(breather) state within a soliton (breather) gas. We remind that for breather
gas $\delta_{0}\neq 0$ and for soliton gas $\delta_{0}=0$.

We also note that the exponential and sub-exponential spectral scalings have
the ‘thermodynamic’ property in the sense that they preserve finiteness of the
total density of waves $K_{N}=\sum_{j=1}^{N}k_{j}$ in the limit $N\to\infty$
so that $\lim_{N\to\infty}K_{N}=\beta$, where $0<\beta<\infty$. Note that for
the super-exponential scaling $\beta\to 0$.

####  3.3.3 Nonlinear dispersion relations and kinetic equation

We first present the results for the general case of breather gas by
evaluating the thermodynamic limit of the nonlinear dispersion relations
(3.61) for the exponential spectral scaling (3.64). Without much loss of
generality we assume that the Stokes band lies on the imaginary axis,
$\gamma_{0}=[-iq,iq]$, $q>0$ (so that $\delta_{0}=iq$) and that
$\eta_{i}\in\text{arc}(a,b)\subset\Gamma^{+}$, and introduce the simultaneous
scaling for the solitonic wavenumbers and frequencies (cf. (3.34))

$k_{j}\sim\frac{\varkappa_{j}}{N},\qquad\omega_{j}\sim\frac{\nu_{j}}{N},\qquad
N\gg 1,$ (3.67)

so that $\varkappa_{j}=\varkappa(\eta_{j})$ and $\nu_{j}=\nu(\eta_{j})$, where
the functions $\varkappa(\lambda)\geq 0$ and $\nu(\lambda)$ are smooth
interpolations of $\varkappa_{j},\nu_{j}$. Similar to the KdV case, the
scaling (3.67) provides a balance of terms in relations (3.61) for $N\gg 1$.

The resulting nonlinear dispersion relations for breather gas have the form
(we refer the reader to [38] for details of the derivation):

$\displaystyle\int_{\Gamma^{+}}D(\lambda,\mu)f(\mu)|\mathrm{d}\mu|+\sigma(\lambda)f(\lambda)=\text{Im}[R_{0}(\lambda)],$
(3.68)
$\displaystyle\int_{\Gamma^{+}}D(\lambda,\mu)v(\mu)|\mathrm{d}\mu|+\sigma(\lambda)v(\lambda)=-2\text{Im}[\lambda
R_{0}(\lambda)],$ (3.69)

where $R_{0}(z)=\sqrt{z^{2}+q^{2}}$ (with the branch cut $[-iq,iq]$, and the
branch of the radical defined by $R_{0}(z)\to z$ as $z\to\infty$), and with a
slight abuse of notation, we denoted
$\int_{a}^{b}\dots|d\mu|\equiv\int_{\Gamma^{+}}\dots|d\mu|$. Further,

$D(\lambda,\mu)=\left[\ln\left|\frac{\mu-\bar{\lambda}}{\mu-\lambda}\right|+\ln\left|\frac{R_{0}(\lambda)R_{0}(\mu)+\lambda\mu+q^{2}}{R_{0}(\bar{\lambda})R_{0}(\mu)+\bar{\lambda}\mu+q^{2}}\right|\right],$
(3.70) $f(\lambda)=\frac{1}{2\pi}\varkappa(\lambda)\varphi(\lambda),\ \
v(\lambda)=\frac{1}{2\pi}\nu(\lambda)\varphi(\lambda),\ \
\sigma(\lambda)=\frac{2\tau(\lambda)}{\varphi(\lambda)}.$ (3.71)

Equations (3.68), (3.69) are the focusing NLS counterparts of the nonlinear
dispersion relations (3.37), (3.38) for the KdV soliton gas with $f(\lambda)$
and $v(\lambda)$ having the meanings of the DOS and the spectral flux density
respectively, and the function $\sigma(\lambda)$ encoding the Zakharov-Shabat
spectrum of the finite gap potentials in the thermodynamic limit.

The soliton gas limit in (3.68), (3.69) is achieved by vanishing the Stokes
band, $q\to 0$, resulting in

$\displaystyle\int_{\Gamma^{+}}\ln\left|\frac{\mu-\bar{\lambda}}{\mu-\lambda}\right|f(\mu)|\mathrm{d}\mu|+\sigma(\lambda)f(\lambda)$
$\displaystyle=$ $\displaystyle\text{Im}\lambda,$ (3.72)
$\displaystyle\int_{\Gamma^{+}}\ln\left|\frac{\mu-\bar{\lambda}}{\mu-\lambda}\right|v(\mu)|\mathrm{d}\mu|+\sigma(\lambda)v(\lambda)$
$\displaystyle=$ $\displaystyle-4\text{Im}\lambda\,\text{Re}\lambda.$ (3.73)

Eliminating the function $\sigma(\lambda)$ from the nonlinear dispersion
relations (3.68), (3.69) and (3.72), (3.73) we obtain the equation of state

$s(\lambda)=s_{0}(\lambda)+\int_{\Gamma^{+}}G(\lambda,\mu)[s(\lambda)-s(\mu)]f(\mu)|\mathrm{d}\mu|,$
(3.74)

where $s(\lambda)=v(\lambda)/f(\lambda)$ is the effective velocity of the
tracer soliton (breather) in the gas, and $s_{0}(\lambda)$, $G(\lambda,\mu)$
are defined as follows.

For breather gas:

$s_{0}(\lambda)=-2\frac{\text{Im}[\lambda
R_{0}(\lambda)]}{\text{Im}[R_{0}(\lambda)]},\qquad
G(\lambda,\mu)=\frac{1}{\text{Im}[R_{0}(\lambda)]}D(\lambda,\mu),$ (3.75)

and for soliton gas:

$s_{0}(\lambda)=-4\text{Re}\lambda,\quad
G(\lambda,\mu)=\frac{1}{\text{Im}\lambda}\ln\left|\frac{\mu-\bar{\lambda}}{\mu-\lambda}\right|.$
(3.76)

One can see that $s_{0}(\lambda)$ in (3.75) coincides with the group velocity
$c_{g}(\lambda)$ of an isolated TW breather (3.56) and $s_{0}(\lambda)$ in
(3.76) coincides with the group velocity of the fundamental soliton of the
focusing NLS equation. Furthermore, the integral kernel $G(\lambda,\mu)$ in
(3.76) coincides with the absolute value of the position shift (2.33) in the
two-soliton collision for the focusing NLS equation. As one may expect, the
expression for $G(\lambda,\mu)$ in eq. (3.75) is identified with the position
shift in two-breather collisions [125], [114] (see the proof in [40]). Thus
the outlined derivation provides the justification of the collision rate
assumption (2.12) for dense soliton and breather gases of the focusing NLS
equation.

