Breather Gases in the Focusing NLSE

• Breather gases are ensembles of coherent nonlinear breathers riding on a finite background defined by the focusing nonlinear Schrödinger equation and its variants.
• They reveal integrable turbulence where macroscopic properties like transport and density of states are encoded in the Zakharov–Shabat spectral framework.
• Experimental, numerical, and deterministic synthesis approaches validate breather gas dynamics through controlled optical fiber tests and kinetic simulations.

Breather gases (BGs) are ensembles of breathers—coherent localized structures on a nonzero background—governed by the focusing nonlinear Schrödinger family of equations. In the contemporary integrable-systems literature, a BG is the finite-background analogue of a soliton gas: its elementary constituents are not zero-background solitons but “solitons on finite background,” such as Akhmediev, Kuznetsov–Ma, and Peregrine breathers. Their significance is twofold. First, BGs provide a canonical realization of integrable turbulence in modulationally unstable media. Second, they admit a spectral thermodynamic description in which macroscopic transport, density of states, and statistical observables are encoded by the underlying Zakharov–Shabat spectrum (El et al., 2019, Copie et al., 7 Jul 2025).

1. Definition and governing equations

In normalized form, the focusing nonlinear Schrödinger equation (fNLS) used in the spectral theory of soliton and breather gases is

$i\,\psi_t+\psi_{xx}+2|\psi|^2\psi=0,$

with a nonzero background for the breather-gas setting (El et al., 2019). In the nonzero-boundary formulation used for inverse-scattering and Riemann–Hilbert analyses, one also encounters

$$i\,q_t+q_{xx}+2\big(|q|^2-q_0^2\big)q=0,\qquad \lim_{x\to \pm \infty}q(x,t)=q_\pm,\quad |q_\pm|=q_0>0,$$

which makes the role of the carrier explicit (Weng et al., 7 Jan 2025). The essential distinction from a soliton gas is therefore spectral and dynamical: a BG contains a finite background, often described as a Stokes mode or permanent spectral band, whereas a soliton gas is obtained when that band collapses.

Breathers are localized nonlinear excitations riding on that background. The literature surveyed here treats Akhmediev breathers as space-periodic and time-localized, Kuznetsov–Ma breathers as time-periodic and space-localized, and Peregrine breathers as doubly localized limits (Roberti et al., 2021). In this sense, a BG is not a gas of linear waves and not a thermodynamic gas of molecules; it is a statistical ensemble of coherent nonlinear structures whose interactions remain elastic at the integrable level.

The same conceptual object appears in physical, unnormalized models. In the optical experiment reported in 2025, the governing equation is the physical one-dimensional NLSE

$$i\frac{\partial A}{\partial z}=\frac{\beta_2}{2}\frac{\partial^2 A}{\partial t^2}-\gamma |A|^2A,$$

with anomalous dispersion $\beta_2<0$ and Kerr nonlinearity $\gamma>0$; the reported single-mode-fiber parameters are $\beta_2=-22\,\mathrm{ps}^2/\mathrm{km}$ and $\gamma=1.3\,\mathrm{W}^{-1}\mathrm{km}^{-1}$ at $1550\,\mathrm{nm}$ (Copie et al., 7 Jul 2025). This optical formulation preserves the same integrable structure in the nearly conservative regime and provides a direct bridge between abstract BG theory and laboratory observation.

2. Spectral thermodynamic formulation

The modern spectral theory of BGs was developed by El and Tovbis through a thermodynamic limit of finite-gap solutions of the fNLS (El et al., 2019). In that framework, the difference between breather and soliton gases is encoded by the presence or absence of an exceptional spectral band $\Gamma_0$, representing the finite background. The key macroscopic quantity is the density of states $u(\eta)$; its temporal counterpart is $$i\,q_t+q_{xx}+2\big(|q|^2-q_0^2\big)q=0,\qquad \lim_{x\to \pm \infty}q(x,t)=q_\pm,\quad |q_\pm|=q_0>0,$$0; and the dressed group velocity is defined by

$$i\,q_t+q_{xx}+2\big(|q|^2-q_0^2\big)q=0,\qquad \lim_{x\to \pm \infty}q(x,t)=q_\pm,\quad |q_\pm|=q_0>0,$$1

