# Dissipation-driven emergence of a soliton condensate in a nonlinear
electrical transmission line

Loic Fache  Univ. Lille, CNRS, UMR 8523 - PhLAM - Physique des Lasers Atomes
et Molécules, F-59 000 Lille, France  Hervé Damart  Univ. Lille, CNRS, UMR
8523 - PhLAM - Physique des Lasers Atomes et Molécules, F-59 000 Lille, France
François Copie  Univ. Lille, CNRS, UMR 8523 - PhLAM - Physique des Lasers
Atomes et Molécules, F-59 000 Lille, France  Thibault Bonnemain  Department of
Mathematics, King’s College, London, United Kingdom  Thibault Congy
Department of Mathematics, Physics and Electrical Engineering, Northumbria
University, Newcastle upon Tyne, NE1 8ST, United Kingdom  Giacomo Roberti
Department of Mathematics, Physics and Electrical Engineering, Northumbria
University, Newcastle upon Tyne, NE1 8ST, United Kingdom  Pierre Suret  Univ.
Lille, CNRS, UMR 8523 - PhLAM - Physique des Lasers Atomes et Molécules, F-59
000 Lille, France  Gennady El  Department of Mathematics, Physics and
Electrical Engineering, Northumbria University, Newcastle upon Tyne, NE1 8ST,
United Kingdom  Stéphane Randoux<EMAIL_ADDRESS>Univ. Lille,
CNRS, UMR 8523 - PhLAM - Physique des Lasers Atomes et Molécules, F-59 000
Lille, France

###### Abstract

We present an experimental study on the perturbed evolution of Korteweg-
deVries soliton gases in a weakly dissipative nonlinear electrical
transmission line. The system’s dynamics reveal that an initially dense, fully
randomized, soliton gas evolves into a coherent macroscopic state identified
as a soliton condensate through nonlinear spectral analysis. The emergence of
the soliton condensate is driven by the spatial rearrangement of the systems’s
eigenmodes and by the proliferation of new solitonic states due to
nonadiabatic effects, a phenomenon not accounted for by the existing
hydrodynamic theories.

In the seminal article from 1965, Zabusky and Kruskal discovered stability and
interaction properties of solitons in numerical simulations of the Korteweg-
deVries (KdV) equation which they had obtained as a long-wave approximation of
the Fermi–Pasta–Ulam-Tsingou oscillator chain Zabusky and Kruskal (1965). Two
years later, Gardner et al. introduced the inverse scattering transform (IST)
method, which provides an analytical framework to solve initial value problems
for the KdV equation with rapidly decaying initial conditions Gardner  _et
al._ (1967). IST theory was then generalized to other nonlinear dispersive
equations of physical relevance, such as the one-dimensional nonlinear
Schrödinger equation (1D-NLSE) Zakharov and Shabat (1972); Novikov  _et al._
(1984); Yang (2010), which is a universal model for the description of many
physical systems.

Solitons play a fundamental role in nonlinear physics due to their remarkable
property of retaining shape, amplitude, and velocity upon interactions with
other solitons Newell (1985); Miura (1976); Drazin and Johnson (1989); Copie
_et al._ (2023). This is reflected in IST theory by the fact that solitons
represent normal modes of integrable nonlinear partial differential equations.
The IST formalism can be seen as a canonical transformation which carries the
physical coordinates, in which the integrable equations are originally given,
to new (IST) coordinates which are action-angle variables. Within the IST
formalism, the amplitude and velocity of each individual soliton are encoded
into a discrete eigenvalue $\eta_{k}$, which represents an action variable
that is time-invariant – the so-called isospectrality property of integrable
evolution Newell (1985); Miura (1976); Drazin and Johnson (1989).

From the earliest days of IST theory, it was understood that the decomposition
of the nonlinear field into a basis of normal modes represents a powerful tool
applicable to the study of perturbed soliton evolutions relevant to physical
applications where purely integrable dynamics are unlikely Karpman and Maslov
(1977); Kaup  _et al._ (1978); Kivshar and Malomed (1989). The evolution of a
KdV soliton under the action of weak dissipation is a prototypical example
where the effects of a small perturbation on an integrable equation have been
studied extensively Madsen and Mei (1969); Grimshaw (1970); Johnson (1973);
Kaup  _et al._ (1978); Kodama and Ablowitz (1981); Kivshar and Malomed (1989);
Grimshaw  _et al._ (1994); Johnson (1994); Ko and Kuehl (1978); Knickerbocker
and Newell (1980, 1985); El and Grimshaw (2002).

On the other hand, the question of emergent, large-scale behaviors in
integrable or nearly-integrable many-body systems, such as soliton gases
(SGs), is one of the main focuses of present-day experimental and theoretical
physics Schemmer  _et al._ (2019); Castro-Alvaredo  _et al._ (2016); De Nardis
_et al._ (2018); Ruggiero  _et al._ (2020); Redor  _et al._ (2019); Marcucci
_et al._ (2019); Suret  _et al._ (2020, 2023); Fache  _et al._ (2024); Mossman
_et al._ (2024); Vecchio  _et al._ (2020); Congy  _et al._ (2024). The out-of-
equilibrium evolution of SGs can be investigated using the complementary tools
of the nonlinear spectral kinetic theory of SGs Zakharov (1971); El and
Kamchatnov (2005); El and Tovbis (2020) and generalized hydrodynamics (GHD),
the hydrodynamic theory for many-body quantum and classical integrable systems
Doyon  _et al._ (2018); Doyon (2020); Castro-Alvaredo  _et al._ (2016);
Bertini  _et al._ (2016); Bonnemain and Doyon (2024). The correspondence
between these two hydrodynamic theories has been recently established in ref.
Bonnemain  _et al._ (2022) for KdV SGs.