The integral equations (3.72), (3.73) have been recently studied in a rather
general spectral geometry [44]. The existence and uniqueness of their
solutions was investigated and the non-negativity of the solutions for the DOS
$f(\lambda)$ was proved.

Spectral kinetic equation for nonequilibrium gas

The derivation of the transport equation for non-equilibrium soliton and
breather gases for the focusing NLS equation follows the general modulation
construction outlined in Section 3.1 and realised for the KdV equation in
Section 3.2.2, albeit with some important differences.

The multiphase Whitham modulation theory for $n$-gap solutions of the focusing
NLS equation [122], [124] includes the system of $n=2N$ conservation laws
${\boldsymbol{k}}_{t}+{\boldsymbol{\omega}}_{x}=0$, which is split into two
distinct subsystems

$\displaystyle\partial_{t}k_{j}({\boldsymbol{\lambda}})+\partial_{x}\omega_{j}({\boldsymbol{\lambda}})=0,\quad
j=1,\dots,N,$ (3.77)
$\displaystyle\partial_{t}\tilde{k}_{j}({\boldsymbol{\lambda}})+\partial_{x}\tilde{\omega}_{j}({\boldsymbol{\lambda}})=0,\quad
j=1,\dots,N$ (3.78)

for the solitonic and carrier components of the fundamental wavenumber and
frequency vectors (see Section 3.3.1). These subsystems are complemented by
the respective nonlinear dispersion relations

$\displaystyle{k}_{j}=K_{j}({\boldsymbol{\lambda}}),\quad\omega_{j}=\Omega_{j}({\boldsymbol{\lambda}}),\quad
j=1,\dots,N,$ (3.79)
$\displaystyle\tilde{k}_{j}=\tilde{K}_{j}({\boldsymbol{\lambda}}),\quad\tilde{\omega}_{j}=\tilde{\Omega}_{j}({\boldsymbol{\lambda}}),\quad
j=1,\dots,N,$ (3.80)

obtained by taking the imaginary and the real part of the basic system (A.4)
(see Appendix). In particular, the dispersion relations for the solitonic
components are explicitly given by the system (3.61).

Application of the thermodynamic limit to the modulation system (3.77), (3.79)
for the solitonic wavenumbers (see Section 3.2.2 for the similar derivation in
the KdV context) yields the spectral transport equation

$f_{t}+(fs)_{x}=0$ (3.81)

for the DOS $f(\lambda,x,t)$. The transport equation (3.81) is complemented by
the equation of state (3.74), (3.75) for breather gas (or (3.74), (3.76) for
soliton gas). One can see that the kinetic equation for soliton gas agrees
with equation (2.34) derived via the phenomenological approach based on the
collision rate assumption (2.12).

The carrier wave modulation system (3.78), (3.80) in the thermodynamic limit
becomes (see [38] for details)

$\tilde{f}_{t}+(\tilde{f}\tilde{s})_{x}=0,\quad\tilde{s}(\lambda)=\tilde{\mathcal{S}}[f(\lambda)],$
(3.82)

where $\tilde{f}(\lambda,x,t)$ and $\tilde{f}\tilde{s}=\tilde{v}(\lambda,x,t)$
are some smooth functions on $\Gamma^{+}$ interpolating
$\tilde{k}_{j},\tilde{\omega}_{j}$, that is,
$\tilde{f}(\eta_{j})=\tilde{k}_{j}$, $\tilde{v}(\eta_{j})=\tilde{\omega}_{j}$,
$j=1,\dots,N$, where $\eta_{j}$ are the centres of the bands (3.59). The
dependencies of $\tilde{s}(\lambda)$ on the DOS $f(\lambda)$ for soliton and
breather gases are given by somewhat lengthy expressions which we do not
present here. We only mention that in the limit of zero density,
$f(\lambda)\to 0$, the quantity $\tilde{s}$ transforms into the phase velocity
$s_{p}(\lambda)=-{2\text{Re}[\lambda^{2}]}/{\text{Re}\lambda}$ of the
fundamental soliton (2.32) if $q=0$ or the phase velocity
$c_{p}(\lambda)=-{2\text{Re}[\lambda
R_{0}(\lambda)]}/{\text{Re}[R_{0}(\lambda)]}$ (3.56) of the TW breather
($q\neq 0$).

Generalisation to 2D Case

The above derivation of the spectral kinetic equation for the focusing NLS
soliton and breather gases involves the basic assumption that the DOS
$f(\lambda)$ is supported on a 1D symmetric curve $\Gamma$ in the complex
plane. This restriction can be readily removed as we show below.

In the case when the shrinking bands $\gamma_{j}$, $j>0$ fill a compact 2D
region $\Lambda^{+}$ of the upper complex half-plane, the counterpart of the
exponential scaling (3.64) is

$|\delta_{j}|\sim e^{-N^{2}\tau{(\eta_{j})}},$ (3.83)

where $\tau(\lambda)$ is a positive smooth function on $\Lambda^{+}$. The
scaling of the gaps remains $\mathcal{O}(1/N)$, where by the gap width we
understand the closest distance between the bands. In this case
$\varphi(\lambda)>0$ is the 2D density of bands (and we also distinguish the
cases of exponential, sub-exponential and super-exponential scalings of bands,
similarly to the 1D case).

We now assume one of the 2D spectral thermodynamic spectral scalings
(exponential, sub-exponential, and super-exponential) when the shrinking bands
$\gamma_{j}$ fill a 2D region $\Lambda$ of the complex plane, see Section III
B. For the wave numbers and frequencies instead of (3.67) we introduce

$k_{j}=\frac{\varkappa_{j}}{N^{2}},\qquad\omega_{j}=\frac{\nu_{j}}{N^{2}},\qquad
N\gg 1,$ (3.84)

where $\varkappa_{j}=\varkappa(\eta_{j})$ and $\nu_{j}=\nu(\eta_{j})$, and the
interpolating functions $\varkappa(\lambda)\geq 0$, $\nu(\lambda)$ are assumed
to be smooth on $\Lambda^{+}$.