For a spatially homogeneous BG, $$i\,q_t+q_{xx}+2\big(|q|^2-q_0^2\big)q=0,\qquad \lim_{x\to \pm \infty}q(x,t)=q_\pm,\quad |q_\pm|=q_0>0,$$2 satisfies the kinetic equation of state

$$i\,q_t+q_{xx}+2\big(|q|^2-q_0^2\big)q=0,\qquad \lim_{x\to \pm \infty}q(x,t)=q_\pm,\quad |q_\pm|=q_0>0,$$3

where $$i\,q_t+q_{xx}+2\big(|q|^2-q_0^2\big)q=0,\qquad \lim_{x\to \pm \infty}q(x,t)=q_\pm,\quad |q_\pm|=q_0>0,$$4 is the free breather group velocity and $$i\,q_t+q_{xx}+2\big(|q|^2-q_0^2\big)q=0,\qquad \lim_{x\to \pm \infty}q(x,t)=q_\pm,\quad |q_\pm|=q_0>0,$$5 is the pairwise interaction kernel determined by the spectral geometry of the nonzero background (El et al., 2019). The corresponding non-homogeneous gas obeys the transport equation

$$i\,q_t+q_{xx}+2\big(|q|^2-q_0^2\big)q=0,\qquad \lim_{x\to \pm \infty}q(x,t)=q_\pm,\quad |q_\pm|=q_0>0,$$6

together with a carrier equation for the background phase component (El et al., 2019).

The 2022 extension of this theory derived average densities, fluxes, thermodynamic limits of quasimomentum and quasienergy differentials, and introduced periodic fNLS gases (Tovbis et al., 2022). In that setting, one starts from finite-gap solutions with a permanent band $$i\,q_t+q_{xx}+2\big(|q|^2-q_0^2\big)q=0,\qquad \lim_{x\to \pm \infty}q(x,t)=q_\pm,\quad |q_\pm|=q_0>0,$$7 and passes to a continuum spectral support $$i\,q_t+q_{xx}+2\big(|q|^2-q_0^2\big)q=0,\qquad \lim_{x\to \pm \infty}q(x,t)=q_\pm,\quad |q_\pm|=q_0>0,$$8 carrying a density of band centers $$i\,q_t+q_{xx}+2\big(|q|^2-q_0^2\big)q=0,\qquad \lim_{x\to \pm \infty}q(x,t)=q_\pm,\quad |q_\pm|=q_0>0,$$9 and a scaled bandwidth function $$i\frac{\partial A}{\partial z}=\frac{\beta_2}{2}\frac{\partial^2 A}{\partial t^2}-\gamma |A|^2A,$$0. The relation $$i\frac{\partial A}{\partial z}=\frac{\beta_2}{2}\frac{\partial^2 A}{\partial t^2}-\gamma |A|^2A,$$1 controls the transition between rarefied and condensate regimes; the condensate is the critical state $$i\frac{\partial A}{\partial z}=\frac{\beta_2}{2}\frac{\partial^2 A}{\partial t^2}-\gamma |A|^2A,$$2 in which macroscopic behavior is determined entirely by pairwise interactions (Tovbis et al., 2022). Periodic gases furnish an especially explicit class: their densities of states, averaged conserved densities, and macroscopic $$i\frac{\partial A}{\partial z}=\frac{\beta_2}{2}\frac{\partial^2 A}{\partial t^2}-\gamma |A|^2A,$$3-functions can be computed directly from the underlying periodic potential (Tovbis et al., 2022).

A partial rigorous foundation for this program was established through a potential-theoretic analysis of the nonlinear dispersion relations for the fNLS soliton gas (Kuijlaars et al., 2021). There, the density of states $$i\frac{\partial A}{\partial z}=\frac{\beta_2}{2}\frac{\partial^2 A}{\partial t^2}-\gamma |A|^2A,$$4 and its temporal analogue $$i\frac{\partial A}{\partial z}=\frac{\beta_2}{2}\frac{\partial^2 A}{\partial t^2}-\gamma |A|^2A,$$5 are characterized as solutions of singular logarithmic integral equations; existence, uniqueness, non-negativity, absolute continuity, and regularity are proved for the soliton-gas case; and exact solutions are given for bound-state condensates. The same paper writes the breather-gas nonlinear dispersion relations with a carrier-modified kernel involving

$$i\frac{\partial A}{\partial z}=\frac{\beta_2}{2}\frac{\partial^2 A}{\partial t^2}-\gamma |A|^2A,$$6

and identifies this analysis as a first step toward a full mathematical foundation for BGs (Kuijlaars et al., 2021). This suggests that the spectral description of BGs is now structurally well defined, while some parts of the rigorous existence theory remain less complete than in the zero-background soliton-gas case.