In this Letter, we describe experiments that explore the perturbed evolution
of KdV SGs in a weakly dissipative nonlinear medium. We conduct our
experiments in an electrical nonlinear transmission line (NLTL), where
diffusion plays a perturbative role, and the dynamics are accurately captured
by the KdV-Burgers (KdVB) equation. Our experiments reveal that an initially
dense KdV SG, manifested as a strongly oscillating random field, slowly
evolves into a macroscopically coherent structure consisting of a broad smooth
nonlinear rarefaction wave connected at its leading edge to an oscillating
diffusive-dispersive shock wave El  _et al._ (2017). From the perspective of
the kinetic theory of SG, the formed composite nonlinear wave represents a
soliton condensate, a critically dense SG whose macroscopic properties are
dominated by soliton interactions while the individual soliton dynamics are
not discernible El and Tovbis (2020); Congy  _et al._ (2023). Soliton
condensates have peculiar spectral properties which we identify by analyzing
the IST eigenfunctions (or normal modes) composing the resulting wavefield. As
was shown in Congy  _et al._ (2023) the integrable KdV SG dynamics do not
permit a spontaneous formation of a soliton condensate from a regular (non-
condensate) SG evolution. Contrastingly, as we demonstrate, the formation of a
soliton condensate from a non-condensate initial state in a perturbed
integrable system becomes possible owing to a complex nonadiabatic process of
the generation of numerous new solitonic modes, a phenomenon not accounted for
by the existing hydrodynamic or kinetic theories of SG.

Figure 1: Experiments. (a) Schematic representation of the NLTL. (b) Space-
time plot showing the emergence of the soliton condensate in the NLTL. (c)
Measured SG at $t=0$ (blue line), $t=175$ (orange line), $t=420$ (green line).
(d) Time evolution of the discrete eigenvalues measured during the evolution
of the SG. The red flow is associated to newly generated eigenvalues arising
from the excitation of the continuous spectrum. (e) Time evolution of the
number $N(t)$ of discrete eigenvalues measured during the evolution of the SG.
The left (resp. right) inset shows the discrete IST spectrum at $t=0$ (resp.
$t=420$). The red line in (e) is obtained from numerical simulations of the
experiment with $\epsilon=0.1$.

Our experimental setup, shown schematically in Fig. 1(a), is an NLTL with an
architecture similar to that used in previous investigations of KdV soliton
propagation Remoissenet (1996); Ricketts and Ham (2018); Jäger (1982);
Elizondo-Decanini  _et al._ (2015). It consists of a ladder network of $160$
identical lumped inductor-varactor sections with inductance $L$ and varactor
capacitance $C(V)$. The varactors are commercial components (Varicap) for
which $C(V)$ is a linearly-decreasing function of the applied voltage $V$. The
varactors have a small series resistance $R$, which results into frequency-
dependent NLTL losses. Hence, propagation in the NLTL in the long-wavelength
limit is not described by the KdV equation but by the KdVB equation Ricketts
and Ham (2018):

$u_{t}+6uu_{x}+u_{xxx}=\epsilon u_{xx},$ (1)

where $u(x,t)$ represents the normalized electrical voltage, see Supplemental
material for details about the experimental setup and the normalization. The
strength $\epsilon$ of the Burgers term in Eq. (1) is determined by the
resistance $R$ of the varactors. In our experiments, the value of $\epsilon$
is $0.1$.

The initial condition $u_{0}(x)=u(x,t=0)$ used in our experiment is a dense SG
numerically synthesized as a $N$-soliton solution of the KdV equation using
the Darboux transformation method Congy  _et al._ (2023); Liao and Huang
(1996); Perego (2024), see Supplemental material for details. The discrete IST
spectrum of the initial condition is composed of $N=200$ discrete eigenvalues
$\eta_{k}$ that are randomly distributed between $0.2$ and $0.6$ in a uniform
way, see Fig. 1(e). As shown in Fig. 1(c), the initial field measured at the
first $LC$ cell of the NLTL is a random oscillating field in which individual
solitons cannot be discerned due to their strong overlap and interactions.

Measuring the voltage along the NLTL, we have built a space-time plot
revealing the evolution of the SG up to $t=420$, see Fig. 1(b). In the absence
of the diffusion term ($\epsilon=0$), the dense SG initially localized in
space would have evolved into a rank-ordered expanding soliton train, see ref.
Karpman (1967) and Fig. 3(g). Contrastingly, we observe that the presence of
the small diffusion term prevents the dense SG from reaching a diluted state
where individual solitons separate, see Fig. 1(b)(c). Physically, weak
dissipation promotes soliton interactions, leading to the emergence at long
time ($t\sim 400$) of a broad, smooth wavefield with a coherent oscillatory
structure identified as a diffusive-dispersive shock wave El  _et al._ (2017);
Ricketts and Ham (2018).

Using the perspective of the IST method, we can gain insights into the
features observed in the experiment. It is well known that the IST spectrum of
the solutions of the KdV equation (Eq. (1) with $\epsilon=0$) can be obtained
by solving the linear Schrödinger equation for an auxiliary function
$\psi(x,t)$

$\psi_{xx}+(u(x,t)-\lambda)\psi=0,$ (2)

where the KdV field $u(x,t)$ plays the role of a potential with $t$ being a
parameter Newell (1985); Miura (1976); Drazin and Johnson (1989). For the
fundamental soliton solution of the KdV equation
$u_{k}(x,t)=2\eta_{k}^{2}\,\text{sech}^{2}(\eta_{k}(x-4\eta_{k}^{2}t-x_{0k}))$,
Eq. (2) admits only one eigenmode that is parameterized by one real discrete
eigenvalue $\lambda_{k}=-\eta_{k}^{2}$ and by one normalized real
eigenfunction $\psi_{k}(x,t)$ that is localized in space
($|\psi_{k}(x,t)|\rightarrow 0$ for $|x|\rightarrow\infty$ and
$\int_{-\infty}^{+\infty}\psi_{k}^{2}(x,t)\,dx=1$) Drazin and Johnson (1989).
Importantly, any $N$-soliton solution of the KdV equation with purely discrete
IST spectrum can be represented using the following reconstruction formula
Gardner  _et al._ ; Miura (1976); Newell (1985)

$u(x,t)=4\sum_{k=1}^{N}\eta_{k}\,\psi_{k}^{2}(x,t).$ (3)

Using a perturbative approach, where the Burgers term in Eq. (1) is considered
to be small, we computed the discrete eigenvalues $\eta_{k}$ associated with
the experimentally measured field $u(x,t)$ by solving Eq. (2) at various times
$t$. If the dynamics were governed by the KdV equation, one would expect an
isospectral evolution, wherein the $N=200$ eigenvalues specified by our
initial condition would remain unchanged over time. Instead, we observe that
the eigenvalues distribute along two distinct “flows”, see Fig. 1(d). In the
first flow (black lines in Fig. 1(d)), the eigenvalues associated with the
eigenmodes composing the initial SG slowly decay with time, as expected from
the soliton perturbation theory Karpman and Maslov (1977); Kaup  _et al._
(1978); Kivshar and Malomed (1989). The second flow (red lines in Fig. 1(d))
consists of the newly formed discrete eigenvalues progressively emerging
during the evolution. These new eigenvalues correspond to the generation of
small-amplitude solitons whose number $N$ grows with time, see Fig. 1(e)
showing that $N$ grows from $N(t=0)=200$ to $N(t=420)=225$.