Then the 2D thermodynamic limit of the nonlinear dispersion relations (3.61)
leads to the same integral equations (3.68)-(3.74) but with the line
integration along $\Gamma^{+}$ replaced by the integration over a 2D compact
domain $\Lambda^{+}$:

$\int\limits_{\Gamma^{+}}\dots|\mathrm{d}\mu|\to\iint\limits_{\Lambda^{+}}\dots\mathrm{d}\xi\mathrm{d}\zeta\,,$
(3.85)

where $\mu=\xi+i\zeta$.

For convenience of the exposition we shall be using the notation
$\int_{\Gamma^{+}}\dots|\mathrm{d}\mu|$ in both 1D and 2D cases keeping in
mind that in the 2D case the meaning of the integral is given by Eq. (3.85).

####  3.3.4 Rarefied soliton gas and soliton condensate

The dispersion relations and the equations of state for soliton and breather
gases were derived under the assumption of the exponential spectral scaling
(3.64) (or (3.83) in 2D case). We now consider the two other scalings of
interest: the superexponential scaling (3.66) and subexponential scaling
(3.65). We will restrict our exposition to soliton gas but will indicate when
the results can be extended to breather gas.

It is convenient to characterise the spectral scalings and the corresponding
soliton gases in terms of the function $\sigma(\lambda)$ parametrising the
nonlinear dispersion relations (3.68) and (3.72). From Eq. (3.68) we have:

$\sigma(\lambda)=\frac{\text{Im}\lambda-\int_{\Gamma^{+}}\ln\left|\frac{\mu-\bar{\lambda}}{\mu-\lambda}\right|f(\mu)|\mathrm{d}\mu|}{f(\lambda)}\geq
0,$ (3.86)

(there is a similar expression for breather gas which we do not present here).
For the exponential scaling $\sigma(\lambda)=\mathcal{O}(1)$, while the
limiting cases $\sigma\to\infty$ and $\sigma\to 0$ correspond to the super-
and sub-exponential spectral scalings respectively.

Rarefied soliton gas

Rarefied soliton gas represents an infinite random ensemble of weakly
interacting solitons characterized by a small DOS, $f\ll 1$, and therefore,
$\sigma\gg 1$ by (3.86). We shall refer to the limit $f\to 0$,
$\sigma\to\infty$, $f\sigma=\mathcal{O}(1)$ as the ideal gas limit as it
corresponds to the gas of non-interacting breathers (solitons). Spectrally
this limit corresponds to the super-exponential spectral scaling (3.66).

For a rarefied gas the interaction (integral) term in the equation of state
(3.74) is sub-dominant so the leading order term $s(\lambda)=s_{0}(\lambda)$
describes the group velocity distribution in an ideal breather (soliton) gas.
Then the first correction to the ideal gas velocity $s_{0}(\lambda)$ is
readily computed to give:

$s(\lambda)\approx
s_{0}(\lambda)+\int\limits_{\Gamma^{+}}G(\lambda,\mu)[s_{0}(\lambda)-s_{0}(\mu)]f(\mu)|\mathrm{d}\mu|.$
(3.87)

Eq. (3.87) represents the focusing NLS counterpart of the equation (2.9) for
the effective soliton velocity in a rarefied KdV soliton gas introduced by
Zakharov [28].

All the above results apply to breather gas as well.

Soliton condensate

The inequality $\sigma(\lambda)\geq 0$ in (3.86) imposes a fundamental
constraint on the DOS $f(\lambda)$. As follows from the discussion in Sec.
3.3.2, the critical value $\sigma(\eta)=0$ corresponds to the sub-exponential
spectral scaling (3.65). One can see from the nonlinear dispersion relations
(3.72), (3.73) that in this case the gas properties are fully determined by
the interaction (integral) terms, while the information about the individual
quasi-particles (described by the non-integral, secular, terms) is completely
lost. By analogy with Bose-Einstein condensation we shall call the soliton gas
at $\sigma=0$ the soliton condensate. From (3.72), (3.73) we obtain the
dispersion relations for the focusing NLS soliton condensate

$\int_{\Gamma^{+}}\ln\left|\frac{\mu-\bar{\lambda}}{\mu-\lambda}\right|f(\mu)|\mathrm{d}\mu|=\text{Im}\lambda,\quad\int_{\Gamma^{+}}\ln\left|\frac{\mu-\bar{\lambda}}{\mu-\lambda}\right|v(\mu)|\mathrm{d}\mu|=-4\text{Im}\lambda\,\text{Re}\lambda.$
(3.88)

Equations (3.88) represent integral equations (the Fredholm equations of the
first kind) for the critical density of states $f=f_{\rm c}(\lambda)$ and the
corresponding spectral flux density $v=v{\rm{}_{c}}(\lambda)$. The existence
and uniqueness of their solutions depend on the geometry of the spectral locus
domain $\Gamma^{+}$ (generally 2D). Below we present a particular case where
an explicit solution is available.

If the spectral support $\Gamma$ of the DOS belongs to a vertical line,
$\Gamma^{+}\subset i\mathbb{R}^{+}$, the second equation (3.88) implies
$v(\lambda)=0$ and, hence, $s(\lambda)=0$. We shall generally call such a non-
propagating gas a bound state soliton gas, and it will be the bound state
soliton condensate in the present context. Let $\Gamma=[-iq,iq]$ for some
$q>0$ and assume that $f(\bar{\mu})=-f(\mu)$ for all $\mu\in[0,iq]$ (an odd
extension of $f(\mu)$ onto $[-iq,0]$). Then, as was shown in [38] the first
equation (3.88) for the DOS reduces to

$\pi H[\hat{f}](\xi):=\int_{-q}^{q}\frac{\hat{f}(y)\mathrm{d}y}{y-\xi}=1,$
(3.89)

where $\hat{f}(y)=f(iy)$, $\xi=\text{Im}\lambda$, and $H[\hat{f}]$ denotes the
finite Hilbert transform (FHT) of $\hat{f}$ over $[-q,q]$ [126], [127].
Inverting the FHT $H$ subject to the additional constraint $H[\hat{f}](0)=0$,
we obtain the DOS $f=f_{\rm c}(\lambda)$ for the bound state soliton
condensate

$f_{\rm
c}(\lambda)=\frac{-i\lambda}{\pi\sqrt{\lambda^{2}+q^{2}}},~{}~{}~{}\lambda\in(-iq,iq).$
(3.90)

One can observe that the DOS (3.90) coincides with the appropriately
normalised semi-classical distribution of discrete Zakharov-Shabat spectrum
for the potential $\psi\in\mathbb{R}$ in the form of a broad rectangular
barrier. Eq. (3.90) (up to a norming factor) is obtained as a derivative of
the corresponding Weyl’s law following from the Bohr-Sommerfeld quantisation
rule for the Zakharov-Shabat operator [84]. A numerical realisation of the
bound state soliton condensate is shown in Fig. 12. It is achieved by building
a dense (as dense as possible) ensemble of $100$ strongly interacting solitons
with the discrete spectrum eigenvalues distributed according to the Bohr-
Sommerfeld rule and random phases of the norming constants (see [26] for the
details of the effective numerical implementation of dense $N$-soliton
ensembles with $N$ large).

