Soliton Gas Sea States: Nonlinear Dynamics

• Soliton gas sea states are complex wave fields where a dense ensemble of solitons interacts, defined through nonlinear spectral measures from integrable models.
• Laboratory and field studies using KdV and focusing NLSE frameworks demonstrate the distinct kinetic and statistical signatures of these nonlinear regimes.
• Observations reveal specific spectral separations and non-Gaussian amplitude statistics, offering advanced diagnostics for characterizing oceanic soliton gases.

Soliton gas sea states are irregular wave fields in which a large random ensemble of interacting solitons dominates the relevant dynamics. In shallow water, the low-frequency component can be interpreted as a dense soliton gas described by the soliton limit of the Korteweg–de Vries equation, whereas in deep water the corresponding framework is the focusing nonlinear Schrödinger equation, its inverse scattering transform, and the density of states in nonlinear spectral space (Costa et al., 2014, Suret et al., 2020). The term covers field observations, laboratory realizations, and kinetic descriptions of sea-like states in which nonlinear coherent structures, rather than only linear Fourier modes, provide the principal macroscopic organization; rare open-ocean cases with very high soliton dominance have also been identified (Lee et al., 6 Oct 2025).

1. Conceptual definition and terminology

In the modern literature, a soliton gas is a many-particle statistical ensemble of interacting solitons in an integrable dispersive PDE, or, in the focusing 1D-NLSE language, a large random ensemble of solitons characterized statistically through the spectral data of the associated integrable equation (Ferapontov et al., 2021, Suret et al., 2020). In ocean-wave usage, the phrase becomes physically meaningful when the wave field is not merely observed to contain isolated solitons, but is shown to be soliton-dominated in either the low-frequency shallow-water sector or the discrete nonlinear spectrum of a deep-water envelope field (Costa et al., 2014, Lee et al., 6 Oct 2025).

Several distinctions structure the subject. “Soliton turbulence” in shallow water is used synonymously with “integrable soliton turbulence,” because the governing KdV dynamics are completely integrable; in the Currituck Sound interpretation, the low-frequency energy is viewed as a dense soliton gas with random finite-gap-theory phases and highly non-Gaussian statistics (Costa et al., 2014). In focusing-NLS settings with a finite carrier, the more general object is a breather gas rather than a zero-background soliton gas, because the exceptional Stokes band represents the carrier-wave background; collapsing that band yields the soliton-gas limit (El et al., 2019). The literature also distinguishes a dense gas, which cannot be represented as a superposition of individual solitons, from a diluted gas of weakly interacting structures, and it introduces condensate limits in which interaction effects dominate the macroscopic state (Suret et al., 2020, Congy et al., 2022).

The density of states is the central macroscopic quantity. In the focusing-NLS formulation, $u\,d\lambda\,dx$ or $u\,d\beta\,d\gamma\,dx$ counts soliton states in a nonlinear spectral phase-space element; in KdV soliton-gas theory, $f(\eta,x,t)$ is the spectral density of states that replaces a linear spectrum as the natural descriptor of a coherent nonlinear sea (Suret et al., 2020, Ferapontov et al., 2021). This suggests that “sea state” classification is broadened from linear power spectra, moments, and directional statistics to nonlinear spectral populations of coherent modes.

2. Governing equations and nonlinear spectral descriptions

The shallow-water branch is organized primarily by KdV and its bidirectional generalizations. In the Currituck Sound field study, the low-frequency component is modeled by the Korteweg–de Vries equation

$$\eta_t + c_0 \eta_x + \alpha \eta \eta_x + \beta \eta_{xxx} = 0,$$

with

$$c_0 = \sqrt{gh}, \qquad \alpha = \frac{3c_0}{2h}, \qquad \beta = \frac{c_0 h^2}{6},$$

while the higher-frequency spectral peak is associated with the nonlinear Schrödinger equation

$$i(\psi_t + C_g \psi_x) + \mu \psi_{xx} + \nu |\psi|^2 \psi = 0.$$

The key point is spectral separation: a low-frequency KdV shallow-water regime coexists with a higher-frequency NLS wave-packet regime, allowing the low-frequency part of the sea state to be isolated and identified as solitonic by finite-gap theory (Costa et al., 2014).

