Energy Scalability Limits of Dissipative Solitons

Alexander Rudenkov; Evgeni Sorokin; Irina T. Sorokina; Vladimir L. Kalashnikov

REVIEW 4 major objections 5 minor 90 references

Energy Scalability Limits of Dissipative Solitons

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper claims that energy scaling of a single dissipative soliton is capped by a negative-temperature instability that splits it into multiple pulses.

desk verdict The paper has a genuinely new quantitative claim—dissipative-soliton energy ceilings set by negative temperature around E*≈20–40—but the causal step is not demonstrated, and the paper itself concedes the evidence points to noise-selected multipulsing rather than spontaneous breakup of a sustained pulse. read the letter →

arxiv 2412.04297 v3 pith:Z47BRXRC submitted 2024-12-05 physics.optics nlin.PS

Vladimir L. Kalashnikov , Alexander Rudenkov , Evgeni Sorokin , Irina T. Sorokina This is my paper · ORCID

classification physics.opticsnlin.PS PACS 42.65.Tg42.60.Fc

keywords dissipativesolitonresonancethermodynamicsnegativetemperaturecomplexGinzburg-Landauequationchirped-pulseoscillatorall-normal-dispersionfiberlasermode-locking

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that a single dissipative soliton—the self-stabilizing pulse that mode-locked lasers emit—has a hard energy ceiling set by its own thermodynamics, not just by the stability of the laser equations. Working in the adiabatic theory of strongly chirped pulses, the authors treat the soliton's truncated Lorentzian spectrum as a probability distribution of quasi-particle microstates and derive an entropy and an effective temperature for the pulse. As energy grows, the two internal correlation scales decouple, the entropy rises, and past a threshold the internal energy starts to fall, so the effective temperature becomes negative; a negative-temperature single-pulse state is 'overheated' and unstable, and the system relaxes by splitting into several lower-energy pulses. The paper identifies the onset of negative temperature near $E^{*}\approx 20$ and the entropy preference for multipulse complexes near $E^{*}\approx 40$, and it reports qualitative agreement with measured spectra from a Cr$^{2+}$:ZnS chirped-pulse oscillator. If true, this gives laser designers a quantitative rule: below those thresholds single-pulse energy can grow safely, beyond them multipulsing is thermodynamically inevitable.

What carries the argument

The carrying object is the truncated Lorentzian spectral profile $p(\omega)\propto (\Xi^2+\omega^2)^{-1}H(\Delta^2-\omega^2)$, interpreted as a normalized probability distribution of soliton quasi-particle microstates. It defines two correlation scales: the short 'graining' scale $l=\pi/\Delta$ and the long 'confining' scale $\Lambda=1/\Xi$. The entropy $H_s$ (Eq. 8 with the renormalization in Eq. D3), internal energy $U$ (Eq. 9/D4), and temperature $\Theta=(\partial H_s/\partial U)^{-1}$ are all derived from this distribution, so the ratio $\Xi/\Delta$ controls the number of microstates. The mechanism that drives the claimed limit is the decoupling of $l$ and $\Lambda$: in dissipative soliton resonance, $\Xi\to0$ while $\Delta$ saturates, which raises the entropy; when $U$ simultaneously stops rising and begins to fall, the temperature goes negative and the single pulse becomes unstable against splitting into multiple $P_0^-$ pulses.