Figure 12: Numerical realisation of the bound state soliton condensate for the
focusing NLS equation. Shown is the central portion of a random $100$-soliton
ensemble.

It was shown in [25] that the bound state soliton condensate dynamics
reproduces with remarkable accuracy the statistical characteristics of
stationary integrable turbulence generated at the nonlinear stage of the
development of the noise-induced modulational instability [8], [10], a
fundamental and ubiquitous physical phenomenon in focusing nonlinear media
[128], [129], [6]. Furthermore, it was demonstrated in [27] that the nonlinear
wave field in the bound state soliton condensate exhibits a spontaneous
emergence of rogue waves, offering a new perspective on the dynamical and
statistical mechanisms underlying this phenomenon observed in water waves,
nonlinear optics and matter waves (see [118] and references therein).

The bound state soliton condensate DOS (3.90) is a particular solution of the
dispersion relations (3.88). Another explicit solution of the soliton
condensate dispersion relations (3.88) is available in the special case when
the spectral support $\Gamma$ of the DOS represents a circle or a circular arc
in the complex plane. This solution with nonzero spectral flux density
$v(\lambda)$ obtained in [38] corresponds to a dynamic (non-bound state)
soliton condensate.

In conclusion we note that the soliton condensate regime is also available for
the KdV soliton gas. Indeed, substitution of $\sigma=0$ in the KdV soliton gas
dispersion relations (3.37), (3.38) yields

$\int\limits_{0}^{1}\ln\left|\frac{\eta+\mu}{\eta-\mu}\right|f(\mu)\mathrm{d}\mu=\eta\,,\quad\int\limits_{0}^{1}\ln\left|\frac{\eta+\mu}{\eta-\mu}\right|v(\mu)\mathrm{d}\mu=4\eta^{3}\,.$
(3.91)

One can see that the soliton condensate dispersion relations (3.88) and (3.91)
represent particular cases of the general conditions (2.15) for singular
solutions of the soliton gas’ equation of state (2.13).

####  3.3.5 Special breather gases

In Section 3.3.1 we presented some properties of breather solutions to the
focusing NLS equation and indicated three special cases: the Akhmediev
breather (AB), the Kuznetsov-Ma (KM) breather and the Peregrine soliton (PS),
which are often considered as mathematical models for rogue waves in fluids
and nonlinear optical media. These breathers can be viewed as quasiparticles
of the special, ‘rogue wave breather gases’.

Figure 13: Adapted from [40]. Numerical synthesis of a generic breather gas
(left column (a), (e), (i)) and of three single-component breather gases: KM
gas (second column (b), (f), (j)), AB gas ( third column (c), (g), (k)) and PS
gas (fourth column (d), (h), (i)). The four breather gases are parametrised by
$N=50$ complex eigenvalues $\lambda_{n}$, see bottom row. The first row
(a)–(d) represents the space-time evolution of the breather gases, with the
second row (e)–(h) being an enlarged view of some restricted region of the
$(x-t)$ plane. The third row (i)–(l) represents the spectral portraits of each
breather gas with the vertical line between $0$ and $+i$ being the branch cut
(Stokes band) associated with the plane wave background. Each point in the
upper complex plane in (i), (j), (k), (l) represents a discrete eigenvalue in
the IST problem with non-zero boundary conditions. The eigenvalues
parametrising the single-component breather gases are densely placed in a
small square region which is centred around a point $\lambda_{0}$ of the
imaginary vertical axis and which is strongly enlarged in the insets shown in
(j), (k), (l). The individual breather ‘positions’ $x_{j}$ are uniformly
distributed in the range $[-1,1]$ for the generic breather gas (a) and for the
PS gas (d) while they are uniformly distributed in the range $[-32,32]$ for
the KM gas (b) and the AB gas (c), see [40] for details.

Assuming the Stokes band $\gamma_{0}=[-iq,iq]$, $q>0$ the AB and KM breather
gases are characterised by the DOS $f(\lambda)$ that is supported on the
appropriate intervals of the imaginary axis, $\lambda\in\Gamma^{+}$:
$\Gamma^{+}=[0,ip]$, $p<q$ for AB gas and $\Gamma^{+}=[ip,ib]$, $b>p>q$ for KM
gas. For PS gas the DOS is given by $f(\lambda)=w\delta(\lambda-iq)$, where
$w>0$ is the PS gas density. The three types of the rogue wave breather gases
were realised numerically in [40] by building a random ensemble of $N\sim 50$
breathers via the Darboux transform recursive scheme in high precision
arithmetic, see Fig. 13. The interaction properties of the constructed
breather gases were investigated by propagating through them a trial generic
breather (TW) and comparing the effective (mean) propagation velocity with the
predictions following from the equation of state (3.74). One of these
predictions is that the propagation through the PS breather gas is ballistic,
i.e. the PS breather gas does not alter the velocity $s_{0}(\lambda)$ (3.75)
of the trial breather due to vanishing of the logarithmic interaction kernel
$G(\lambda,\mu)$ (3.75) evaluated at $\mu=iq$ [38]. Other predictions,
confirmed by detailed quantitative comparisons with numerical simulations in
[40], are that the effective velocity of propagation $s(\lambda)$ of a trial
TW breather through a rogue wave gas is greater than $s_{0}$ for the KM gas
and less than $s_{0}$ for the AB gas.