Deep-water laboratory sea states are formulated in the focusing 1D-NLSE. In physical variables,

$$\frac{\partial A}{\partial Z} + \frac{1}{C_g} \frac{\partial A}{\partial T}= i \frac{k_0}{\omega_0^2} \frac{\partial^2 A}{\partial T^2} + i \alpha k_0^3 |A|^2 A,$$

with $\omega_0^2=k_0g$ and $C_g=g/(2\omega_0)$, and in canonical form

$$i \frac{\partial \psi}{\partial t} + \frac{1}{2} \frac{\partial^2 \psi}{\partial x^2} + |\psi|^2 \psi=0$$

or, in an equivalent normalization used for interacting gas jets,

$u\,d\beta\,d\gamma\,dx$0

Each discrete Zakharov–Shabat eigenvalue $u\,d\beta\,d\gamma\,dx$1 parametrizes an envelope soliton, with $u\,d\beta\,d\gamma\,dx$2 related to amplitude and $u\,d\beta\,d\gamma\,dx$3 to velocity (Suret et al., 2020, Fache et al., 2023).

Bidirectional shallow-water and directional two-dimensional settings require broader integrable models. The Kaup–Boussinesq system

$u\,d\beta\,d\gamma\,dx$4

is used for bidirectional shallow-water soliton gases, because it accommodates both overtaking and head-on collisions (Redor et al., 2019, Congy et al., 2020). For stationary 2D gases, the time-independent reduction of KPII,

$u\,d\beta\,d\gamma\,dx$5

coincides with the integrable good Boussinesq equation in the $u\,d\beta\,d\gamma\,dx$6-plane and supports a kinetic theory of stationary KPII line-soliton gases (Bonnemain et al., 2024).

The spectral description depends on the integrable equation. KdV field data in shallow water were analyzed by finite-gap theory as a nonlinear Fourier transform for periodic or quasiperiodic records (Costa et al., 2014). Deep-water reflectionless random $u\,d\beta\,d\gamma\,dx$7-soliton states are synthesized and diagnosed by the inverse scattering transform of the Zakharov–Shabat problem; the corresponding density of states in $u\,d\beta\,d\gamma\,dx$8 space becomes the central macroscopic descriptor of the gas (Suret et al., 2020). For focusing NLS with a nonzero Stokes band, the thermodynamic limit of finite-gap spectra yields a breather-gas theory; collapsing the Stokes band recovers the soliton-gas limit (El et al., 2019).

3. Shallow-water realizations

The clearest field-based shallow-water sea-state example remains the Currituck Sound storm record. Measurements in very shallow water at $u\,d\beta\,d\gamma\,dx$9 showed a storm-driven wind-wave field with a low-frequency power spectrum behaving as $f(\eta,x,t)$0, a spectral minimum at $f(\eta,x,t)$1, and a distinct separation between a KdV low-frequency regime and an NLS spectral-peak regime. The record beginning at 21:00 on 4 February 2002 contained 8192 points, had duration 27.96 minutes, sampling interval $f(\eta,x,t)$2, and significant wave height $f(\eta,x,t)$3; near storm peak, $f(\eta,x,t)$4. Finite-gap analysis showed that the low-frequency $f(\eta,x,t)$5 region was soliton dominated, that the solitons had random FGT phases on $f(\eta,x,t)$6, and that the probability density of soliton amplitudes was highly non-Gaussian. For each of 14 time series near the storm peak there were about 120 solitons, with average full width at half maximum $f(\eta,x,t)$7 and average soliton height $f(\eta,x,t)$8; this was the basis for calling the state a dense soliton gas (Costa et al., 2014).