What would settle it

The claim would be falsified by a statistical self-start experiment: run many noise-seeded starts of the same chirped-pulse oscillator at a series of pump energies and record the number of pulses that emerge. The theory predicts that single-pulse probability drops below about 95% as the dimensionless energy $E^{*}$ passes roughly 20 and that multipulse states become entropically preferred past roughly 40; if single-pulse output remains the rule well beyond those thresholds, the thermodynamic instability is not the limiting mechanism. As a spectral check, the DSR spectrum must show a Lorentzian central spike of width $\Xi$ shrinking toward zero while the cutoff $\Delta$ saturates; a spectrum that keeps broadening without a narrowing spike contradicts the probability-distribution assumption underlying the negative temperature.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that dissipative soliton resonance does not lead to unlimited energy accumulation: the same spectral condensation that lets a strongly chirped pulse stretch and store more energy also creates the instability that ends it. The soliton spectrum is a Lorentzian of width $\Xi$ cut off at $\Delta$ (Eq. 2), and the paper reads it as a Rayleigh-Jeans distribution of quasi-particle microstates. From this distribution it computes an entropy $H_s$ and an internal energy $U$; with increasing dimensionless energy $E^{*}$, $\Xi$ shrinks toward zero while $\Delta$ saturates, so the ratio of the confining scale $\Lambda=1/\Xi$ to the graining scale $l=\pi/\Delta$ grows and the entropy rises. At the point where $U$ begins to decrease with $E^{*}$, the reciprocal temperature $\Theta^{-1}=\partial H_s/\partial U$ becomes negative. The single-pulse $P_0^+$ solution then is only metastable, and the system 'thermalizes' into a set of lower-energy $P_0^-$ pulses; the paper locates the negative-temperature onset near $E^{*}\approx20$ and the entropy-preference crossover near $E^{*}\approx40$, and it connects these thresholds to the experimentally observed multipulsing and spectral signatures in a Cr$^{2+}$:ZnS chirped-pulse oscillator.

Load-bearing premise

The argument stands on treating the soliton's truncated Lorentzian spectrum as a physical probability distribution of quasi-particle microstates, so the entropy and temperature derived from it are real; the paper itself notes the laser is far from equilibrium and the thermodynamic analogy is not literal, so if that spectral-probability reading fails, the negative temperature and the $E^{*}\approx20$ and $E^{*}\approx40$ thresholds are formal artifacts rather than a physical ceiling.

Editorial extensions

If this is right

A single dissipative soliton is thermodynamically safe only below the negative-temperature onset near $E^{*}\approx20$; designs that push past this point should expect multipulsing to become increasingly probable.
Above $E^{*}\approx40$, a complex of two or more equal $P_0^-$ pulses has higher entropy than a single $P_0^+$ pulse of the same total energy, so energy added to the laser goes into more pulses rather than a stronger single pulse.
The same three spectral signatures—growth of a Lorentzian central spike, saturation of the spectral half-width $\Delta$, and asymptotic temporal stretching—mark both the entry into dissipative soliton resonance and the approach to the breakup limit, which is what makes the threshold observable.
The fidelity condition $\Xi=\Delta$ coincides with the best compressibility of the chirped pulse; beyond it the spectral chirp becomes strongly frequency-dependent and compression quality degrades, so operating near the fidelity curve optimizes both energy and pulse quality.
Near the breakup threshold the probability distribution over pulse number broadens and shifts to higher $n$, meaning the laser shows hysteresis and multistability: the same pump conditions can sustain a single pulse or several pulses depending on history.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Inference: because the breakup mechanism depends only on the truncated-Lorentzian spectral structure and the decoupling of the two correlation scales, the same negative-temperature threshold near $E^{*}\approx20$ should appear in all-normal-dispersion fiber lasers and other strongly chirped soliton lasers, not only in solid-state chirped-pulse oscillators.
Inference: if the effective temperature is a genuine control parameter, reshaping the spectral dissipation profile—rather than lowering the pump energy—is a testable way to 'cool' the soliton ensemble and shift multipulsing to higher energies.
Inference: the entropy definition in Eq. (D3) is renormalized by an additive shift to remove an infinity as $\Xi\to0$, so the precise values $E^{*}\approx20$ and $E^{*}\approx40$ should be read as crossover locations; the robust content of the paper is the existence of an entropy-driven multipulsing transition, not the exact numbers.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Read it. The genuinely new piece is the claim that energy scalability of dissipative solitons is capped by a thermodynamic transition: entropy grows with energy, internal energy eventually decreases, temperature goes negative, and the single-pulse state becomes unstable around E*~20–40. That specific quantitative threshold is not in their earlier papers, and it is a useful, falsifiable target. The paper also ships commented Mathematica code and shows real Cr:ZnS CPO spectra, which is more than most theory papers. The adiabatic spectrum and DSR machinery are largely self-borrowed, but that is legitimate; the control-parameter diagram is a solid framework.

