation rather than a universal representation of dissipation.

Within its own published scope, however, V-HONLS is not presented as a memory-kernel or fractional closure. It is a weakly viscous, local-in-time dissipative extension of HONLS designed for deep-water wave trains. Its defining features are the pair of viscous terms $\Gamma=\beta=0$7 and $\Gamma=\beta=0$8, its tendency to suppress coherent soliton-like spectral organization, and its ability to produce robust spectral downshifting without requiring a tight dynamical linkage between rogue-wave events and permanent carrier downshift (Schober et al., 27 Jul 2025).