Controlled shallow-water laboratory realizations complement that field observation. In a 34 m wave flume of width $f(\eta,x,t)$9 and depth $$\eta_t + c_0 \eta_x + \alpha \eta \eta_x + \beta \eta_{xxx} = 0,$$0, continuous sinusoidal piston forcing generated solitons through nonlinear steepening and soliton fission, while reflections at a vertical wall and the wavemaker created a stationary nonequilibrium bidirectional gas. High-resolution space-time measurements showed straight coherent ridges, overtaking and head-on collisions, and low-order statistics $$\eta_t + c_0 \eta_x + \alpha \eta \eta_x + \beta \eta_{xxx} = 0,$$1 and $$\eta_t + c_0 \eta_x + \alpha \eta \eta_x + \beta \eta_{xxx} = 0,$$2 for a representative dataset with $$\eta_t + c_0 \eta_x + \alpha \eta \eta_x + \beta \eta_{xxx} = 0,$$3, in good agreement with numerical KdV values $$\eta_t + c_0 \eta_x + \alpha \eta \eta_x + \beta \eta_{xxx} = 0,$$4 and $$\eta_t + c_0 \eta_x + \alpha \eta \eta_x + \beta \eta_{xxx} = 0,$$5. The collision time scale was about $$\eta_t + c_0 \eta_x + \alpha \eta \eta_x + \beta \eta_{xxx} = 0,$$6, whereas the damping e-fold time was about $$\eta_t + c_0 \eta_x + \alpha \eta \eta_x + \beta \eta_{xxx} = 0,$$7, so the stationary state was interpreted as being well described by pure integrable dynamics despite weak dissipation (Redor et al., 2019).

Video-based wave measurements in a similar 34 m shallow-water flume made the soliton-gas interpretation operational at the level of individual interactions. Seven synchronized monochrome cameras covered a 14 m central observation window at about $$\eta_t + c_0 \eta_x + \alpha \eta \eta_x + \beta \eta_{xxx} = 0,$$8, with estimated surface-elevation accuracy better than $$\eta_t + c_0 \eta_x + \alpha \eta \eta_x + \beta \eta_{xxx} = 0,$$9. Continuous monochromatic forcing at $$c_0 = \sqrt{gh}, \qquad \alpha = \frac{3c_0}{2h}, \qquad \beta = \frac{c_0 h^2}{6},$$0 and amplitudes $$c_0 = \sqrt{gh}, \qquad \alpha = \frac{3c_0}{2h}, \qquad \beta = \frac{c_0 h^2}{6},$$1 produced a transition from standing-wave behavior to a random state in which weak and strong interactions were detected in Radon space, solitary pulses were identified as KdV or Rayleigh solitons, and phase shifts were singled out as the seminal mechanism for disorganization and soliton gas formation (Redor et al., 2020).

Two-dimensional shallow-water experiments extended the subject beyond quasi-1D geometry. In a 2D wave tank, stereoscopic surface mapping over a large $$c_0 = \sqrt{gh}, \qquad \alpha = \frac{3c_0}{2h}, \qquad \beta = \frac{c_0 h^2}{6},$$2 domain with centimetre-resolution was used to build and monitor 2D shallow-water soliton gas. Random 2D states were produced both from multiple line solitons with random incidence $$c_0 = \sqrt{gh}, \qquad \alpha = \frac{3c_0}{2h}, \qquad \beta = \frac{c_0 h^2}{6},$$3 and from irregular random waves forced with a JONSWAP spectrum $$c_0 = \sqrt{gh}, \qquad \alpha = \frac{3c_0}{2h}, \qquad \beta = \frac{c_0 h^2}{6},$$4. In both cases Mach reflections and Mach expansions produced solitons that mainly propagated in directions perpendicular to the wave-makers, and the study described these experiments as the first observations of random 2D soliton gas for gravity waves (Leduque et al., 2024).