##  4 Hydrodynamic reductions and integrability

In this section we review some general mathematical properties of the kinetic
equation for soliton gas studied in [41], [42]. For convenience we reproduce
here the generalised kinetic equation (2.10), (2.13) for
unidirectional/bidirectional isotropic soliton gas:

$f_{t}+(fs)_{x}=0,\quad
s(\lambda,x,t)=s_{0}(\lambda)+\int_{\Gamma}G(\lambda,\mu)f(\mu,x,t)[s(\lambda,x,t)-s(\mu,x,t)]\mathrm{d}\mu,$
(4.1)

where $f(\lambda,x,t),s(\lambda,x,t),s_{0}(\lambda)$ and $G(\lambda,\mu)$ are
real functions and the parameters $\lambda,\mu$ can be real or complex. For
unidirectional gas the spectral support $\Gamma$ of the DOS $f(\lambda,x,t)$
is an interval in ${\mathbb{R}}$. For bidirectional isotropic gas $\Gamma$ is
either a symmetric interval $[-a,a]\subset{\mathbb{R}}$ (cf. (2.26)) or a
Schwarz symmetric compact domain in ${\mathbb{C}}$ (1D or 2D, see (3.85) for
the formal correspondence between the two cases).

Consider a soliton gas composed of a finite ($M\in{\mathbb{N}}$) number of
distinct components, termed monochromatic, or cold, components modelled by the
DOS in the form of a linear combination of the Dirac delta-functions centred
at distinct spectral points $\zeta_{j}\in\Gamma$,

$f(\lambda,x,t)=\sum\limits_{j=1}^{M}w_{j}(x,t)\delta(\lambda-\zeta_{j}),$
(4.2)

where $w_{j}(x,t)>0$ are the components’ weights, and
$\\{\zeta_{j}\\}_{j=1}^{M}\subset\Gamma$, $(\zeta_{j}\neq\zeta_{k}\iff j\neq
k)$. Substitution of (4.2) into the kinetic equation (4.1) reduces it to a
system of hydrodynamic conservation laws

$(w_{i})_{t}+(w_{i}s_{i})_{x}=0,\qquad i=1,\dots,M\,,$ (4.3)

where the component densities $w_{i}(x,t)$ and the transport velocities
$s_{j}(x,t)\equiv s(\zeta_{j},x,t)$ are related algebraically:

$s_{j}=s_{0j}+\sum\limits_{m=1,m\neq j}^{M}G_{jm}w_{m}(s_{j}-s_{m}),\quad
j=1,2,\dots M.$ (4.4)

Here we used the notation

$s_{0j}\equiv s_{0}(\zeta_{j}),\quad G_{jm}\equiv G(\zeta_{j},\zeta_{m}),\quad
j\neq m.$ (4.5)

We note that, although we derived the hydrodynamic reduction (4.3), (4.4) in
the context of unidirectional/isotropic soliton gas, it can be readily
generalised to the bidirectional anisotropic case, see [37]. In the latter
case the ansatz (4.2) is introduced separately for the slow and fast soliton
branches so that the pair of distributions

$f_{-}(\lambda,x,t)=\sum_{i=1}^{M_{-}}w_{i}(x,t)\delta(\lambda-\zeta_{i}),\quad
f_{+}(\lambda,x,t)=\sum_{i=M_{-}+1}^{M_{-}+M_{+}}w_{i}(x,t)\delta(\lambda-\zeta_{i}),$
(4.6)

is transformed into a $M$-dimensional vector
${\boldsymbol{w}}=(w_{1},\dots,w_{M})$, where $M=(M_{-}+M_{+})$.

For $M=2$ system (4.4) can be solved to give explicit expressions for
$s_{1,2}(w_{1},w_{2})$:

$s_{1}=s_{01}+\frac{G_{12}w_{2}(s_{01}-s_{02})}{1-(G_{12}w_{2}+G_{21}w_{1})},\quad
s_{2}=s_{02}-\frac{G_{21}w_{1}(s_{01}-s_{02})}{1-(G_{12}w_{2}+G_{21}w_{1})}.$
(4.7)

An important remark is in order on the meaning of the delta-function ansatz
(4.2) for the DOS $f(\lambda)$. As a matter of fact, the representation (4.2)
is a mathematical idealisation, which has a formal sense in the context of the
integral equation of state (4.1), but cannot be applied to the original
dispersion relations where it appears in both the integral and the secular
terms (cf. (3.37), (3.38) for the KdV equation). In a physically realistic
description the delta-functions in (4.2) should be replaced by some narrow
distributions around the spectral points $\zeta_{j}$, i.e. we first take the
thermodynamic limit $N\to\infty$ and then allow the distributions to become
sharply peaked. As a result, the non-condensate condition $\sigma>0$ in the
dispersion relation for the DOS (equation (3.37) for KdV or equation (3.68)
for focusing NLS) would impose a constraint
$\int_{\Gamma}G(\lambda,\mu)f(\lambda)\mathrm{d}\mu<1$ on $f(\lambda)$ which,
among other things, implies that the denominators in (4.7) must be positive.
It is interesting to note that a variant of hydrodynamic system (4.3), (4.7)
has been recently derived in [130] in the context of $T\bar{T}$-deformed
conformal field theories out of equilibrium.

We now assume that the integral kernel $G(\lambda,\mu)$ in the equation of
state (4.1) can be represented as

$G(\lambda,\mu)=a(\mu)\epsilon(\lambda,\mu),\ \ \text{where}\ \
\epsilon(\lambda,\mu)=\epsilon(\mu,\lambda)>0,\ \ \lambda\neq\mu$ (4.8)

for some non-singular real function $a(\mu)\neq 0$. This structure of the
soliton interaction kernel is typical for integrable dispersive hydrodynamics
and is indeed shared by the equations of state of the KdV, defocusing and
focusing NLS unidirectional/isotropic soliton gases (see (3.44), (2.26),
(2.34)). E.g. for KdV we have $a(\mu)=\mu$,
$\epsilon(\lambda,\mu)=\frac{1}{\lambda\mu}\ln\left|\frac{\lambda+\mu}{\lambda-\mu}\right|$,
see (2.11).

