Soliton Energy Ratio

• Soliton Energy Ratio is a normalized measure quantifying the energy carried by soliton modes relative to the total energy, with definitions varying by governing equations.
• It spans diverse contexts—from spectral fractions in ocean waves to peak density ratios in soliton collisions—displaying bounded, linear, or super-unit scaling based on the setup.
• This metric aids in diagnosing soliton-dominated regimes and evaluating energy transfer and amplification in both conservative and dissipative systems.

In the cited literature, the expression soliton energy ratio denotes a family of normalized energy measures attached to nonlinear coherent structures. The numerator may be the energy carried by soliton modes in a spectral decomposition, the energy contained in a soliton-dominated frequency band, the peak local energy density generated in a multi-soliton collision, or the energy of a moving, dissipative, or quantum-corrected soliton state relative to a reference configuration. Accordingly, the quantity may be bounded in $[0,1]$, may scale linearly with a control parameter, or may exceed unity; its precise meaning is determined by the governing equation and by the decomposition used to identify the soliton sector (Costa et al., 2014, Lee et al., 6 Oct 2025, Saadatmand et al., 2015, Kalashnikov et al., 2023).

1. Principal definitions and scope

Across the cited works, the same label is used for several distinct ratios. In ocean-wave applications, it usually means the fraction of total wave energy carried by soliton modes. In integrable field theory, it may instead compare the maximum local energy density attained during a collision to the total energy of the incoming solitons. In optical-fibre, dissipative, and relativistic settings, it frequently denotes a transport or parameter-continuation quotient rather than a spectral fraction.

Setting	Definition	Characteristic behavior
Shallow-water KdV waves	$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$ with $$E_{\text{soliton}}=\int_0^{\omega_c} S_{\text{soliton}}(\omega)\,d\omega$$ (Costa et al., 2014)	Typically $R_{\text{soliton}}\simeq 0.6\pm0.1$ near storm peak
Deep-water NLS-NFT sea states	$$R_{\text{soliton}}=E_{\text{sol}}/E_{\text{total}}=(E_{\text{total}}-E_{\text{rad}})/E_{\text{total}}\in[0,1]$$ (Lee et al., 6 Oct 2025)	Thresholds $R_{\text{soliton}}\ge 0.90$ raw, $R_{\text{soliton}}>0.50$ after directional filtering
Sine-Gordon multi-soliton collisions	$R(N)=e_{\max}(N)/E_{\rm tot}(N)$ (Saadatmand et al., 2015)	$R(N)=N/4$ for even $N$, $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$0 for odd $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$1
Lossy optical fibres	$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$2 (DalľAgnol et al., 2021)	Exponential under pure linear loss; logistic under linear plus cubic terms
CQGLE dissipative solitons	$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$3 (Kalashnikov et al., 2023)	Becomes arbitrarily large as $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$4
Reduced QED soliton-like state	$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$5 (Skoromnik et al., 2016)	$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$6
Quantum-corrected topological solitons	$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$7; $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$8 (Graham et al., 6 Mar 2025)	Controlled by linear-in-$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$9 slopes and offsets

A common source of ambiguity is that only some of these ratios are literal “fractions of total energy.” In the NLS-NFT sea-state formulation the ratio is explicitly constrained to $$E_{\text{soliton}}=\int_0^{\omega_c} S_{\text{soliton}}(\omega)\,d\omega$$0, whereas in the sine-Gordon collision problem the numerator is a peak density and the ratio grows as $$E_{\text{soliton}}=\int_0^{\omega_c} S_{\text{soliton}}(\omega)\,d\omega$$1, so values above unity are expected rather than anomalous.

2. Shallow-water KdV formulation and finite-gap extraction

In the shallow-water analysis of Currituck Sound, the total mean wave energy per unit span is taken, in linear theory, to be proportional to the integral of the measured power spectrum,

$$E_{\text{soliton}}=\int_0^{\omega_c} S_{\text{soliton}}(\omega)\,d\omega$$2

The soliton contribution is defined as the low-frequency component attributed to the KdV soliton band,

$$E_{\text{soliton}}=\int_0^{\omega_c} S_{\text{soliton}}(\omega)\,d\omega$$3

The measured spectrum exhibits $$E_{\text{soliton}}=\int_0^{\omega_c} S_{\text{soliton}}(\omega)\,d\omega$$4 for $$E_{\text{soliton}}=\int_0^{\omega_c} S_{\text{soliton}}(\omega)\,d\omega$$5 and a higher-frequency wind-wave cascade $$E_{\text{soliton}}=\int_0^{\omega_c} S_{\text{soliton}}(\omega)\,d\omega$$6; in practice $$E_{\text{soliton}}=\int_0^{\omega_c} S_{\text{soliton}}(\omega)\,d\omega$$7 is chosen at the observed spectral minimum separating the KdV band from the higher-frequency wind-wave band (Costa et al., 2014).