4. Deep-water realizations and open-ocean detections

Controlled deep-water synthesis established that soliton-gas sea states need not be inferred only from statistics of measured records; they can be designed in nonlinear spectral space. In a 148 m long, 5 m wide, 3 m deep wave flume with 20 equally spaced resistive wave gauges at $$c_0 = \sqrt{gh}, \qquad \alpha = \frac{3c_0}{2h}, \qquad \beta = \frac{c_0 h^2}{6},$$5, reflectionless $$c_0 = \sqrt{gh}, \qquad \alpha = \frac{3c_0}{2h}, \qquad \beta = \frac{c_0 h^2}{6},$$6-soliton solutions of the focusing 1D-NLSE were generated from prescribed discrete IST spectra. The main dense-gas case used $$c_0 = \sqrt{gh}, \qquad \alpha = \frac{3c_0}{2h}, \qquad \beta = \frac{c_0 h^2}{6},$$7, $$c_0 = \sqrt{gh}, \qquad \alpha = \frac{3c_0}{2h}, \qquad \beta = \frac{c_0 h^2}{6},$$8, $$c_0 = \sqrt{gh}, \qquad \alpha = \frac{3c_0}{2h}, \qquad \beta = \frac{c_0 h^2}{6},$$9, $$i(\psi_t + C_g \psi_x) + \mu \psi_{xx} + \nu |\psi|^2 \psi = 0.$$0, nonlinear length $$i(\psi_t + C_g \psi_x) + \mu \psi_{xx} + \nu |\psi|^2 \psi = 0.$$1, and $$i(\psi_t + C_g \psi_x) + \mu \psi_{xx} + \nu |\psi|^2 \psi = 0.$$2. The resulting wave field extended over $$i(\psi_t + C_g \psi_x) + \mu \psi_{xx} + \nu |\psi|^2 \psi = 0.$$3, corresponding to $$i(\psi_t + C_g \psi_x) + \mu \psi_{xx} + \nu |\psi|^2 \psi = 0.$$4, and was identified as a dense SG “which cannot be represented as superposition of individual solitons.” Nonlinear spectral analysis showed that the density of states evolved slowly along the tank under higher-order perturbative effects that break integrability (Suret et al., 2020).

A complementary deep-water experiment addressed macroscopic gas–gas interaction. In a 140 m water tank, two random 50-soliton jets with nearly identical amplitudes but opposite velocities were synthesized around $$i(\psi_t + C_g \psi_x) + \mu \psi_{xx} + \nu |\psi|^2 \psi = 0.$$5 and $$i(\psi_t + C_g \psi_x) + \mu \psi_{xx} + \nu |\psi|^2 \psi = 0.$$6, more generally $$i(\psi_t + C_g \psi_x) + \mu \psi_{xx} + \nu |\psi|^2 \psi = 0.$$7 with $$i(\psi_t + C_g \psi_x) + \mu \psi_{xx} + \nu |\psi|^2 \psi = 0.$$8. For the representative case, $$i(\psi_t + C_g \psi_x) + \mu \psi_{xx} + \nu |\psi|^2 \psi = 0.$$9, mean peak soliton amplitude $$\frac{\partial A}{\partial Z} + \frac{1}{C_g} \frac{\partial A}{\partial T}= i \frac{k_0}{\omega_0^2} \frac{\partial^2 A}{\partial T^2} + i \alpha k_0^3 |A|^2 A,$$0, steepness $$\frac{\partial A}{\partial Z} + \frac{1}{C_g} \frac{\partial A}{\partial T}= i \frac{k_0}{\omega_0^2} \frac{\partial^2 A}{\partial T^2} + i \alpha k_0^3 |A|^2 A,$$1, and $$\frac{\partial A}{\partial Z} + \frac{1}{C_g} \frac{\partial A}{\partial T}= i \frac{k_0}{\omega_0^2} \frac{\partial^2 A}{\partial T^2} + i \alpha k_0^3 |A|^2 A,$$2. The measured interaction-region density of each species decreased from about $$\frac{\partial A}{\partial Z} + \frac{1}{C_g} \frac{\partial A}{\partial T}= i \frac{k_0}{\omega_0^2} \frac{\partial^2 A}{\partial T^2} + i \alpha k_0^3 |A|^2 A,$$3 to $$\frac{\partial A}{\partial Z} + \frac{1}{C_g} \frac{\partial A}{\partial T}= i \frac{k_0}{\omega_0^2} \frac{\partial^2 A}{\partial T^2} + i \alpha k_0^3 |A|^2 A,$$4 as $$\frac{\partial A}{\partial Z} + \frac{1}{C_g} \frac{\partial A}{\partial T}= i \frac{k_0}{\omega_0^2} \frac{\partial^2 A}{\partial T^2} + i \alpha k_0^3 |A|^2 A,$$5 decreased from about $$\frac{\partial A}{\partial Z} + \frac{1}{C_g} \frac{\partial A}{\partial T}= i \frac{k_0}{\omega_0^2} \frac{\partial^2 A}{\partial T^2} + i \alpha k_0^3 |A|^2 A,$$6 to $$\frac{\partial A}{\partial Z} + \frac{1}{C_g} \frac{\partial A}{\partial T}= i \frac{k_0}{\omega_0^2} \frac{\partial^2 A}{\partial T^2} + i \alpha k_0^3 |A|^2 A,$$7, and Radon-transform velocity extraction showed interaction-induced speed shifts in good quantitative agreement with spectral kinetic theory despite higher-order effects (Fache et al., 2023).