Figure 2: Experiments. (a) Space-time evolution of a solitary wave in the
NLTL. (b) Measured solitary wave at $t=0$ (blue line), $t=5.8$ (orange line),
$t=16.6$ (green line). (c) Time evolution of the discrete eigenvalues measured
during the evolution of the solitary wave. The red lines in (c) are obtained
from numerical simulations of the experiment ($\epsilon=0.1$). (g) Numerical
simulations showing the oscillatory evolution of the masses $M_{DS}$ and
$M_{CS}$ of the discrete and continous spectra.

The described complex process of the proliferation of new soliton states
represents a counterpart of a simpler phenomenon observed in the individual
evolution of a solitary wave in the NLTL. As seen in Fig. 2(a)(b), the
amplitude and velocity of the soliton, taken as the initial condition, slowly
decay during the propagation in the NLTL. This gradual evolution of the
solitary wave is accompanied by the formation of a trailing shelf, which is
analogous to a prominent feature of the propagation of solitary waves in
shallow water with slowly varying depth Knickerbocker and Newell (1985); Kaup
_et al._ (1978); Newell (1985). Nonlinear spectral analysis, performed by
solving Eq. (2) for the measured field $u(x,t)$, reveals that the discrete
eigenvalue linked to the initial condition gradually decays over time, while
two new discrete eigenvalues are created during the shelf formation, as seen
in Fig. 2(c).

This is a non-adiabatic process because, although the soliton’s amplitude
changes slowly, the shelf’s extent does not, resulting in an $O(1)$ mass of
the shelf despite its small amplitude. Consequently, the solitary wave excites
radiative modes (associated with the continuous IST spectrum), potentially
leading to the creation of new solitonic eigenmodes Kaup  _et al._ (1978);
Newell (1985); Wright (1980); El and Grimshaw (2002). In our experiment
described by the KdVB equation, the excitation of the continuous spectrum by
the solitary wave is observed through opposite oscillations between the masses
$M_{DS}$ and $M_{CS}$ of the discrete and continuous spectra, such that the
total mass $M=\int u(x,t){\rm d}x=M_{DS}+M_{CS}$ is conserved Wright (1980),
see Fig. 2(d) and Supplemental Material for details.

Figure 3: Numerical simulations. (a) SG used as initial condition in the
experiment. (b) Spatial distribution of the discrete eigenmodes represented as
$\phi_{k}(x)=\eta_{k}+s\,\psi_{k}(x)$, where $\eta_{k}$ (resp. $\psi_{k}(x)$)
($k=1,..200$) represent the eigenvalues (resp. eigenfunctions) computed by
solving Eq. (2). (c) DOS of the SG shown in (a). (d), (e), (f) Same as (a),
(b), (c) but for a simulation of Eq. (1) at $t=420$ with $\epsilon=0.1$. (g),
(h), (i) Same as (a), (b), (c) but for a simulation of Eq. (1) at $t=4000$
with $\epsilon=0$. (j), (k), (k) Same as (a), (b), (c) but for a simulation of
Eq. (1) at $t=4000$ with $\epsilon=2.10^{-3}$. The inset in (f) (resp. in (l))
shows the DOS measured at $x=500$ (resp. $x=2900$).

Using nonlinear spectral analysis, we now examine how these nonadiabatic
effects impact the evolution of the perturbed KdV SG. Fig. 3(a) shows the SG
used as initial condition in our experiment. The associated distribution of
the discrete eigenvalues $\eta_{k}$ and eigenfunctions $\psi_{k}(x)$
parameterizing this SG is plotted in a single representation showing
$\phi_{k}(x)=\eta_{k}+s\,\psi_{k}(x)$ ($k=1,..200$), see Fig. 3(b). The
eigenvalues and eigenfunctions are computed by solving numerically Eq. (2) for
$u(x,t)=u_{0}(x)$. We introduced a scaling factor, $0<s\ll 1$, to prevent
overlap among the eigenfunctions in Fig. 3(b).

As shown in the insets of Fig. 3(b), the degree of localization in space of
the eigenfunctions composing the random initial condition $u_{0}(x)$ depends
of their associated eigenvalues. Nonlinear discrete eigenmodes with the
largest eigenvalues ($\eta_{k}\sim 0.6$) are more localized in space than the
modes with the smallest eigenvalues ($\eta_{k}\sim 0.2$). Taking $u_{0}(x)$ as
the initial condition and solving numerically Eq. (1) for $\epsilon=0.1$, we
obtain the field shown in Fig. 3(d), in good quantitative agreement with the
experimental measurements. Fig. 3(e) reveals that the distributions of the
eigenvalues and of the eigenfunctions composing the field have been profoundly
modified by the nonlinear evolution. In addition to the creation of new
eigenfunctions with spectral parameters between $0$ and $0.2$, all the
eigenfunctions composing the wavefield at $t=420$ have lost their random
structure and adopted similar uniform oscillatory shape.

This large-scale behavior is in sharp contrast with that observed in the
absence of diffusion, for $\epsilon=0$. As shown in Fig. 3(g), for a pure KdV
evolution, a rank-ordered soliton train forms at long evolution time, as a
result of an isospectral evolution where the eigenfunctions composing the
nonlinear wave field become spatially localized as solitonic peaks, see Fig.
3(h). An intermediate situation between the experimental case ($\epsilon=0.1$
in Fig. 3(d)) and the purely integrable case ($\epsilon=0$ in Fig. 3(g)) is
observed for a very small diffusion parameter ($\epsilon=2.10^{-3}$). In this
situation, the SG exhibits a slowed-down expansion, in which a random soliton
train propagates over a non-zero background, see Fig. 3(j). Fig. 3(k) shows
that the associated eigenfunctions are divided into two classes: the localized
modes and the delocalized modes, the latter originating from the excitation of
the continuous spectrum.