The soft spots are in the load-bearing step. First, the negative-temperature result rests on an entropy rescaling in Appendix D: they start with Shannon entropy H for the normalized spectral distribution (D2), then drop the normalization term to get Hs (D3). Sign is convention-dependent; a different shift changes the sign of dHs/dU and the location of the Θ=0 crossing. You cannot drop the constant from one side and keep it in the other and call the result physical. They do not test robustness to that choice.

Second, the thermodynamics analogy is explicitly non-literal, and they concede in Section IV that sustained single pulses do not spontaneously decay; multipulsing appears during noise-driven formation. That means the simulations support a noise-selected multi-pulse outcome, not the disintegration of an established soliton at E*≈20 or 40. The abstract and conclusions state the causal claim more strongly than the evidence. The free-energy minimization disavowal in Appendix D sits awkwardly with the Maxwell-point wording in the text.

Third, the experimental comparison is qualitative and partly circular: the DSR signatures (Lorentzian spike, spectral saturation, stretching) are properties of the ansatz, so agreement with those signatures confirms the model but not the thermodynamic instability.

The stress-test was mostly right. The central causal step—negative temperature causes breakup—is asserted, not shown. The paper is still coherent and honest; it flags several of its own limitations. The citation pattern is heavy on the group's prior work because it is developing its own framework; no red flag there.

Who is this for: people designing chirped-pulse oscillators and all-normal-dispersion fiber lasers who want a rule of thumb for energy ceilings, and theorists interested in thermodynamic analogies in CGLE systems. It deserves a serious referee—the analytic machinery is nontrivial and the topic is timely—but only with a referee who asks for a linear-stability analysis or a rewritten claim that separates heuristic thermodynamic language from what the numerics actually establish. I would not cite the specific E* thresholds in my own work yet.

Referee Report

4 major / 5 minor

Summary. The paper proposes a thermodynamic description of strongly chirped dissipative solitons governed by the complex cubic-quintic Ginzburg-Landau equation. Starting from the adiabatic truncated-Lorentzian spectral ansatz (Eq. 2), the authors define an entropy Hs, internal energy U, and temperature Θ (Eqs. 8-10, Appendix D) and use these to argue that energy scalability in the dissipative-soliton-resonance (DSR) regime is limited by an entropy increase and by a transition to negative temperature at dimensionless energy E*≈20, which drives breakup into multiple lower-energy P-0 solitons. The paper also reports numerical simulations of noise-driven pulse formation and experimental spectra from a Cr2+:ZnS chirped-pulse oscillator, claiming qualitative agreement with the predicted DSR signatures: a growing central Lorentzian spike, saturation of spectral broadening, and pulse stretching.

Significance. If the thermodynamic interpretation were rigorously established, the paper would provide a practical design rule for the maximum single-pulse energy in chirped-pulse oscillators and all-normal-dispersion fiber lasers. The manuscript's strengths are its systematic adiabatic derivation of the DS spectrum and master diagram (Appendix A, Fig. 1), the clear identification of the P+0/P-0 branches and their distinct energy-scaling behavior, the explicit numerical noise-seeded self-start statistics (Fig. 5), and the availability of commented Mathematica code for the thermodynamic quantities. The experimental spectra (Fig. 2) do show the expected Lorentzian spike and spectral saturation. However, the central negative-temperature instability and the quantitative thresholds E*≈20 and E*≈40 are not currently supported: they depend on an ad hoc entropy rescaling, an unspecified thermodynamic path, and a conjectural mapping from the spectral ansatz to a probability distribution, while the paper itself reports no spontaneous decay of sustained single pulses. The quantitative claims therefore require substantial revision before the paper can be accepted.