We then have

$G_{jk}=a_{k}\epsilon_{jk},\quad\epsilon_{jk}=\epsilon_{kj},\ j\neq k,$ (4.9)

where $a_{k}=a(\zeta_{k})$, $\epsilon_{jk}=\epsilon(\zeta_{j},\zeta_{k})$.
Solving equations (4.7) for $w_{1,2}$ gives

$a_{1}w_{1}=\frac{1}{\epsilon_{12}}\frac{s_{2}-s_{02}}{s_{2}-s_{1}},\quad
a_{2}w_{2}=\frac{1}{\epsilon_{12}}\frac{s_{1}-s_{01}}{s_{1}-s_{2}}.$ (4.10)

Substituting (4.10) in (4.3) for $M=2$ we obtain a diagonal system [36]:

$\frac{\partial s_{1}}{\partial t}+s_{2}\frac{\partial s_{1}}{\partial
x}=0,\qquad\frac{\partial s_{2}}{\partial t}+s_{1}\frac{\partial
s_{2}}{\partial x}=0$ (4.11)

with $s_{1,2}$ being the Riemann invariants. One can see that system (4.11) is
linearly degenerate because its characteristic velocities do not depend on the
corresponding Riemann invariants. System (4.11) represents the diagonal form
of the so-called Chaplygin gas equations, the system of isentropic gas
dynamics with the equation of state $p=-{A}/{\rho}$, where $p$ is the
pressure, $\rho$ is the gas density and $A>0$ is a constant. The Chaplygin gas
equations occur in certain theories of cosmology (see e.g. [131]) and is also
equivalent to the 1D Born-Infeld equation [132], [33] arising in nonlinear
electromagnetic field theory.

As any two-component quasilinear hyperbolic system, system (4.11) is
integrable (linearisable) via the classical hodograph transform. However, for
any $M\geq 3$ integrability of the original system (4.3), (4.4) is no longer
obvious. As a matter of fact, a $M$-component hydrodynamic type system is
generally not integrable for $M\geq 3$. Below we present the theorem, proved
in [41], which summarises the mathematical properties of the hydrodynamic
reductions (4.3), (4.4) subject to the structure (4.9) of the interaction
matrix $G_{jk}$.

Theorem [41].  $M$-component reductions (4.3), (4.4) of the generalized
kinetic equation (4.1) with the interaction kernel satisfying (4.8) are
hyperbolic, linearly degenerate integrable hydrodynamic type systems for any
$M\in\mathbb{N}$.

Specifically, it was shown in [41] that the following fundamental properties
are satisfied for the system (4.3), (4.4):

(i) Riemann invariants.

Let $u_{j}=a_{j}w_{j}$, where the symmetrising coefficients $a_{j}$ are
defined by (4.8), (4.9). Then the invertible point transformation
${\boldsymbol{w}}\to{\boldsymbol{r}}$ specified by

$r_{k}=-\frac{1}{u_{j}}\left(1+\sum_{m\neq k}u_{m}\epsilon_{mk}\right),\
k=1,2,...,M,$ (4.12)

reduces the system (4.3), (4.4) to the diagonal (Riemann invariant) form

$\frac{\partial r_{i}}{\partial t}+V_{i}({\boldsymbol{r}})\frac{\partial
r_{i}}{\partial x}=0,\quad i=1,2,\dots,M,$ (4.13)

where the characteristic velocities $V_{i}$ are expressed in terms of the
Riemann invariants $r_{1},r_{2},\dots,r_{N}$ as

$V_{i}({\boldsymbol{r}})=s_{i}(w_{1}({\boldsymbol{r}}),w_{2}({\boldsymbol{r}}),\dots,w_{M}({\boldsymbol{r}})),\quad
i=1,2,\dots,M.$ (4.14)

Here $s_{i}(w_{1},\dots,w_{M})$ are solutions of the system (4.4),

(ii) Linear degeneracy.

The characteristic velocities $V_{j}$ of the diagonal system (4.13) satisfy

${\partial_{j}V_{j}}=0,\quad
j=1,2,\dots,M,\quad\text{where}\quad\partial_{j}\equiv\partial/\partial
r_{j},$ (4.15)

so that system (4.3), (4.4) is linearly degenerate in the Lax sense [133].

(iii) Semi-Hamiltonian property.

In addition to (4.15) the characteristic velocities $V_{i}({\boldsymbol{r}})$
satisfy the overdetermined system

$\partial_{j}\frac{\partial_{k}V_{i}}{V_{k}-V_{i}}=\partial_{k}\frac{\partial_{j}V_{i}}{V_{j}-V_{i}}\text{,
\ \ \ }i\neq j\neq k$ (4.16)

for each three distinct characteristic velocities
($\partial_{k}\equiv\partial/\partial r_{k}$). Diagonal hydrodynamic type
systems satisfying (4.16) are called semi-Hamiltonian [66].

(iv) Generalised Hodograph solution.

A semi-Hamiltonian hydrodynamic type system can be locally integrated via the
generalised hodograph transform due to Tsarev [66], [134]. Its general local
solution for $\partial_{x}r_{i}\neq 0$, $i=1,\dots,M$ is given by the formula

$x-V_{i}({\boldsymbol{r}})t=W_{i}({\boldsymbol{r}})\,,\quad i=1,2,\dots,M,$
(4.17)

where functions $W_{i}({\boldsymbol{r}})$ are found from the linear system of
PDEs:

$\frac{\partial_{i}W_{j}}{W_{i}-W_{j}}=\frac{\partial_{i}V_{j}}{V_{i}-V_{j}}\,,\quad
i,j=1,\dots,N,\quad i\neq j.$ (4.18)

Properties of linearly degenerate semi-Hamiltonian (integrable) hydrodynamic
type systems have been studied in [135], [136]. Using the results of [135],
[136] it was shown in [41] that the general local solution of the hydrodynamic
reductions (4.3), (4.4) of the kinetic equation can be represented in the form

$x-s_{0i}t=\overset{r_{i}}{\int}{\xi\phi_{i}(\xi)}\mathrm{d}\xi+\underset{m\neq
i}{\sum}\epsilon_{im}\overset{r_{m}}{\int}{\phi_{m}(\xi)}\mathrm{d}\xi\,,\quad
i=1,2,\dots,M,$ (4.19)

where $\phi_{1}(\xi),\dots,\phi_{M}(\xi)$ are arbitrary functions. Some
interesting particular solutions (self-similar, quasiperiodic) following from
(4.19) can be found in [41]. The Hamiltonian structure of the hydrodynamic
reductions (4.3), (4.4) was investigated in [43]. The generalised hydrodynamic
reductions arising if one allows the discrete spectral points $\zeta_{j}$ in
(4.2) be functions of $(x,t)$ were studied in [42].