Now we show that the squared eigenfunctions $\psi_{k}^{2}(x,t)$ are directly
associated with the density of states (DOS) $f(\eta_{k};x,t)$ in the SG
spectral kinetic theory, revealing their fundamental role in this framework.
The SG spectral kinetic theory is a statistical theory describing the
transport in space and time of the DOS. The DOS $f(\eta;x,t)$ is defined as
the joint distribution over the spectral eigenvalues and the soliton
positions, so that $f(\eta;x,t){\rm d}\eta{\rm d}x$ represents the number of
soliton states found at time $t$ in the element of the phase space
$[\eta,\eta+{\rm d}\eta]\times[x,x+{\rm d}x]$ El and Tovbis (2020); El (2021);
Suret  _et al._ (2020, 2023); Fache  _et al._ (2024). In particular, knowledge
of the DOS provides the statistical moments of the field $u(x,t)$, including
its mean value El (2021); Bonnemain  _et al._ (2022):

$\langle u\rangle(x,t)=4\int_{0}^{1}\eta f(\eta;x,t){\rm d}\eta.$ (4)

The brackets in Eq. (4) denote ensemble averaging over a large number of
realizations of the SG.

By comparing Eq. (3) with Eq. (4), one can infer that the DOS can be computed
using the equality

$f(\eta_{k};x,t){\rm d}\eta=\sum_{j=k}^{k+\Delta
k}\langle\psi_{j}^{2}(x,t)\rangle,$ (5)

where $k$ and $k+\Delta k$ are integer indices bounding the discrete summation
over the squared eigenfunctions associated with eigenvalues between $\eta_{k}$
and $\eta_{k}+{\rm d}\eta$. Equation (5) provides a local expression for the
DOS of a SG, akin to formulas used in other branches of physics such as
electron microscopy García de Abajo and Kociak (2008); Tersoff and Hamann
(1985).

Fig. 3(c), (f), (i), (l) show the DOS of the SGs from our numerical
simulations. These plots were created using Eq. (5) and $1000$ realizations of
each SG shown in Fig. 3(a), (d), (g), (j). Fig. 3(c) shows that the DOS of the
initial SG is not uniform in space. The fact that the KdV evolution leads to
the emergence of a rank-ordered soliton train (Fig. 3(g)), with eigenfunctions
becoming localized in space (Fig. 3(h)), translates into the DOS being
localized around the curve shown in Fig. 3(i).

Conversely, the SG formed in the experiment under the perturbative influence
of diffusion exhibits a very specific DOS, locally described by the Weyl
distribution $f(x,\eta)=\eta/(\pi\sqrt{\lambda_{1}^{2}(x)-\eta^{2}})$ Congy
_et al._ (2023), with the function $\lambda_{1}(x)$ being slowly modulated in
space, see Fig. 3(f). This DOS corresponds to the peculiar type of SG known as
a genus zero soliton condensate in the spectral kinetic theory of SG Congy
_et al._ (2023); Kuijlaars and Tovbis (2021); Gelash  _et al._ (2021); El and
Tovbis (2020). Here this soliton condensate can be seen as the SG of minimal
entropy or of zero variance Bonnemain  _et al._ (2022).

For a diffusion strength smaller than the experimental one
($\epsilon=2.10^{-3}$), the DOS at a given position $x>0$ is supported on two
disjoint spectral intervals, see Fig. 3(l). The corresponding coherent
oscillatory structure, identified above as a dispersive-diffusive shock wave,
is also characterized by the soliton condensate DOS within the SG spectral
framework, however, now it is the genus one condensate, see Supplemental
Material and Congy  _et al._ (2023) for details. As shown in Congy  _et al._
(2023), the transitions from a non-condensate to the condensate state and vice
versa are not possible in an integrable system. Our experiment demonstrates
that soliton condensate can spontaneously emerge in a nearly integrable
physical system due to the presence of a perturbative dissipative effect. This
observation suggests the possibility of performing further experiments
equivalent of Joule expansion, by evolving a SG according to KdVB equation and
suddenly setting the dissipation parameter $\epsilon$ to zero.

In summary, we have reported the emergence of a soliton condensate in a NLTL,
a nearly integrable physical system described by the KdVB equation. This
phenomenon arises from the dissipation-driven rearrangement of solitonic
eigenmodes within the system, accompanied by the excitation of the continuous
spectrum, resulting in the formation of new soliton states. The coupling
between solitons and continuous spectrum radiation has been recently
highlighted in deep-water models less restrictive than the integrable 1D-NLSE
Gelash  _et al._ (2024). Notably, current hydrodynamic theories like GHD have
yet to incorporate the excitation of the continuous spectrum by a statistical
ensemble of solitons. We anticipate that our work will inspire further
research in this area.

###### Acknowledgements.

This work has been partially supported by the Agence Nationale de la Recherche
through the StormWave (ANR-21-CE30-0009) and SOGOOD (ANR-21-CE30-0061)
projects, the LABEX CEMPI project (ANR-11-LABX-0007), the Ministry of Higher
Education and Research, Hauts de France council and European Regional
Development Fund (ERDF) through the Nord-Pas de Calais Regional Research
Council and the European Regional Development Fund (ERDF) through the Contrat
de Projets Etat-Région (CPER Photonics for Society P4S). The authors would
like to thank the Centre d’Etudes et de Recherche Lasers et Application
(CERLA) for technical support and the Isaac Newton Institute for Mathematical
Sciences for support and hospitality during the programme “Dispersive
hydrodynamics: mathematics, simulation and experiments, with applications in
nonlinear waves”. TB was supported by the Engineering and Physical Sciences
Research Council (EPSRC) under grant EP/W010194/1. GE’s and GR’s work was also
supported by the EPSRC, grant number EP/W032759/1. GR thanks the Simons
Foundation for partial support.