The dynamical basis is the Korteweg-deVries equation

$$E_{\text{soliton}}=\int_0^{\omega_c} S_{\text{soliton}}(\omega)\,d\omega$$8

A single-soliton solution in a moving frame is written as

$$E_{\text{soliton}}=\int_0^{\omega_c} S_{\text{soliton}}(\omega)\,d\omega$$9

and its energy is defined by the $R_{\text{soliton}}\simeq 0.6\pm0.1$0 norm

$R_{\text{soliton}}\simeq 0.6\pm0.1$1

Under periodic or quasiperiodic boundary conditions, the exact KdV solution is characterized by $R_{\text{soliton}}\simeq 0.6\pm0.1$2 real branch points $R_{\text{soliton}}\simeq 0.6\pm0.1$3. In the soliton limit each spectral band-gap collapses to repeated eigenvalues, yielding $R_{\text{soliton}}\simeq 0.6\pm0.1$4 discrete real eigenvalues $R_{\text{soliton}}\simeq 0.6\pm0.1$5, each mapped one-to-one onto a soliton amplitude $R_{\text{soliton}}\simeq 0.6\pm0.1$6, speed $R_{\text{soliton}}\simeq 0.6\pm0.1$7, and energy $R_{\text{soliton}}\simeq 0.6\pm0.1$8. Numerically, finite-gap theory is applied to the low-frequency portion of the time series to recover the set $R_{\text{soliton}}\simeq 0.6\pm0.1$9.

The empirical significance of the ratio in this setting is twofold. First, finite-gap analysis shows that the low-frequency $$R_{\text{soliton}}=E_{\text{sol}}/E_{\text{total}}=(E_{\text{total}}-E_{\text{rad}})/E_{\text{total}}\in[0,1]$$0 region is soliton dominated. Second, the solitons have random FGT phases, supporting a soliton random phase approximation. The probability density of the solitons demonstrates that they are dense in time and highly non-Gaussian. Although the study did not tabulate $$R_{\text{soliton}}=E_{\text{sol}}/E_{\text{total}}=(E_{\text{total}}-E_{\text{rad}})/E_{\text{total}}\in[0,1]$$1 explicitly, the supplied reconstruction states that all of the low-frequency power is carried by approximately $$R_{\text{soliton}}=E_{\text{sol}}/E_{\text{total}}=(E_{\text{total}}-E_{\text{rad}})/E_{\text{total}}\in[0,1]$$2 solitons in each $$R_{\text{soliton}}=E_{\text{sol}}/E_{\text{total}}=(E_{\text{total}}-E_{\text{rad}})/E_{\text{total}}\in[0,1]$$3 min record near storm peak, and that integrating the observed $$R_{\text{soliton}}=E_{\text{sol}}/E_{\text{total}}=(E_{\text{total}}-E_{\text{rad}})/E_{\text{total}}\in[0,1]$$4 law up to $$R_{\text{soliton}}=E_{\text{sol}}/E_{\text{total}}=(E_{\text{total}}-E_{\text{rad}})/E_{\text{total}}\in[0,1]$$5 against the full measured spectrum gives typically

$$R_{\text{soliton}}=E_{\text{sol}}/E_{\text{total}}=(E_{\text{total}}-E_{\text{rad}})/E_{\text{total}}\in[0,1]$$6

so that over half of the surface-wave energy is contained in a dense gas of KdV solitons at peak storm conditions in shallow water with $$R_{\text{soliton}}=E_{\text{sol}}/E_{\text{total}}=(E_{\text{total}}-E_{\text{rad}})/E_{\text{total}}\in[0,1]$$7 (Costa et al., 2014).

3. Deep-ocean soliton gases and the nonlinear Fourier transform

In deep water, the same phrase is given a more explicitly spectral meaning through the nonlinear Schrödinger nonlinear Fourier transform. Let $$R_{\text{soliton}}=E_{\text{sol}}/E_{\text{total}}=(E_{\text{total}}-E_{\text{rad}})/E_{\text{total}}\in[0,1]$$8 be the complex envelope of a normalized surface-elevation record, extended by zero outside the measurement interval. The total energy is the standard $$R_{\text{soliton}}=E_{\text{sol}}/E_{\text{total}}=(E_{\text{total}}-E_{\text{rad}})/E_{\text{total}}\in[0,1]$$9 norm,