3. Breather-gas fission and statistical laws

A central development after the original spectral theory was the identification of breather-gas fission as a mechanism by which structured initial data evolve into randomized BG states. In the semiclassical study of elliptic potentials, the initial condition is the periodic Jacobi elliptic profile

$$i\frac{\partial A}{\partial z}=\frac{\beta_2}{2}\frac{\partial^2 A}{\partial t^2}-\gamma |A|^2A,$$7

evolved under the focusing NLS with weak additive noise (Biondini et al., 2024). The analysis shows that the noisy finite-band elliptic solution relaxes into a fully randomized, spatially homogeneous, statistically stationary breather gas. The thermodynamic spectrum splits into a regular soliton-gas component and a soliton condensate, so the BG is naturally interpreted as a composite gas. The same paper derives an exact kurtosis law for the resulting bound-state turbulence: $$i\frac{\partial A}{\partial z}=\frac{\beta_2}{2}\frac{\partial^2 A}{\partial t^2}-\gamma |A|^2A,$$8 and, for the $$i\frac{\partial A}{\partial z}=\frac{\beta_2}{2}\frac{\partial^2 A}{\partial t^2}-\gamma |A|^2A,$$9 family,

$\beta_2<0$0

A key consequence is that $\beta_2<0$1 for all $\beta_2<0$2, implying non-Gaussian statistics throughout the nontrivial family (Biondini et al., 2024).

The first optical experiment specifically targeting BGs realized a closely related scenario in a recirculating fiber loop and reported the first experimental observation of breather gases in optics (Copie et al., 7 Jul 2025). The injected field was a slowly modulated square pulse,

$\beta_2<0$3

with $\beta_2<0$4, $\beta_2<0$5, $\beta_2<0$6, and typical mean input power $\beta_2<0$7–$\beta_2<0$8 (Copie et al., 7 Jul 2025). Weak optical noise and slow modulation trigger modulational instability and a nonlinear fission process. For $\beta_2<0$9 and $\gamma>0$0, the background remains quasi-stationary up to about $\gamma>0$1 and then fissures over about $\gamma>0$2–$\gamma>0$3 into a random ensemble of $\gamma>0$4 structures; for $\gamma>0$5 and $\gamma>0$6, fission occurs earlier, around $\gamma>0$7–$\gamma>0$8 (Copie et al., 7 Jul 2025).

The same experiment provided a direct statistical test of the kurtosis-doubling law. With

$\gamma>0$9

and $\beta_2=-22\,\mathrm{ps}^2/\mathrm{km}$0, the deterministic input yields $\beta_2=-22\,\mathrm{ps}^2/\mathrm{km}$1 when the exponential window is neglected (Copie et al., 7 Jul 2025). Across six modulation depths $\beta_2=-22\,\mathrm{ps}^2/\mathrm{km}$2, the measured kurtosis at $\beta_2=-22\,\mathrm{ps}^2/\mathrm{km}$3 is approximately doubled. In the reported representative cases, $\beta_2=-22\,\mathrm{ps}^2/\mathrm{km}$4 evolves from about $\beta_2=-22\,\mathrm{ps}^2/\mathrm{km}$5 to about $\beta_2=-22\,\mathrm{ps}^2/\mathrm{km}$6 for $\beta_2=-22\,\mathrm{ps}^2/\mathrm{km}$7, and from about $\beta_2=-22\,\mathrm{ps}^2/\mathrm{km}$8 to about $\beta_2=-22\,\mathrm{ps}^2/\mathrm{km}$9 for $\gamma=1.3\,\mathrm{W}^{-1}\mathrm{km}^{-1}$0 (Copie et al., 7 Jul 2025). Numerical simulations track the experiment closely and indicate that exact asymptotic doubling would be reached at longer propagation, about $\gamma=1.3\,\mathrm{W}^{-1}\mathrm{km}^{-1}$1 for the reported parameters (Copie et al., 7 Jul 2025).