Open-ocean evidence was reported from Taiwan waters by applying NFT-based diagnostics to 20,523 qualified buoy records from Eluanbi, Gueishandao, and Xiaoliuqiu. The key scalar metric was the soliton energy ratio $$\frac{\partial A}{\partial Z} + \frac{1}{C_g} \frac{\partial A}{\partial T}= i \frac{k_0}{\omega_0^2} \frac{\partial^2 A}{\partial T^2} + i \alpha k_0^3 |A|^2 A,$$8, where 0 means no solitons, 1 corresponds to a pure soliton gas, values above 0.5 indicate that solitons dominate, and values at least 0.9 identify very high-ratio candidates. Eleven sea states with ratio $$\frac{\partial A}{\partial Z} + \frac{1}{C_g} \frac{\partial A}{\partial T}= i \frac{k_0}{\omega_0^2} \frac{\partial^2 A}{\partial T^2} + i \alpha k_0^3 |A|^2 A,$$9 were found, representing only 0.054 percent of the dataset; after probabilistic directional filtering, three records from the Eluanbi station remained entirely above the $\omega_0^2=k_0g$0 threshold and were presented as confirmed deep open-ocean soliton gases. For the highlighted Eluanbi record from 2019/05/16 at 14:00, the NFT discrete spectrum contained 64 solitons, the largest soliton amplitude was $\omega_0^2=k_0g$1, and the soliton energy ratio was 0.96 (Lee et al., 6 Oct 2025).

The deep-water evidence also clarifies what such states are not. The open-ocean candidates were characterized by relatively small significant wave heights, short peak periods, high steepness, high Benjamin–Feir Index, no unusual abnormality index, no large kurtosis, and skewness close to zero. This explicitly separates a soliton-gas sea state from a rogue-wave sea state defined only by one or a few exceptional crests (Lee et al., 6 Oct 2025).

5. Kinetic, hydrodynamic, and condensate descriptions

At the macroscopic level, the core KdV soliton-gas model is El’s kinetic equation

$\omega_0^2=k_0g$2

with $\omega_0^2=k_0g$3 and $\omega_0^2=k_0g$4 for KdV. This equation arises as the thermodynamic limit of the Whitham equations and treats the velocity of each spectral component as collision-renormalized by the whole gas. Delta-functional reductions

$\omega_0^2=k_0g$5

produce finite-component “cold” soliton gases described by a $\omega_0^2=k_0g$6 quasilinear system with $\omega_0^2=k_0g$7 Jordan blocks of size $\omega_0^2=k_0g$8, and these reductions possess commuting hydrodynamic flows, conservation laws, and a generalized hodograph solution formula (Ferapontov et al., 2021).