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Supplemental material for :
”Dissipation-driven emergence of a soliton condensate in a nonlinear
electrical transmission line”

Loic Fache,1 Hervé Damart,1 François Copie,1 Thibault Bonnemain, 2, Giacomo
Roberti,3, Thibault Congy,3 Pierre Suret,1 Gennady El,3 Stéphane Randoux1

1 Univ. Lille, CNRS, UMR 8523 - PhLAM -Physique des Lasers Atomes et
Molécules, F-59 000 Lille, France

2 Department of Mathematics, King’s College, London, United Kingdom

3 Department of Mathematics, Physics and Electrical Engineering, Northumbria
University, Newcastle upon Tyne, NE1 8ST, United Kingdom

The purpose of this Supplemental Material is to provide some details about the
experimental setup and about the experimental methododology. All equations,
figures, reference numbers within this document are prepended with “S” to
distinguish them from corresponding numbers in the Letter.

###### Contents

  1. I  Description of the experimental setup and of its associated dynamical equations
  2. II  Nonlinear spectral synthesis of soliton gases using the Darboux method
  3. III  Solitary Wave Propagation, Mass Conservation, and Continuous Spectrum Excitation in the KdVB Equation
  4. IV  Soliton condensates of genus 0 and genus 1

##  I  Description of the experimental setup and of its associated dynamical
equations

Figure S4: (a) Schematic representation of the experimental setup consisting
into a ladder network of $160$ lumped inductor-varactor sections. The
inductors have a constant inductance $L\simeq 15\mu$H while the varactors have
a capacitance that decreases with the applied voltage. The varactors have also
a small but non-negligible series resistance $R\simeq 5\,\Omega$. (b)
Photograph of the experimental setup. (c) Typical characteristic of the
varicap used in the NLTL showing the evolution of the capacitance $C$ as a
function of the applied voltage. The characteristic is nearly linear for
positive voltages below $\sim 3$ V. For this range of voltage, it can be
approximated by $C(V_{n})=C_{0}(1-2bV_{n})$ with $C_{0}=140$ pF and $b=0.154$
V-1.

Our experimental setup is the electrical nonlinear transmission line (NLTL)
schematically shown in Fig. S1(a). It consists into a ladder network of $160$
lumped inductor-varactor sections with each individual inductance (KEMET
SBCP-47HY150B) being $L\simeq 15\mu$ H and each varactor capacitance
$C(V_{n})$ being a decreasing function of the voltage $V_{n}$ applied to it.
The varactors are commercial components (Varicap: NXP-BB201). Fig. S1(c) shows
the measured capacitance $C(V_{n})$ as the function of the voltage $V_{n}$.
For positive voltages below $\sim 3$ V, the capacitance decreases
approximately linearly with the voltage and the curve plotted in Fig. S1(c)
can be well approximated by $C(V_{n})=C_{0}(1-2bV_{n})$ with $C_{0}=140$ pF
and $b=0.154$ V-1. Finally and importantly, the varactors have a small but
non-negligible resistance $R\sim 5\,\Omega$.

The theoretical derivation of the dynamical equations describing the NLTL
shown in Fig. S1(a) has been reported in details in ref. Ricketts and Ham
(2018). For the sake of clarity and completeness, we report here the main
steps showing that nonlinear propagation in the NLTL in the long-wavelength
limit is described by the KdV-Burgers equation. The nodal equations for the
voltages and currents shown in Fig. S1(a) read:

$I_{n}-I_{n-1}=\frac{\partial Q(V_{n})}{\partial t},$ (S1) $L\frac{\partial
I_{n}}{\partial t}=\left[V_{n-1}+R\frac{\partial Q(V_{n-1})}{\partial
t}\right]-\left[V_{n}+R\frac{\partial Q(V_{n})}{\partial t}\right],$ (S2)
$L\frac{\partial I_{n+1}}{\partial t}=\left[V_{n}+R\frac{\partial
Q(V_{n})}{\partial t}\right]-\left[V_{n+1}+R\frac{\partial
Q(V_{n+1})}{\partial t}\right],$ (S3)

Combining Eqs. (S1), (S2), (S3), we obtain:

$L\frac{\partial^{2}Q(V_{n})}{\partial
t^{2}}=V_{n-1}+V_{n+1}-2V_{n}+R\frac{\partial}{\partial
t}\left[Q(V_{n+1})+Q(V_{n-1})-2Q(V_{n})\right]$ (S4)

Changing $V_{n}(t)$ into $V(n,t)$ and Taylor-expanding the voltages $V(n,t)$
as

$V_{n\pm 1}=V\pm\frac{\partial V}{\partial
n}+\frac{1}{2!}\frac{\partial^{2}V}{\partial
n^{2}}\pm\frac{1}{3!}\frac{\partial^{3}V}{\partial
n^{3}}+\frac{1}{4!}\frac{\partial^{4}V}{\partial n^{4}}\pm O(n^{5}),$ (S5)

we obtain

$V_{n+1}+V_{n-1}-2V_{n}=\frac{\partial^{2}V(n,t)}{\partial
n^{2}}+\frac{1}{12}\frac{\partial^{4}V(n,t)}{\partial n^{4}}+O(n^{5})$ (S6)

Substituting the charge-voltage relation of the varactor
$Q(V_{n})=C_{0}(V_{n}-bV_{n}^{2})$ in Eq. (S4), we find that propagation in
the NLTL in the long-wavelength limit is described at leading order by the
following partial differential equation Ricketts and Ham (2018):

$V_{tt}-b(V^{2})_{tt}=v_{0}^{2}\left(V_{nn}+\frac{V_{nnnn}}{12}\right)+\frac{R}{L}V_{nnt},$
(S7)

where $v_{0}=1/\sqrt{LC_{0}}$. The subscripts in Eq. (S7) denote derivation
with respect to $n$ and $t$. Following ref. Ricketts and Ham (2018) a small
dimensionless parameter $\alpha$ is introduced for scaling linear and
nonlinear effects. Changing $V$ into $\alpha V$, $R/L$ into $\alpha^{1/2}R/L$
and introducing the following new variables:

$\begin{cases}s=\alpha^{1/2}(n+(\alpha-1)v_{0}t),\\\
\tau=\alpha^{3/2}t,\end{cases}$

we obtain the following equation that describes the nonlinear propagation of
waves in the NLTL in physical variables