4. Deterministic constructions, synthesis, and shielding

BGs are not confined to stochastic fission scenarios. They can also be constructed directly from discrete spectral data. A numerically precise realization was given through recursive Darboux-transform synthesis of random $\gamma=1.3\,\mathrm{W}^{-1}\mathrm{km}^{-1}$2-breather ensembles with $\gamma=1.3\,\mathrm{W}^{-1}\mathrm{km}^{-1}$3 on a unit-amplitude plane-wave background (Roberti et al., 2021). Three one-component gases were synthesized by clustering eigenvalues in the spectral regions associated with Akhmediev, Kuznetsov–Ma, and Peregrine breathers. Because direct split-step evolution rapidly destroys the background through modulational instability, the space-time evolution was obtained by repeated spectral resynthesis rather than direct NLS integration. High-precision arithmetic, implemented with about $\gamma=1.3\,\mathrm{W}^{-1}\mathrm{km}^{-1}$4 digits, was essential; standard double precision becomes unstable for breather order $\gamma=1.3\,\mathrm{W}^{-1}\mathrm{km}^{-1}$5 (Roberti et al., 2021).

That synthesis work also supplied a direct kinetic validation of the El–Tovbis equation of state. A trial Tajiri–Watanabe breather propagated through one-component KM-, AB-, and PB-gases exhibits, respectively, acceleration, deceleration, and essentially ballistic motion, in agreement with the kinetic predictions. For a trial breather in a one-component KM gas,

$\gamma=1.3\,\mathrm{W}^{-1}\mathrm{km}^{-1}$6

where $\gamma=1.3\,\mathrm{W}^{-1}\mathrm{km}^{-1}$7 and $\gamma=1.3\,\mathrm{W}^{-1}\mathrm{km}^{-1}$8 is the two-breather position-shift kernel (Roberti et al., 2021). For a one-component AB gas, the appropriate formulation uses the spectral flux $\gamma=1.3\,\mathrm{W}^{-1}\mathrm{km}^{-1}$9 because the free AB group velocity diverges on the branch cut, and the observed correction is a deceleration (Roberti et al., 2021). These results establish that BGs can be assembled as controlled quasi-particle ensembles, not merely as late-time turbulence.

A complementary deterministic viewpoint is provided by shielding. Falqui, Grava, and Puntini constructed deterministic BGs as continuum limits of $1550\,\mathrm{nm}$0-breather Riemann–Hilbert problems in which discrete spectra fill compact domains and norming constants scale as $1550\,\mathrm{nm}$1 (Falqui et al., 2024). For quadrature domains of the form

$1550\,\mathrm{nm}$2

with a specially chosen interpolating density $1550\,\mathrm{nm}$3, the limiting BG coincides exactly with a finite $1550\,\mathrm{nm}$4-breather solution (Falqui et al., 2024). In the simplest case $1550\,\mathrm{nm}$5, the domain is a disk and the shielded gas reproduces a one-breather Kuznetsov–Ma profile. For $1550\,\mathrm{nm}$6, the same construction produces either a KM-type or Tajiri–Watanabe-type two-breather, depending on the sign of $1550\,\mathrm{nm}$7 (Falqui et al., 2024).

The 2025 extension of this program reformulated shielding for the focusing NLS with nonzero boundary conditions and emphasized the geometric mechanism of spectral coagulation (Weng et al., 7 Jan 2025). In the reflectionless NZBC setting, a BG is defined as the limit of $1550\,\mathrm{nm}$8-breather solutions whose discrete eigenvalues populate prescribed spectral domains. For a circular concentration domain, the gas coagulates into a single breather located at the center eigenvalue of the domain; for higher-degree quadrature concentration domains, it coagulates into an $1550\,\mathrm{nm}$9-breather solution whose eigenvalues solve $\Gamma_0$0 (Weng et al., 7 Jan 2025). The same paper also derives a pure-jump Riemann–Hilbert problem for line-domain concentrations and shows that a uniform distribution inside an ellipse is equivalent, at the RH level, to a line-domain BG supported on the focal segment (Weng et al., 7 Jan 2025). This deterministic branch of the theory shows that the term “gas” does not require stochasticity: in suitable continuum limits, a dense breather ensemble can behave exactly as a few-breather solution.