The densest KdV limit is the soliton condensate, defined spectrally by $\omega_0^2=k_0g$9. In that limit, the kinetic description reduces exactly to the $C_g=g/(2\omega_0)$0-phase KdV–Whitham modulation equations

$C_g=g/(2\omega_0)$1

with $C_g=g/(2\omega_0)$2. This establishes a direct bridge between dense-gas kinetic theory, finite-gap modulation, and classical dispersive hydrodynamics, including Riemann problems that generate generalized rarefaction waves and dispersive shock waves (Congy et al., 2022).

For focusing NLS, the analogous spectral-kinetic structure is built from the density of states of a breather or soliton gas. The thermodynamic limit of finite-gap spectra yields the equation of state

$C_g=g/(2\omega_0)$3

together with

$C_g=g/(2\omega_0)$4

The breather-gas formulation includes a Stokes spectral band for the carrier wave; collapsing that band recovers the soliton-gas limit. The theory also identifies ideal-gas, finite-density-gas, and condensate regimes, with the condensate defined by the condition that the average collision-induced position shift equals the average separation between quasiparticles (El et al., 2019).

Bidirectional shallow-water sea states require two coupled densities of states. In bidirectional dispersive hydrodynamics,

$C_g=g/(2\omega_0)$5

and the effective velocities satisfy coupled integral equations involving both overtaking and head-on collision shifts. The paper distinguishes isotropic gases, in which head-on and overtaking shifts have the same sign, from anisotropic gases, in which they have opposite signs. The resonant NLS gas is shown to be equivalent to the Kaup–Boussinesq shallow-water bidirectional soliton gas, thereby supplying an explicit kinetic theory for random shallow-water sea states composed of interacting left- and right-going solitary waves (Congy et al., 2020).

A foundational issue in dense KdV sea states is the definition of soliton positions when individual humps are not visibly separated. A recent resolution introduces effective positions, a fluid-cell projection that removes all solitons outside a mesoscopic interval while leaving the KdV field unchanged inside that interval, and a density-of-states interpretation based directly on those effective positions. On large scales the positions satisfy semi-classical Bethe equations, and a non-rigorous derivation reproduces the KdV kinetic equation. This supplies a microscopic underpinning for the hydrodynamic notion that mesoscopic observables are functions of the local density of states (Doyon, 18 May 2026).

6. Statistical signatures, scope, and open directions

Across the literature, soliton-gas sea states are identified not by one criterion but by a bundle of spectral and statistical signatures. In shallow water these include low-frequency $C_g=g/(2\omega_0)$6 spectra, nonlinear finite-gap saturation of that band by solitons, random FGT phases, high soliton density in time, and highly non-Gaussian amplitude statistics (Costa et al., 2014). In deep-water NLS settings the signatures shift toward discrete nonlinear spectrum, density of states in IST space, weak change in the linear Fourier spectrum despite strong nonlinear organization, and, in the open-ocean NFT analysis, high soliton energy ratio after correction for directional interference (Suret et al., 2020, Lee et al., 6 Oct 2025).

The statistics of integrable turbulence can also be expressed directly in terms of the density of states. For semiclassical focusing NLS in regimes dominated by a bound-state soliton gas, the asymptotic intensity PDF is represented by

$C_g=g/(2\omega_0)$7

and, for a broad class of slowly varying nonzero-background initial data, by the explicit mixture formula

$C_g=g/(2\omega_0)$8

In this framework, different densities of states produce either exponential intensity statistics or strongly non-Gaussian heavy tails; the latter are described as a signature of the rogue waves presence (Congy et al., 7 May 2026).