$\frac{\partial V}{\partial\tau}+v_{0}\frac{\partial V}{\partial
s}+bv_{0}V\frac{\partial V}{\partial
s}+\frac{v_{0}}{24}\frac{\partial^{3}V}{\partial
s^{3}}=\frac{R}{2L}\frac{\partial^{2}V}{\partial s^{2}}.$ (S8)

Considering that $V_{s}\sim-\frac{1}{v_{0}}V_{\tau}$, the equation describing
the space-time evolution of the voltage along the LC oscillator chain reads:

$V_{s}+\frac{1}{v_{0}}V_{\tau}-\frac{b}{c_{0}}VV_{\tau}-\frac{1}{24v_{0}^{3}}V_{\tau\tau\tau}-\frac{R}{2Lc_{0}^{3}}V_{\tau\tau}=0.$
(S9)

Finally introducing the following normalized space and time variables together
with the normalized field $u(x,t)$

$\begin{cases}t=\frac{1}{3}\sqrt{b^{3}V_{0}^{3}}s,\\\
x=-2v_{0}\sqrt{bV_{0}}\left(\tau-\frac{s}{v_{0}}\right),\\\
u=\frac{V}{V_{0}},\end{cases}$ (S10)

we obtain the KdV-Burgers equation:

$u_{t}+6uu_{x}+u_{xxx}=\epsilon u_{xx},$ (S11)

where $\epsilon=6R/(Lv_{0}\sqrt{bV_{0}})$, $V_{0}$ being the value of the
voltage used for normalization.

Fig. S2 compares the space-time diagram recorded in the experiment (left
column) with the space-time diagram converted to dimensionless units (right
column) using the transformations specified by Eq. (S10). The initial
condition is identical to the one used in the Letter: it is a soliton gas (SG)
parameterized by $200$ discrete eigenvalues $\eta_{k}$ that are randomly
distributed between $0.2$ and $0.6$ in an uniform way. After the step of the
numerical spectral synthesis of the SG (see Sec. II), it is generated by a
programmable electric waveform generator having a bandwidth of $100$ MHz and a
sampling rate of $1$ GS/s. The SG has an initial duration of $\sim 28\mu$s.
The typical timescale associated with voltage fluctuations, which are linked
to the randomness of the SG, is around $150$ ns. The experimental space-
diagram is recorded by measuring the voltage every $5$ LC cells using an
oscilloscope having a bandwidth of $200$ MHz.

The normalized time interval over which the evolution of the SG is observed is
as large as $420$. Such an extended evolution time is achieved not in a single
pass through the NLTL, but in $6$ passes. Each pass corresponds to a
normalized time interval $\Delta t=70$, which is equivalent to physical
propagation through the $160$ LC cells composing the NLTL. To achieve multiple
passes in the NLTL, the electrical voltage recorded at the final LC cell after
each pass is used as the initial condition for the subsequent pass.

Due to the small series resistance of the inductors ($\sim 0.09\,\Omega$),
there is slight damping in the NLTL. Consequently, the mass, defined as
$M=\int u(x,t)dx$, is not perfectly conserved in a single pass. In the
experiment, the mass decays by $\sim 5\%$ over $\Delta t=70$. This mass decay
is compensated for after each pass by a slight amplification of the signal
recorded at the final LC cell before it is reinjected in the NLTL. This
compensation results in small periodic oscillations of the mass, as shown in
Fig. S2(c). However, the mass is well conserved on average over the entire
evolution of the SG between $t=0$ and $t=420$.

Figure S5: (a) Space-time diagram recorded in the NLTL by using a $200$ MHz
oscilloscope. The voltage is measured every $5$ LC cells. $n$-index indicates
the LC cell at which the voltage is probed. (b) Normalized experimental space-
time diagram computed by using Eq. (S10). (c) Time evolution of the measured
mass $M(t)=\int u(x,t)dx$ showing small oscillations due to the weak damping
and its compensation at each pass in the NLTL. (d) Time evolution of the
voltage measured at $n=0$ and $n=960$, after $6$ passes in the NLTL. (e)
Normalized field $u(x,t)$ at $t=0$ and $t\sim 420$.

##  II  Nonlinear spectral synthesis of soliton gases using the Darboux method

The algorithm to generate N-soliton solution of the KdV equation with $N=200$
is based on the Darboux method. The Darboux transform is a recursive
transformation scheme where the fundamental soliton solution of the KdV
equation is used as a building block for the construction of higher-order
$N$-soliton solutions through the iterative addition of new discrete
eigenvalues $\eta_{k}$. The algorithm for the generation of KdV SG has been
developed in ref. Congy  _et al._ (2023). Refering the reader to ref. Congy
_et al._ (2023); Liao and Huang (1996) for more details about the mathematical
description of the Darboux machinary, we only provide here the algorithmic
recipe to generate the N-soliton solution in a practical way.

\- The first step consists is generating an ensemble of $N$ discrete
eigenvalues $\eta_{k}$ ($k=1-N$) ranked in ascending order, from the smallest
to the largest. An ensemble of $N$ position parameters $x_{0k}$ must also be
created. For generating the SG used as initial condition in our experiment,
the position parameters $x_{0k}$ are randomly and uniformely distributed over
the interval $[-25,25]$. The spectral parameters $\eta_{k}$ associated with
this SG are randomly distributed between $0.2$ and $0.6$ in an uniform way.

\- A numerical box of size $L$ and a given time $t$ must then be defined is
such a way that the quantities:

$\Theta_{k}(x,t)=\eta_{k}(x-x_{0k}-4\eta_{k}^{2}t)$ (S12)

can be computed for $x\in[-L/2,L/2]$ for all the values of $k$ ($k=1-N$).