5. Experimental observation and cross-platform realizations

The optical realization of BGs relied on a recirculating fiber loop designed to approximate conservative propagation over very long effective distances (Copie et al., 7 Jul 2025). The loop contained about $\Gamma_0$1 of single-mode fiber closed by a $\Gamma_0$2 coupler, so that $\Gamma_0$3 of the circulating power was retained and $\Gamma_0$4 tapped for measurement each roundtrip. Backward Raman pumping at $\Gamma_0$5, gated with an acousto-optic modulator and adjusted with a variable optical attenuator, compensated the roundtrip losses and produced “virtually lossless” propagation over $\Gamma_0$6 roundtrips, about $\Gamma_0$7 (Copie et al., 7 Jul 2025). A fast photodetector and real-time oscilloscope sampled the field each roundtrip at $\Gamma_0$8; the combined detection bandwidth was $\Gamma_0$9 (Copie et al., 7 Jul 2025). This architecture enabled single-shot spatiotemporal maps of $u(\eta)$0 and, crucially, long-range statistics on a stable finite background.

The principal experimental observables were therefore not inverse-scattering quantities but direct real-space diagnostics: space-time maps, mean power, and kurtosis (Copie et al., 7 Jul 2025). The nearly constant mean power over the full $u(\eta)$1 was essential because BG integrity depends on the persistence of the background; significant dissipation would alter the carrier and bias breather formation and statistics (Copie et al., 7 Jul 2025). Minor discrepancies between experiment and simulation were attributed primarily to the finite $u(\eta)$2 detection bandwidth rather than to dissipative effects (Copie et al., 7 Jul 2025).

A related matter-wave setting appears in one-dimensional Bose gases with attractive interactions, where higher-order solitons act as breathers under the same integrable mean-field equation (Opanchuk et al., 2017). There the many-body Hamiltonian is the attractive Lieb–Liniger model, while the mean-field limit obeys the focusing Gross–Pitaevskii/NLSE. The protocol considered in the mesoscopic regime prepares a localized condensate with coherent amplitude

$u(\eta)$3

quenches the interaction from $u(\eta)$4 to $u(\eta)$5, and generates a higher-order soliton with period

$u(\eta)$6

in dimensionless units (Opanchuk et al., 2017). Truncated-Wigner simulations at $u(\eta)$7–$u(\eta)$8 predict gradual damping and fragmentation of the breather oscillations, with weaker relaxation as $u(\eta)$9 increases (Opanchuk et al., 2017). The same source states that, in a gas containing multiple breathers, ensemble-averaged signals should show cycle-to-cycle damping without sharp fragmentations, again with weaker damping at higher $$i\,q_t+q_{xx}+2\big(|q|^2-q_0^2\big)q=0,\qquad \lim_{x\to \pm \infty}q(x,t)=q_\pm,\quad |q_\pm|=q_0>0,$$00 (Opanchuk et al., 2017).

Platform	Reported realization	Principal outcome
Optical fiber loop	Recirculating SMF loop with Raman loss compensation over $$i\,q_t+q_{xx}+2\big(|q|^2-q_0^2\big)q=0,\qquad \lim_{x\to \pm \infty}q(x,t)=q_\pm,\quad |q_\pm|=q_0>0,$$01	First experimental observation of optical BGs and kurtosis doubling (Copie et al., 7 Jul 2025)
Darboux spectral synthesis	High-precision construction of $$i\,q_t+q_{xx}+2\big(|q|^2-q_0^2\big)q=0,\qquad \lim_{x\to \pm \infty}q(x,t)=q_\pm,\quad |q_\pm|=q_0>0,$$02 AB-, KM-, and PB-gases	Validation of kinetic transport laws via trial-breather propagation (Roberti et al., 2021)
1D Bose gas	Interaction quench from $$i\,q_t+q_{xx}+2\big(|q|^2-q_0^2\big)q=0,\qquad \lim_{x\to \pm \infty}q(x,t)=q_\pm,\quad |q_\pm|=q_0>0,$$03 to $$i\,q_t+q_{xx}+2\big(|q|^2-q_0^2\big)q=0,\qquad \lim_{x\to \pm \infty}q(x,t)=q_\pm,\quad |q_\pm|=q_0>0,$$04	Mesoscopic breather relaxation and fragmentation in TWA (Opanchuk et al., 2017)

These realizations span three complementary regimes: direct experimental integrable turbulence, exact numerical spectral synthesis, and mesoscopic quantum relaxation. Together they indicate that BGs are not tied to a single experimental modality but to a common finite-background focusing-NLSE structure.