In conclusion of this section we note that, somewhat counter-intuitevely, the
above results on the hyperbolic structure and integrability of hydrodynamic
reductions of the kinetic equation for soliton gas apply to the multi-
component soliton gas of the focusing NLS equation, whose Whitham modulation
system is known to be elliptic for a generic set of modulation parameters and
for any genus [137], [122]. This apparent paradox is resolved by noticing that
the Whitham system for the focusing NLS equation exhibits real eigenvalues
(characteristic speeds) in the soliton limit. E.g. for the genus 1 case, the
two pairs of complex conjugate eigenvalues of the modulation matrix collapse
into a single, quadruply degenerate real eigenvalue, corresponding to the
velocity of the fundamental soliton, see e.g. [63].

##  5 Riemann problem for soliton gas

Having obtained the hydrodynamic description of multi-component soliton gas it
is natural to consider the Riemann problem that plays the fundamental role in
classical gas and fluid dynamics [138]. The classical Riemann problem consists
in finding solution to a system of hyperbolic conservation laws subject to
piecewise constant initial conditions exhibiting discontinuity at $x=0$. The
distribution solution of the Riemann problem generally consists of a
combination of constant states, simple (rarefaction) waves and strong
discontinuities (shocks or contact discontinuities) [133]. The discontinuous
solutions satisfy the Rankine-Hugoniot jump conditions. If a hyperbolic system
is linearly degenerate it does not support simple waves and classical shocks,
and the solution of Riemann problem contains only constant states and contact
discontinuities [139]. Here, following [75], [37], we present such solutions
for different types of soliton gases.

###  5.1 General solution

We consider the hyperbolic system (4.3),

$(w_{i})_{t}+(s_{i}(\boldsymbol{w})w_{i})_{x}=0,\quad i=1,\dots,n$ (5.1)

subject to the initial data corresponding to two $n$-component unidirectional
or bidirectional soliton gases prepared in the respective uniform states
$\boldsymbol{w}^{\rm L}\in\mathbb{R}^{n}$ and $\boldsymbol{w}^{\rm
R}\in\mathbb{R}^{n}$, that are initially separated:

$\boldsymbol{w}(x,0)=\begin{cases}\boldsymbol{w}^{\rm L},&\text{if }x<0,\\\
\boldsymbol{w}^{\rm R},&\text{if }x\geq 0.\end{cases}$ (5.2)

We emphasize here that the Riemann problem (5.1), (5.2) is formulated for the
kinetic equation (via the substitution (4.2)) but not for the original
dispersive hydrodynamic system (1.1) or (1.2). For this reason we shall call
it the spectral Riemann problem as it essentially describes the spatiotemporal
evolution of the spectral DOS of the soliton gas. The spectral Riemann problem
for the KdV and focusing NLS two-component soliton gases ($n=2$) was
investigated in [36], [75], [38] and for rather general isotropic and
anisotropic dispersive hydrodynamic soliton gases with an arbitrary number of
spectral components in [37]. The Riemann problem of this type has also been
considered in the context of generalised hydrodynamics [55], [56], [140],
[141], [142], [143].

The spectral Riemann problem (5.1), (5.2) corresponds to the dispersive
hydrodynamic Riemann problem that can be viewed as a soliton gas shock tube
problem, an analog of the shock tube problem of classical gas dynamics [138],
[33]. The shock tube problem represents a good benchmark for the spectral
kinetic theory of soliton gas where one can investigate both overtaking and
head-on collisions by choosing the appropriate components of solitonic spectra
and then comparing the predictions of the kinetic theory with direct numerical
simulations of dispersive hydrodynamics.

Following [75], [37] we consider the spectral Riemann problem (analytically)
and the corresponding soliton gas shock tube problem (numerically) for
$n$-component unidirectional and bidirectional soliton gases. Due to the scale
invariance of the problem (the kinetic equation (5.1) and the initial
condition (5.2) are both invariant with respect to the scaling transformation
$x\to Cx$, $t\to Ct$), the spectral solution is a self-similar distribution
${\boldsymbol{w}}(x/t)$. Because of the linear degeneracy (4.15) of the
hydrodynamic system (5.1) the only admissible similarity solutions are
constant states separated by propagating contact discontinuities, cf. for
instance [139]. Discontinuous, weak, solutions are physically acceptable here
since the kinetic equation describes the conservation of the number of
solitons within any given spectral interval, and the Rankine-Hugoniot type
conditions can be imposed to ensure the conservation of the number of solitons
across the discontinuities. As a result, the solution of the Riemann problem
(4.3), (5.2) for each component $w_{i}(x,t)$ is composed of $n+1$ constant
states, or plateaus, separated by $M$ discontinuities (see e.g. [133]):

$w_{i}(x,t)=\begin{cases}w_{i}^{1}=w_{i}^{\rm L},&x/t<z_{1},\\\ \dots\\\
w_{i}^{j},&z_{j-1}\leq x/t<z_{j}\\\ \dots\\\ w_{i}^{n+1}=w_{i}^{\rm
R},&z_{n}\leq x/t,\end{cases}\quad$ (5.3)

where the lower index $i$ indicates the $i$-th component of the vector
$\boldsymbol{w}$, and the upper index $j$ is the index of the plateau. For
clarity we labeled the superscripts $j=1$ as “L” (left boundary condition) and
$j=n+1$ as “R” (right boundary condition). The contact discontinuities
propagate at the characteristic velocities [133]:

$z_{j}=s_{j}(w_{1}^{j},\dots,w_{n}^{j})=s_{j}(w_{1}^{j+1},\dots,w_{n}^{j+1}),$
(5.4)

where the plateaus’ values $w_{i}^{j}$ are found from the Rankine-Hugoniot
jump conditions [33]:

$\displaystyle-
z_{j}\Big{[}w_{i}^{j+1}-w_{i}^{j}\Big{]}+\Big{[}s_{i}(w_{1}^{j+1},\dots,w_{n}^{j+1})w_{i}^{j+1}-s_{i}(w_{1}^{j},\dots,w_{n}^{j})w_{i}^{j}\Big{]}=0,$
(5.5)

where $i,j=1\dots n$. The Rankine-Hugoniot conditions with $i=j$ are trivially
satisfied by the definition of contact discontinuity (5.4). We also note that
conditions (5.4) are compatible with (5.5) for linearly degenerate systems of
conservation laws.

Note that, if the solitons were not interacting, the initial step distribution
$w_{i}(x,0)$ for the component $\lambda=\zeta_{i}$ would have propagated at
the free soliton velocity $s_{0i}$:

$w^{\rm free}_{i}(x,t)=\begin{cases}w_{i}^{\rm L},&x/t<s_{0i},\\\ w_{i}^{\rm
R},&s_{0i}\leq x/t,\end{cases}\quad i=1\dots n,$ (5.6)

which is very different compared to the solution (5.3).