$R_{\text{soliton}}\ge 0.90$0

and the nonlinear Parseval relation decomposes it into discrete and continuous parts,

$R_{\text{soliton}}\ge 0.90$1

Here $R_{\text{soliton}}\ge 0.90$2 is the energy carried by discrete eigenvalues, each corresponding to an NLS soliton, and $R_{\text{soliton}}\ge 0.90$3 is the energy carried by the continuous spectrum. The soliton energy ratio is then

$R_{\text{soliton}}\ge 0.90$4

For the normalization used in the study,

$R_{\text{soliton}}\ge 0.90$5

where $R_{\text{soliton}}\ge 0.90$6 are discrete Zakharov-Shabat eigenvalues and $R_{\text{soliton}}\ge 0.90$7 is the reflection coefficient (Lee et al., 6 Oct 2025).

The extraction pipeline is applied to each $R_{\text{soliton}}\ge 0.90$8-minute record sampled at $R_{\text{soliton}}\ge 0.90$9. The raw elevation is detrended and windowed, the carrier frequency $R_{\text{soliton}}>0.50$0 is estimated and removed by multiplication with $R_{\text{soliton}}>0.50$1, and the envelope is normalized to unit-coefficient focusing NLS scaling. The forward scattering problem

$R_{\text{soliton}}>0.50$2

is then solved to locate all discrete eigenvalues with $R_{\text{soliton}}>0.50$3 and to compute the continuous reflection coefficient.

In this formulation, the soliton energy ratio is used as a criterion for the closeness of a sea state to a soliton gas. A record is flagged as “very high soliton energy” when $R_{\text{soliton}}>0.50$4. These cases are associated with wave steepness $R_{\text{soliton}}>0.50$5 exceeding $R_{\text{soliton}}>0.50$6 and Benjamin-Feir index $R_{\text{soliton}}>0.50$7 exceeding $R_{\text{soliton}}>0.50$8. Because directional interference can artificially increase the ratio, the study applies a probabilistic directional-filtering method: a retention angle $R_{\text{soliton}}>0.50$9 about the mean direction is selected, off-axis bands are removed from $R(N)=e_{\max}(N)/E_{\rm tot}(N)$0, random uniform phases are assigned, $R(N)=e_{\max}(N)/E_{\rm tot}(N)$1 synthetic realizations are reconstructed, and a distribution of $R(N)=e_{\max}(N)/E_{\rm tot}(N)$2 is formed. After this correction, $R(N)=e_{\max}(N)/E_{\rm tot}(N)$3 is used to declare a soliton gas in the principal propagation direction.

The field results are unusually specific. The dataset comprises $R(N)=e_{\max}(N)/E_{\rm tot}(N)$4 deep-water, unimodal-spectrum $R(N)=e_{\max}(N)/E_{\rm tot}(N)$5-min records from three buoys in Taiwan waters, with $R(N)=e_{\max}(N)/E_{\rm tot}(N)$6–$R(N)=e_{\max}(N)/E_{\rm tot}(N)$7, $R(N)=e_{\max}(N)/E_{\rm tot}(N)$8, $R(N)=e_{\max}(N)/E_{\rm tot}(N)$9, $R(N)=N/4$0, and directional spread $R(N)=N/4$1. Eleven events had raw $R(N)=N/4$2, corresponding to $R(N)=N/4$3 of the dataset. These records had typically $R(N)=N/4$4, $R(N)=N/4$5–$R(N)=N/4$6, $R(N)=N/4$7, $R(N)=N/4$8, nearly symmetric statistics with skewness approximately zero and kurtosis $R(N)=N/4$9, and no large rogue-wave indices. One Eluanbi example at 2019-05-16 14:00 UTC had $N$0, $N$1, $N$2, $N$3, $N$4, and $N$5 discrete eigenvalues reaching $N$6 in amplitude. After directional filtering, three Eluanbi cases retained mean $N$7 for both $N$8 and $N$9, and are therefore reported as confirmed deep-ocean soliton gases (Lee et al., 6 Oct 2025).