A complementary NLS prototype is adiabatically grown integrable turbulence. Starting from random-phase homogeneous noise and weak linear pumping, the post-pump focusing-NLSE state can become almost purely solitonic: in the base case at $C_g=g/(2\omega_0)$9, $$i \frac{\partial \psi}{\partial t} + \frac{1}{2} \frac{\partial^2 \psi}{\partial x^2} + |\psi|^2 \psi=0$$0, $$i \frac{\partial \psi}{\partial t} + \frac{1}{2} \frac{\partial^2 \psi}{\partial x^2} + |\psi|^2 \psi=0$$1, $$i \frac{\partial \psi}{\partial t} + \frac{1}{2} \frac{\partial^2 \psi}{\partial x^2} + |\psi|^2 \psi=0$$2, $$i \frac{\partial \psi}{\partial t} + \frac{1}{2} \frac{\partial^2 \psi}{\partial x^2} + |\psi|^2 \psi=0$$3, $$i \frac{\partial \psi}{\partial t} + \frac{1}{2} \frac{\partial^2 \psi}{\partial x^2} + |\psi|^2 \psi=0$$4, and 64 solitons, i.e. $$i \frac{\partial \psi}{\partial t} + \frac{1}{2} \frac{\partial^2 \psi}{\partial x^2} + |\psi|^2 \psi=0$$5 of all solitons containing $$i \frac{\partial \psi}{\partial t} + \frac{1}{2} \frac{\partial^2 \psi}{\partial x^2} + |\psi|^2 \psi=0$$6 of total wave action, belong to the main bound state. The same state has $$i \frac{\partial \psi}{\partial t} + \frac{1}{2} \frac{\partial^2 \psi}{\partial x^2} + |\psi|^2 \psi=0$$7 and rogue-wave probability $$i \frac{\partial \psi}{\partial t} + \frac{1}{2} \frac{\partial^2 \psi}{\partial x^2} + |\psi|^2 \psi=0$$8, compared with the Rayleigh value $$i \frac{\partial \psi}{\partial t} + \frac{1}{2} \frac{\partial^2 \psi}{\partial x^2} + |\psi|^2 \psi=0$$9. This identifies a dense bound-state soliton gas as a sea-state-like stationary random field with strong extreme-wave statistics, although still within an idealized 1D-NLSE setting (Agafontsev et al., 2022).

The role of weak non-integrable perturbations remains active territory. In a weakly dissipative nonlinear electrical transmission line realizing KdV–Burgers dynamics, an initially dense, fully randomized soliton gas with $u\,d\beta\,d\gamma\,dx$00, $u\,d\beta\,d\gamma\,dx$01, and $u\,d\beta\,d\gamma\,dx$02 evolved not into a rank-ordered soliton train but into a condensate-like coherent state; the number of discrete eigenvalues increased from $u\,d\beta\,d\gamma\,dx$03 to $u\,d\beta\,d\gamma\,dx$04, and the long-time DOS was fitted by a genus-0 Weyl distribution with $u\,d\beta\,d\gamma\,dx$05. This suggests that weak dissipation can enable discrete–continuous spectral exchange and macroscopic condensate formation not captured by existing hydrodynamic theories, but the result is presently an analog-model result rather than a water-wave field observation (Fache et al., 2024).

The scope of present evidence remains sharply delimited. Shallow-water KdV interpretations do not claim that deep-water wind seas are soliton gases, deep-water NLS analyses apply to narrowband and effectively unidirectional conditions, and the open-ocean NFT study itself treats only records satisfying deep-water and unimodal-spectrum criteria (Costa et al., 2014, Suret et al., 2020, Lee et al., 6 Oct 2025). A plausible implication is that soliton-gas sea states are best understood not as a universal replacement for conventional sea-state theory, but as a distinct nonlinear regime—rare in the open ocean, more accessible in shallow water and controlled experiments, and most naturally diagnosed by nonlinear spectral quantities rather than by linear spectra alone.