\- For each $k$ between $1$ and $N$, compute :

$\displaystyle
q_{1}(x,t,\eta_{k})=\eta_{k}\tanh(\Theta_{k}(x,t))\quad\mathrm{for\,\,k\,\,odd}$
(S13) $\displaystyle
q_{1}(x,t,\eta_{k})=\eta_{k}(\tanh(\Theta_{k}(x,t)))^{-1}\quad\mathrm{for\,\,k\,\,even}$

\- For $N\geq k\geq 2$, compute:

$q_{k}(x,t,\eta_{k})=\frac{\eta_{k}^{2}-\eta_{k-1}^{2}}{q_{k-1}(x,t,\eta_{k})-q_{k-1}(x,t,\eta_{k-1})}-q_{k-1}(x,t,\eta_{k-1})$
(S14)

\- For $N\geq k\geq 2$, compute iteratively

$u_{k}(x,t)=-u_{k-1}(x,t)-2\eta_{k}^{2}+2q_{k}(x,t,\eta_{k})^{2}$ (S15)

by starting the iteration from
$u_{1}(x,t)=-2\eta_{1}^{2}\,\text{sech}^{2}(\Theta_{1}(x,t))$. The N-soliton
solution $u(x,t)$ of the KDV equation associated with the distribution of the
spectral parameters $\eta_{k}$ initially chosen is obtained by changing
$u_{N}(x,t)$ into $-u_{N}(x,t)$.

##  III  Solitary Wave Propagation, Mass Conservation, and Continuous Spectrum
Excitation in the KdVB Equation

The question of the evolution of a KdV soliton under the influence of
dissipative perturbations has been extensively studied in the literature
Madsen and Mei (1969); Grimshaw (1970); Johnson (1973); Kaup  _et al._ (1978);
Kodama and Ablowitz (1981); Kivshar and Malomed (1989); Grimshaw  _et al._
(1994); Johnson (1994); Ko and Kuehl (1978); Knickerbocker and Newell (1980,
1985); El and Grimshaw (2002). In particular, the concept that new discrete
eigenvalues can be generated due to the excitation of continuous spectrum
radiation has been discussed in references Kaup  _et al._ (1978); Wright
(1980); Newell (1985) in the context of solitary wave propagation in shallow
water with slowly varying depth. In this section, we present numerical
simulations to describe the long-term evolution of a solitary wave, aiming to
synthesize and illustrate the nonadiabatic features previously identified in
the literature within the framework of KdVB dynamics.

Figure S6: Numerical simulations of Eq. (S16) with $\epsilon=0.1$. The initial
condition is a soliton given by
$u(x,t=0)=2\eta_{1}^{2}\,sech^{2}(\eta_{1}\,x)$ with $\eta_{1}=0.707$. (a)
Field $u(x,t)$ numerically computed at $t=0$ (blue line) and at $t=100$
(orange line). (b) Space-time plot showing the evolution of the solitary wave.
(c) Corresponding time evolution of the mass $M$, of the momentum $P$ and of
the energy $E$. (d) Time evolution of the discrete eigenvalues successively
created during the time evolution of the solitary wave. (e) Time evolution of
the discrete masses defined as $m_{i}=4\eta_{i}$ and of the discrete mass
$M_{DS}=\sum\limits_{i=1}^{5}4\eta_{i}$. The dashed black line represents the
total mass defined as $M=\int u(x,t)dx$. (f) Time evolution of the discrete
mass $M_{DS}$ (black line) and of the mass of the continuous spectrum defined
as
$M_{cs}=\frac{1}{\pi}\int_{-\infty}^{+\infty}\log\left(1-\frac{|b(\xi)|^{2}}{|a(\xi)|^{2}}\right)d\xi$.
The red line is the total mass defined as $M=M_{DS}+M_{CS}$ according to Eq.
(S22).

As in the Letter, we consider the KdVB equation

$u_{t}+6uu_{x}+u_{xxx}=\epsilon u_{xx},$ (S16)

and its associated Schrödinger equation in the IST theory

$\psi_{xx}+(u(x,t)-\lambda)\psi=0.$ (S17)

The mass $M$ of the field $u(x,t)$ is defined as :

$M=\int_{-\infty}^{+\infty}u(x,t)dx.$ (S18)

The mass $M$ is a conserved quantity both in KdV and in KdVB equations:
$dM/dt=0$. The momentum $P$ is defined as :

$P=\int_{-\infty}^{+\infty}u^{2}(x,t)dx.$ (S19)

The momentum $P$ is a conserved quantity in KdV equation ($dP/dt=0$) but it is
not a conserved quantity in KdVB equation. For the KdVB equation, it is easy
to show that :

$\frac{dP}{dt}=-2\epsilon\int_{-\infty}^{+\infty}u_{x}^{2}(x,t)dx,$ (S20)

which means that the rate at which this quantity decays is determined by the
gradients $u_{x}$ of the field in space. The energy $E$ is defined as :

$E=\int_{-\infty}^{+\infty}\left(u^{3}(x,t)-\frac{u_{x}^{2}(x,t)}{2}\right)dx.$
(S21)

The energy is a conserved quantity in KdV but not in KdVB. In KdVB, the energy
changes in time but does not necessarily decrease.

In the IST formalism, the mass $M$ and the momentum $P$ have the following
expressions:

$M=4\sum_{k}\eta_{k}+\frac{1}{\pi}\int_{-\infty}^{+\infty}\log\left(1-\frac{|b(\xi)|^{2}}{|a(\xi)|^{2}}\right)d\xi$
(S22)
$P=\frac{16}{3}\sum_{k}\eta_{k}^{3}-\frac{4}{\pi}\int_{-\infty}^{+\infty}\xi^{2}\log\left(1-\frac{|b(\xi)|^{2}}{|a(\xi)|^{2}}\right)d\xi$
(S23)

where $\eta_{k}$ are the discrete eigenvalues that can be computed by solving
Eq. (S17) for the potential $u(x,t)$. $\rho(\xi)=b(\xi)/a(\xi)$ is the
reflection coefficient defined in the IST theory, see e.g. ref. Kaup  _et al._
(1978); Wright (1980); Newell (1985). In Eq. (S22) (resp. Eq. (S23)), the
discrete summation over the $\eta_{k}$ (resp. $\eta_{k}^{3}$) is related to
the discrete part of the spectrum. On the other hand the integral term in Eq.
(S22) (resp. (S23)) represents the contribution to the mass (resp. to the
momentum) of the continuous part of the spectrum that is associated with the
radiative modes.