major comments (4)

[Section III E and Appendix D, Eq. (D3)] The entropy Hs used to compute the temperature is obtained from the full Shannon entropy H in Eq. (D2) by dropping the term ln(2Ξ tan^-1(Δ/Ξ)). This term is not a constant: it depends on Ξ and Δ, and therefore on U through Eq. (D4). Consequently ∂Hs/∂U differs from ∂H/∂U by the derivative of a state-dependent quantity, so the sign of the resulting temperature is a property of the chosen entropy convention rather than an invariant thermodynamic prediction. The paper provides no argument that the omitted term is thermodynamically irrelevant; it is dropped merely to make Hs positive and finite as Ξ→0. Since the negative-temperature threshold E*≈20 in Fig. 11 and the 'overheated' instability claim in Section III E are the central conclusions of the paper, the temperature should be recomputed with the full H, or the rescaling must be justified physically. As it stands, the negative-temperature result may be an artifact of the rescaling.
[Section IV and Fig. 5] The causal link between negative Θ and pulse splitting is not demonstrated. Section IV states that 'spontaneous transitions or decay are not seen when the regime is sustained. Such transitions appear during the initial stage of DS formation.' Figure 5 presents statistics from 150 stochastic samples of noise-driven formation, showing that multipulsing becomes more probable at higher E*_cw. This supports a noise-selected outcome during self-start, not the disintegration of an established single P+0 DS when Θ crosses zero. To substantiate the breakup claim, the authors should either simulate a sustained single-pulse initial condition perturbed above the predicted threshold, or explicitly restrict the claim to self-start statistics. As written, the assertion in Section III E that an 'overheated' DS 'tends to relax' into multiple pulses is unsupported by the numerical evidence presented.
[Section III B and Appendix D, Eqs. (D1)-(D4)] The interpretation of the DS spectrum p(ω) as a normalized probability distribution of quasi-particle microstates is a conjecture that is not independently tested. The three DSR 'signatures' (Lorentzian spike, spectral saturation, pulse stretching) are direct consequences of the truncated-Lorentzian ansatz Eq. (2) and the definitions of Ξ and Δ, so their observation in Fig. 2 does not validate the thermodynamic interpretation. The negative temperature and the entropy-preference threshold E*≈40 are derived from this spectral-probability mapping and therefore inherit its assumptions. The manuscript should explicitly identify which predictions are robust to abandoning the quasi-particle interpretation, or provide an independent test—for example, connecting Hs to the measured multipulsing statistics via fluctuation relations. The caveats in footnote 1 and Appendix D ('the analogy to thermodynamics is not literal', 'free energy minimization cannot be considered a criterion of dynamic stability') should be reflected in the abstract's claims.
[Eq. (10) and Fig. 11] The definition Θ = (∂Hs/∂U)^{-1} does not specify which variables are held fixed in the derivative. Hs and U are functions of both Ξ and Δ (Eqs. D3-D4), and along the DS branch these are not independent (Eqs. A6-A8). Without specifying the thermodynamic path—for example, constant C and Σ, or constant Δ—the temperature is not uniquely defined, and the Θ=0 crossing may depend on that choice. The authors should state the constraints used to generate Figs. 10 and 11 and verify that the negative-temperature onset is independent of the path chosen.

minor comments (5)

[Section III C] The section title 'DS "quantization" and thermolization' contains a typo: 'thermolization' should be 'thermalization'.
[Section IV vs. Fig. 9(a)] Section IV states that the entropy difference becomes positive at E*≈30, whereas Fig. 9(a) and the surrounding text state E*≈40. These numbers should be reconciled.
[Fig. 5 caption] The caption says 'Percentage of multiple DSs' but does not define whether this is the fraction of samples with more than one pulse, the average pulse number, or another quantity; the y-axis label should be specified.
[Eq. (5)] In Eq. (5), the argument of the sinc function is Δ(τ-t), which is dimensionless only if Δ is understood as an angular frequency; please state the convention used for spectral half-widths.
[General] The abbreviation 'CPO' is used both for 'chirped-pulse oscillator' and in the compound 'CPO-CPA'; please define it at first use and keep the usage consistent.