6. Scope, limitations, and terminological ambiguity

Several limitations recur across the current literature. In optics, the experimentally accessible propagation length is finite: the reported loop reaches $$i\,q_t+q_{xx}+2\big(|q|^2-q_0^2\big)q=0,\qquad \lim_{x\to \pm \infty}q(x,t)=q_\pm,\quad |q_\pm|=q_0>0,$$05, while simulations indicate that exact asymptotic kurtosis doubling would require about $$i\,q_t+q_{xx}+2\big(|q|^2-q_0^2\big)q=0,\qquad \lim_{x\to \pm \infty}q(x,t)=q_\pm,\quad |q_\pm|=q_0>0,$$06 for the same parameters (Copie et al., 7 Jul 2025). The optical measurements also do not resolve individual breather identities, phase relations, or collision laws; the emphasis is on ensemble statistics rather than on fitting explicit Akhmediev, Kuznetsov–Ma, or Peregrine formulas to each localized event (Copie et al., 7 Jul 2025). In the semiclassical elliptic-potential theory, the BG is a bound-state gas with zero density of fluxes, which simplifies the thermodynamic description but does not exhaust the full variety of possible finite-velocity gases (Biondini et al., 2024). In the rigorous theory, existence and uniqueness are fully developed for the soliton-gas nonlinear dispersion relations, whereas the corresponding breather-gas existence theory is presented as an extension in progress (Kuijlaars et al., 2021).

The distinction between random and deterministic BGs is also substantive. Noise-seeded fission from periodic or slowly modulated backgrounds produces randomized, spatially homogeneous states whose macroscopic statistics are governed by the underlying unperturbed spectrum (Biondini et al., 2024, Copie et al., 7 Jul 2025). By contrast, shielding constructions produce deterministic continuum limits that exactly replicate finite-breather solutions (Falqui et al., 2024, Weng et al., 7 Jan 2025). A plausible implication is that “breather gas” now denotes a broader spectral-statistical class rather than a uniquely stochastic object.

A further point of ambiguity is terminological rather than mathematical. In an unrelated biomedical instrumentation paper, the acronym “BGs” is used for exhaled “Breather Gases,” meaning a synchronized set of respiratory measurements including $$i\,q_t+q_{xx}+2\big(|q|^2-q_0^2\big)q=0,\qquad \lim_{x\to \pm \infty}q(x,t)=q_\pm,\quad |q_\pm|=q_0>0,$$07, $$i\,q_t+q_{xx}+2\big(|q|^2-q_0^2\big)q=0,\qquad \lim_{x\to \pm \infty}q(x,t)=q_\pm,\quad |q_\pm|=q_0>0,$$08, $$i\,q_t+q_{xx}+2\big(|q|^2-q_0^2\big)q=0,\qquad \lim_{x\to \pm \infty}q(x,t)=q_\pm,\quad |q_\pm|=q_0>0,$$09, temperature, humidity, and bidirectional mass flow (Lacouture et al., 1 Dec 2025). That usage belongs to breath analysis and compact mixed-signal sensing, not to integrable turbulence or focusing-NLSE spectral theory. In the nonlinear-wave literature surveyed above, however, “breather gases” refers specifically to ensembles of finite-background breathers and the macroscopic states generated by their thermodynamic, kinetic, or experimental evolution.

Within that nonlinear-wave meaning, BGs now occupy a distinct place between exact coherent structures and statistical wave turbulence. They inherit the spectral exactness of inverse scattering, the transport structure of kinetic equations, the heavy-tailed statistics of integrable turbulence, and, in the most recent experiments, direct observability in nearly conservative optical systems (El et al., 2019, Copie et al., 7 Jul 2025).