Fig. S3 shows numerical simulations of Eq. (S16) with $\epsilon=0.1$ for an
initial condition being under the form of a soliton given by
$u(x,t=0)=2\eta_{1}^{2}\,sech^{2}(\eta_{1}\,x)$ with $\eta_{1}=0.707$, like in
the experiment shown in Fig. 2 of the Letter. The maximum evolution time
reached in the numerical simulation shown in Fig. S3 is $100$, which is
significantly larger than the experimental evolution time ($t\simeq 16$). The
presence of diffusion described by the term on the right-hand side of Eq.
(S16) induces a slow decay in the soliton’s amplitude and a gradual decrease
in its velocity, as illustrated in Fig. S3(a)(b). This evolution is
accompanied by the formation of a shelf and of small oscillations behind the
solitary wave. The mass $M$ is conserved during the evolution while the
momentum $P$ and the energy $E$ decrease over time, see Fig. S3(c).

Fig. S3(d) shows that the amplitude $\eta_{1}$ of the discrete eigenvalue
linked to the initial condition gradually decays over time, while $4$ new
discrete eigenvalues are created during the shelf formation. Fig. S3(d) shows
the evolution of the masses $m_{i}=4\eta_{i}$ of each of the $4$ discrete
eigenmodes, see Eq. (S22). Of course the discrete masses
$m_{i}(t)=4\eta_{i}(t)$ $(i=1-4)$ follow the same time evolution as the
discrete eigenvalues $\eta_{i}(t)$. The interesting point is that the sum of
the discrete masses $M_{DS}(t)=\sum\limits_{i=1}^{8}4\eta_{i}(t)$ is not
constant but slightly fluctuates above the constant value of the total mass
$M=\int u(x,t)dx$, see the full and dotted black lines in Fig. S3(e).

The decay in time of $\eta_{1}$ is associated to a decrease in mass, which
must be compensated by an increase to preserve the fact that the total mass
$M=\int u(x,t)dx$ must be conserved Kivshar and Malomed (1989). The missing
mass is provided by the generation of the new discrete eigenvalue $\eta_{2}$.
However the decay of $\eta_{1}(t)$ and the growing of $\eta_{2}(t)$ can never
be balanced in such an exact way that the mass associated with these two
discrete eigenvalues is constant in time :
$m_{1,2}(t)=m_{1}(t)+m_{2}(t)=4(\eta_{1}(t)+\eta_{2}(t))\neq cte$. For the
total mass conservation condition $dM/dt=0$ to be fulfilled, there is no
choice but for the mass $M_{cs}$ associated with the radiative modes to
fluctuate in the same way as $M_{d}$ but with an opposite sign
($M_{CS}=\frac{1}{\pi}\int_{-\infty}^{+\infty}\log\left(1-\frac{|b(\xi)|^{2}}{|a(\xi)|^{2}}\right)d\xi$).
This is illustrated in Figure S3(f), which shows that the negative mass
fluctuations of the continuous spectrum exactly compensate for the positive
fluctuations of the discrete mass $M_{DS}$, so that the total mass
$M=M_{DS}+M_{CS}$ is constant over time.

##  IV  Soliton condensates of genus 0 and genus 1

The notion of soliton condensate as the ‘densest possible soliton gas’ for a
given spectral support was introduced in El and Tovbis (2020) in the context
of the focusing NLS equation. A detailed theory of soliton condensates for the
KdV equation has been developed in Congy  _et al._ (2023). Here, we refer the
reader to these papers and only provide the mathematical expressions for the
densities of state (DOS) of soliton condensates of genus $0$ and $1$. We also
show that these mathematical expressions fit very well the DOS found in
numerical simulations reported in Fig. 3 of the Letter.

Figure S7:  (a) The blue crosses represents the spectral distribution shown in
the inset of Fig. 3(f). The orange line fits the blue crosses with the model
(S24) with $\lambda_{1}=0.533$. (b) The blue crosses represents the spectral
distribution shown in the inset of Fig. 3(l). The orange line fits the blue
crosses with the model (S25) with $\lambda_{1}=0.131$, $\lambda_{2}=0.295$,
$\lambda_{3}=0.359$.

The DOS of a KdV soliton condensate of genus $0$ is given by the so-called
Weyl distribution that reads:

$f(\eta)=\frac{\eta}{\pi\sqrt{\lambda_{1}^{2}-\eta^{2}}};\qquad\eta\in[0,\lambda_{1}].$
(S24)

Fig. S4(a) shows that the DOS of the SG measured at $t=420$ in the electrical
NLTL ($\epsilon=0.1$) is very well fitted by the Weyl distribution with
$\lambda_{1}=0.533$, see also Fig. 3(f) in the Letter.

The DOS of a KdV soliton condensate of genus $1$ lives on a spectral support
not composed of one band but of two bands with their endpoints being given by
the three real parameters $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$. The DOS
of the soliton condensate of genus 1 reads

$f(\eta)=\frac{i\eta(\eta^{2}-w^{2})}{\pi\sqrt{(\eta^{2}-\lambda_{1}^{2})(\eta^{2}-\lambda_{2}^{2})(\eta^{2}-\lambda_{3}^{2})}};\qquad\eta\in[0,\lambda_{1}]\cup[\lambda_{2},\lambda_{3}]$
(S25)

with $w^{2}=\lambda_{3}^{2}-(\lambda_{3}^{2}-\lambda_{1}^{2})\chi(m)$,
$\chi(m)=E(m)/K(m)$ and
$m=(\lambda_{2}^{2}-\lambda_{1}^{2})/(\lambda_{3}^{2}-\lambda_{1}^{2})$.
$E(m)$ (resp. $K(m)$) is the complete elliptic integral of the second (resp.
first) kind.

Fig. S4(b) shows that the DOS of the SG measured at ($x=2900$, $t=4000$) in
the numerical simulations of the KdvB equation with small diffusion
($\epsilon=2.10^{-3}$) is remarkably well fitted by Eq. (S25) with
$\lambda_{1}=0.131$, $\lambda_{2}=0.295$, $\lambda_{3}=0.359$, see also Fig.
3(l) in the Letter. It was conjectured and numerically verified in Congy  _et
al._ (2023) that realizations of the genus $1$ condensate with DOS (S25)
almost surely coincide with periodic wave solutions of KdV parameterized by
$\lambda_{1}$, $\lambda_{2}$ and $\lambda_{3}$. These solutions locally
describe diffusive-dispersive shock waves of the KdVB equation El  _et al._
(2017).