Circularity Check

1 steps flagged · score 6.0 of 10

The negative-temperature breakup threshold is an artifact of the state-dependent entropy re-scaling in Eq. (D3), not a stability result from the CGLE dynamics.

self definitional [Appendix D, Eqs. (D2)-(D4); Section III E, Eq. (10) and Fig. 11]
"Now, we need to re-scale the expression for entropy to make it positive and exclude an infinity limΞ→0 H = −∞. The shifted expression for entropy reads as: Hs = ... The points of the Θ = 0 – crossing with the transition to negative temperatures mean that the DS with larger energies becomes unstable, i.e., 'overheated'"

Eq. (D3) is not a constant entropy shift: it deletes the state-dependent term ln(2Ξ tan−1(Δ/Ξ)) from the Shannon entropy H in Eq. (D2), so Hs = H + ln(2Ξ tan−1(Δ/Ξ)). Since the deleted term depends on Ξ, ∂Hs/∂U is not ∂H/∂U, and the sign of Θ is a property of that ad hoc re-scaling rather than of the underlying CGLE solution. The paper then equates Θ<0 with the DS being 'unstable' and 'overheated,' which is the central breakup claim. Yet the paper itself concedes that 'spontaneous transitions or decay are not seen when the regime is sustained,' so the threshold E*≈20 is a consequence of the chosen entropy convention, not a demonstrated instability.

full rationale

The derivation chain is self-contained and mostly non-circular: the truncated-Lorentzian spectrum (Eq. 2) comes from an adiabatic stationary-phase treatment of Eq. (1), the stochastic simulations in Fig. 5 are independent numerical evidence for noise-selected multipulsing, and the experimental spectra provide external qualitative support. The main circularity is concentrated in Appendix D and Section III E: the thermodynamic temperature is defined through an entropy Hs obtained by dropping a state-dependent normalization term from the standard Shannon entropy. Because that dropped term depends on the spectral width parameter Ξ, the resulting negative-temperature crossover is a property of the chosen entropy convention, not an invariant thermodynamic result. The authors' own appended caveats that free-energy minimization is not a dynamic stability criterion and that no spontaneous decay of sustained single pulses is observed reinforce that the negative-temperature 'overheating' step is an interpretation imposed on the model rather than a mechanism extracted from it. The self-citations to prior work by the same group are ordinary use of earlier derivations and are not by themselves load-bearing circularity.

Assumptions & free parameters 4 free parameters · 5 assumptions · 2 invented entities

The central claim rests on several model-specific choices: the CGLE, the adiabatic approximation, the truncated-Lorentzian spectral ansatz, the probability interpretation of the spectrum, and an ad hoc entropy shift. The only genuinely new content is the thermodynamic interpretation of energy-scaling limits; the underlying dynamics are prior theory.

free parameters (4)

SAM coefficient kappa = 0.5 or 1 MW^-1
Chosen to be close to Ref [37]; not independently measured in this paper. The physical energy thresholds depend on it through the normalization E* = E kappa sqrt(zeta/(beta gamma)).
SAM saturation coefficient zeta = 1 or 2 MW^-1
Chosen by hand to produce different splitting curves in Fig. 1; not measured. Affects the DSR region and the physical energy scale.
SPM coefficient gamma = 5.1 MW^-1
Fixed for the Cr:ZnS example; a model input rather than a fitted parameter, but not derived in this paper.
spectral filter bandwidth alpha = 200 nm gain bandwidth
Set from Cr:ZnS material data; the paper uses it to fix the master diagram and the value of C.