4. Collision, amplification, and transport ratios in optical systems

In dispersion-mapped optical fibres, the relevant quantity is an energy-transfer ratio between colliding solitons. Two well-separated fundamental solitons are launched into a fibre governed by a generalized nonlinear Schrödinger equation with piecewise-constant group-velocity dispersion, anomalous outside a short segment of length $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$00 and normal within it. Each input pulse has energy

$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$01

with $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$02. After a collision in the normal-dispersion section, the output energies are $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$03 and $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$04. The transfer into soliton 1 is quantified by

$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$05

and the corresponding gain ratio is

$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$06

For fixed physical parameters, the transfer is fitted semi-analytically by

$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$07

with phase offset $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$08, and amplitudes scaling as power laws in $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$09, $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$10, $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$11, $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$12, $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$13, and $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$14. The quoted best-fit parameters are

$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$15

$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$16

At $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$17, $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$18, $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$19, $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$20, $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$21, and $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$22, the maximum transfer is approximately $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$23 at $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$24, while the minimum is approximately $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$25 at $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$26. A representative single collision yields $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$27, and a second engineered collision raises this to $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$28, corresponding to a cumulative gain of approximately $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$29. The mechanism relies on the dispersion-sign change and does not require third-order dispersion or Raman terms (Savojardo et al., 2017).

A different optical meaning appears in lossy fibres, where the ratio tracks propagation-induced depletion or compensation. The fundamental-mode energy is

$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$30

From the perturbed CPDE model,

$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$31

Assuming a $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$32-shaped fundamental soliton,

$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$33

one obtains $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$34 and therefore

$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$35

with $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$36 and $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$37. The corresponding ratio satisfies

$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$38

with logistic-form solution

$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$39

where $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$40 and $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$41. The subcases are explicit: $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$42 for pure linear loss and $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$43 for pure cubic gain or loss. For simulations with $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$44, the first $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$45 where $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$46 occurs at approximately $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$47, $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$48, $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$49, $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$50, and $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$51 for $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$52, $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$53, $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$54, $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$55, and $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$56, respectively. In this sense, a gain of $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$57 triples the propagation distance when the dissipation rate is $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$58 (DalľAgnol et al., 2021).

5. Relativistic and dissipative formulations

In the reduced-QED soliton-like construction, the energy ratio compares a moving state to its rest state rather than separating solitonic and radiative sectors. After neglecting transverse photons and adopting a self-consistent mean-field ansatz in the Coulomb gauge, the rest-state energy is obtained from coupled nonlinear Dirac equations and a Poisson-type self-consistent potential. The lowest-energy solution is reported as

$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$59

A canonical separation of total momentum then yields the relativistic dispersion relation

$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$60

The corresponding ratio is

$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$61

with the physical positive branch written as $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$62. The small-momentum expansion is

$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$63

while in the ultra-relativistic limit $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$64 one has $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$65 (Skoromnik et al., 2016).

In the cubic-quintic Ginzburg-Landau equation, the energy ratio is instead a parameter-continuation quotient on a master diagram. The dissipative-soliton energy is

$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$66

and in the adiabatic strongly chirped limit with $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$67 the spectrum has Rayleigh-Jeans form

$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$68

which integrates to

$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$69

The family is controlled by the universal parameter

$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$70

At the vacuum-stability threshold $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$71,

$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$72

The soliton-energy ratio between two parameter sets is then

$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$73

Dissipative soliton resonance is identified by the condition $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$74 while the peak power remains finite at $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$75, which occurs at

$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$76

At this threshold, $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$77, $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$78, and $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$79. The same formulation admits a thermodynamic interpretation with inverse spectral temperature $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$80, chemical potential $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$81, entropy $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$82, and free energy $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$83 (Kalashnikov et al., 2023).

6. Topological charge, quantum corrections, and comparative interpretation

For two-component solitons in one spatial dimension, the relevant ratios are organized by the topological charge $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$84. Both the classical energy and the vacuum polarization energy are fitted to very high numerical accuracy by straight lines,

$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$85

For the parameter choice $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$86, $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$87, the reported values are

$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$88

With loop-counting parameter $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$89, the total energy becomes

$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$90

Two ratios are then natural:

$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$91

and

$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$92

Because both energies are linear in $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$93, the binding properties are controlled by the offsets rather than curvature. The critical condition is set by solving

$$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$94

For small $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$95, all higher-charge solitons are bound; once $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$96 exceeds the critical value, all higher-charge states become unbound. The reported numerical behavior is that $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$97 is typically of order $$R_{\text{soliton}} = E_{\text{soliton}}/E_{\text{total}}$$98 for the parameter sets studied (Graham et al., 6 Mar 2025).

A common misconception is to treat soliton energy ratio as a single invariant fraction of soliton content. The cited literature does not do so. It uses the same term for spectral soliton fractions in KdV and NLS sea states, for localization efficiency in sine-Gordon collisions, for propagation and amplification quotients in optical fibres, for relativistic dispersion in a reduced-QED soliton-like state, and for parameter- or charge-dependent quotients in dissipative and quantum-corrected soliton theories. This suggests that the term identifies a problem-dependent diagnostic rather than a universal observable.