assumptions (5)

domain assumption The complex cubic-quintic Ginzburg-Landau equation (1) adequately models the mode-locked laser.
The entire analysis is based on this equation, standard in the field but not derived from first principles.
domain assumption Adiabatic assumptions: normal GDD (beta > 0), nondissipative factors dominate (gamma >> kappa, beta >> alpha), large chirp psi >> 1, and slowly varying envelope (Section II, assumptions i-iii).
These justify the truncated-Lorentzian spectrum (Eq. 2). If violated, the thermodynamic quantities are not valid.
ad hoc to paper The DS spectrum (2) can be interpreted as a probability distribution of quasi-particle microstates (Section III B, Appendix D).
The authors call this a conjecture; it is not derived from microscopic dynamics.
ad hoc to paper The entropy re-scaling in Eq. (D3) does not change the physical conclusions.
Appendix D subtracts a divergent term to make entropy positive; the paper does not prove the sign of dHs/dU (the temperature) is invariant under this shift.
domain assumption The soliton condition C near 1 (or C = 2/3 at DSR) and a Gibbs-like steady-state distribution apply (Section II, [47]).
Used to justify the statistical interpretation of the spectrum.

invented entities (2)

DS quasi-particles (microstates)
purpose: To interpret the truncated-Lorentzian spectrum as a statistical ensemble and define entropy and temperature.
No direct measurement of these microstates; they are a theoretical construct tied to the correlation scales l and Lambda in Eq. (5).
Negative-temperature phase for DS
purpose: To characterize the instability that leads to multipulsing.
Negative temperature is a formal consequence of the chosen entropy and internal-energy definitions, not directly observed. The experiment shows multipulsing but not a negative temperature.

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Cite this review

Pith. "Pith review of Energy Scalability Limits of Dissipative Solitons." pith.science (2026). https://pith.science/paper/Z47BRXRC

@misc{pith2026241204297,
  author       = {Pith},
  title        = {Pith review of: Energy Scalability Limits of Dissipative Solitons},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z47BRXRC}},
  note         = {Machine review of arXiv:2412.04297}
}

abstract

In this study, we apply a thermodynamical approach to elucidate the primary constraints on the energy scaling of dissipative solitons (DS). We rely on the adiabatic theory of strongly chirped DS and define the DS energy scaling in terms of dissipative soliton resonance (DSR). Three main experimentally verifiable signatures identify a transition to DSR: i) growth of a Lorentzian spike at the centrum of the DS spectrum, which resembles a spectral condensation in Bose-Einstein condensate (BEC), ii) saturation of the spectrum broadening, and iii) asymptotical DS stretching. We connect the DSR breakup with three critical factors: i) decoupling of two correlation scales inherent in strongly chirped DS, ii) resulting rise of the DS entropy with energy, which provokes its disintegration, and iii) transition to a nonequilibrium phase, which is characterized by negative temperature. The breakup results in multiple stable DSs with lower energy. Theoretical results are in good qualitative agreement with the experimental data from a Kerr-lens mode-locked Cr$^{2+}$:ZnS chirped-pulse oscillator (CPO) that paves the way for optimizing high-energy femtosecond pulse generation in solid-state CPO and all-normal-dispersion fiber lasers.

Figures

Figures reproduced from arXiv: 2412.04297 by the authors.

**Figure 2.** Experimental spectra (solid lines, left scale) from [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Dimensionless spectral chirp ψ(ω) ∝ ∂ 2 ∂ω2 ω 2 (Ξ2+ω2)(∆2−ω2) vs. ω expressed in the units of ∆ for three different relations between Ξ and ∆. half maximum) region of the input DS to that of the compressed pulse [62]. The quality parameter remains close to unity, and the compressed pulse width decreases monotonously with anomalous dispersion till the value of the latter approaches −3γ 2/2βκζ∆4 ≈ −0.02 ps2 [39, 40]… view at source ↗

Figures from the paper (10 more)

**Figure 4.** Figure 4: (a): Dependencies of the FHWM width (black) [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: Percentage of multiple DSs gathered from 150 independent stochastic samples. (a): [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 7.** Figure 7: (a) Illustration of the DS energy pairing (see [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: Curves demonstrate Hs-dependencies on C and E∗ for fixed Σ (white curves), C (green curve), and the fidelity line (magenta). The entropy value for the lat- [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]

**Figure 8.** Figure 8: The DS entropy Hs versus C and the logarithm of E ∗ for the DS solutions P + 0 (right side of the surface) and P − 0 (left side of the surface). The yellow curve shows the border between them. The black curve with Σ = 0 confines a region of DS existence. The magenta li…

**Figure 11.** Figure 11: The temperature Θ evolution with E ∗ for the “quantized” P + 0 -complexes in [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]

**Figure 9.** Figure 9: (a) The entropy differences ∆Hs between the multipulse P − 0 -complexes (orange – two, blue – three, green – four, and magenta – five pulses) and their single pulse P + 0 - counterparts with the same Σ in dependence on E ∗ (see [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]

**Figure 10.** Figure 10: The DS internal energy U versus C and the logarithm of E ∗ for the DS solutions P + 0 (right side of the surface) and P − 0 (left side of the surface). The curves correspond to those in [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]

**Figure 12.** Figure 12: The absolute values of the successive terms [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]

**Figure 13.** Figure 13: Free energy F for the multipulse complexes (twopulse - black dashed curve, three - orange, and four - blue) and their P + 0 -counterparts (corresponding solid curves). ACKNOWLEDGMENTS This work was supported by Norges Forskningsråd (#303347 (UNLOCK), #326241 (Lammo-…

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Works this paper leans on

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[1]

quasi-particles

confined by the conditions Ξ < ∆, that means Σ < 3 16 1 − C 2 , and C ∈ h 2 − 4 √ Σ, 2/3 i . The en- ergy scaling within this region does not require more GDD because the C-parameter could be constant. A decrease of Ξ compared to ∆ means a transition to the spectral condensation regime, in which the DS harvests the energy by becoming longer with the simul...
[2]

vacuum temperature

This modulation does not influence DS stability or other properties. A. DS fidelity and compressibility The energy scaling provided by the spectral conden- sation, i.e., the Ξ-decrease, decouples the correlation scales l and Λ. The first consequence of a shift from the fidelity curve with an asymptotic energy growth causing l/Λ → 0 is a spectral chirp inh...
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and (b) 0.5 MW−1 (dashed curves 1′, 2′, 3′). C=0.54, γ=5.1 MW−1, ζ = 1 MW−1, and spectral filter bandwidth equals 200 nm. Points show< τ >for the dual DSs,κ = 1 MW−1. The inset in (a) shows the connection betweenE∗ cw and E∗ in Fig. 1 through the order parameter σ/ϑ. The solid and dashed lines correspond toκ = 1 and 0.5 MW−1, respectively. Defining p(n) a...
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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

quasi-particles

confined by the conditions Ξ < ∆, that means Σ < 3 16 1 − C 2 , and C ∈ h 2 − 4 √ Σ, 2/3 i . The en- ergy scaling within this region does not require more GDD because the C-parameter could be constant. A decrease of Ξ compared to ∆ means a transition to the spectral condensation regime, in which the DS harvests the energy by becoming longer with the simul...

[2] [2]

vacuum temperature

This modulation does not influence DS stability or other properties. A. DS fidelity and compressibility The energy scaling provided by the spectral conden- sation, i.e., the Ξ-decrease, decouples the correlation scales l and Λ. The first consequence of a shift from the fidelity curve with an asymptotic energy growth causing l/Λ → 0 is a spectral chirp inh...

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scatters

and (b) 0.5 MW−1 (dashed curves 1′, 2′, 3′). C=0.54, γ=5.1 MW−1, ζ = 1 MW−1, and spectral filter bandwidth equals 200 nm. Points show< τ >for the dual DSs,κ = 1 MW−1. The inset in (a) shows the connection betweenE∗ cw and E∗ in Fig. 1 through the order parameter σ/ϑ. The solid and dashed lines correspond toκ = 1 and 0.5 MW−1, respectively. Defining p(n) a...

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