###  5.2 Riemann problem: Examples

KdV soliton gas

Figure 14: (adapted from [75]). Soliton gas shock tube problem: numerical
solution of the KdV equation for one typical realisation of the collision of
two cold soliton gases. The expanding interaction region is shown in blue. The
piece-wise constant solution for the total density of solitons $\beta$ (5.18)
is shown above the KdV solution plot.

We now consider the simplest concrete Riemann problem for soliton gas. Namely,
we shall study the collision of two single-component KdV soliton gases [75].
To be consistent with the notations of Sections 2.1 and 3.2 we shall be using
the spectral variable $\eta=\sqrt{-\lambda}$. Also we simplify the general
notation of the previous section to make the specific results more
transparent.

We consider the two-component ansatz (4.2) for the DOS,

$f(\eta;x,t)\ =\ w_{1}(x,t)\delta(\eta-\eta_{1})\ +\
w_{2}(x,t)\delta(\eta-\eta_{2})\,,$ (5.7)

where

$0<\eta_{2}<\eta_{1}<1\,,\quad w_{1,2}\ \geqslant\ 0\,.$ (5.8)

Substitution of (5.7) into the kinetic equation (2.10), (2.11) for KdV soliton
gas yields the system of two conservation laws

$\partial_{t}w_{1}\ +\ \partial_{x}(w_{1}s_{1})\ =\
0\,,\qquad\partial_{t}w_{2}\ +\ \partial_{x}(w_{2}s_{2})\ =\ 0,$ (5.9)

where the transport velocities $s_{1,2}\equiv s(\eta_{1,2},x,t)$ are given by
(cf. (4.7))

$s_{1}=4\eta_{1}^{2}+\frac{4(\eta_{1}^{2}-\eta_{2}^{2})\alpha_{2}w_{2}}{1-\alpha_{1}w_{1}-\alpha_{2}w_{2}},\
\
s_{2}=4\eta_{2}^{2}-\frac{4(\eta_{1}^{2}-\eta_{2}^{2})\alpha_{1}w_{1}}{1-\alpha_{1}w_{1}-\alpha_{2}w_{2}},$
(5.10)

where

$\alpha_{1,2}\ =\ \frac{1}{\eta_{1,2}}\;\ln\left|\frac{\eta_{1}\ +\
\eta_{2}}{\eta_{1}\ -\ \eta_{2}}\right|\ >\ 0\,.$ (5.11)

It is assumed that $w_{1}\alpha_{1}+w_{2}\alpha_{2}<1$ (see the remark after
eq. (4.7)).

We now consider the system (5.9), (5.10) subject to the Riemann data (5.2)
with ${\boldsymbol{w}}^{L}=(w_{10},0)$, ${\boldsymbol{w}}^{R}=(0,w_{20})$,
i.e.

$\left\\{\begin{array}[]{ll}w_{1}(x,0)=w_{10},\ \
w_{2}(x,0)=0\,,&x<0,\\\\[6.0pt] w_{2}(x,0)=w_{20},\ \ w_{1}(x,0)=0,&x\geq
0,\end{array}\right.$ (5.12)

where $w_{10},w_{20}>0$ are some constants satisfying
$w_{i0}\leqslant\eta_{i}/3$ (see (2.43)). Note that the initial velocities of
the colliding soliton gases are fully determined, via Eq. (5.10), by the
density distribution (5.12). An example of one realisation of direct numerical
simulation of the collision of two single-component KdV soliton gases modelled
by the spectral Riemann problem (5.9), (5.12) is shown in Fig. 14. Details of
the numerical implementation of the KdV soliton gas in the simulations can be
found in [75].

The required solution for each component $w_{i}$ represents a combination of
three constant states (plateaus) separated by two contact discontinuities
given by equations (5.3) - (5.5) for $n=2$. In the notations of this section,

$w_{i}(x,t)=\begin{cases}w_{i0}\,\delta_{i,1},&x/t<c^{-},\\\ w_{ic},&c^{-}\leq
x/t<c^{+},\\\ w_{i0}\,\delta_{i,2},&c^{+}\leq x/t,\end{cases}\quad$ (5.13)

where $i\in\\{1,2\\}$ and $\delta_{ij}$ is the Kronecker delta. The values
$w_{1c}$ and $w_{2c}$ of the component densities in the middle plateau region,
where the interaction of soliton gases occurs, together with the velocities
$c^{\pm}$ of the contact discontinuities, are found from the Rankine–Hugoniot
jump conditions,

$\begin{array}[]{l}-c^{-}(w_{10}-w_{1c})\ +\ (w_{10}s_{10}-w_{1c}s_{1c})\ =\
0\,,\\\ -c^{-}(0-w_{2c})\ +\ (0-w_{2c}s_{2c})\ =\ 0\,;\end{array}$ (5.14)
$\begin{array}[]{l}-c^{+}(w_{1c}-0)\ +\ (w_{1c}s_{1c}-0)\ =\ 0\,,\\\
-c^{+}(w_{2c}-w_{20})\ +\ (w_{2c}s_{2c}-w_{20}s_{20})\ =\ 0\,,\end{array}$
(5.15)

where the velocities $s_{1c}=s_{1}(w_{1c},w_{2c})$ and
$s_{2c}=s_{2}(w_{1c},w_{2c})$ are determined by relations (5.10).

Figure 15: (adapted from [75]). Comparisons for the shock tube problem: the
speeds $c^{\pm}$ of the edges of the interaction region for two different sets
of parameters. Figure 16: (adapted from [75]). (a) The equilibrium total
density $\beta_{c}=w_{1c}+w_{2c}$ in the interaction region as a function of
the density $w_{10}$ of the ‘left’ gas. The density of the ‘right’ gas
$w_{20}=0.047$; (b) the moments $\langle u\rangle_{c},\langle
u^{2}\rangle_{c}$ of the random wave field in the interaction region.

Solving (5.14), (5.15) we obtain:

$w_{1c}\ =\
\frac{w_{10}(1-\alpha_{2}w_{20})}{1-\alpha_{1}\alpha_{2}w_{10}w_{20}}\,,\quad
w_{2c}\ =\
\frac{w_{20}(1-\alpha_{1}w_{10})}{1-\alpha_{1}\alpha_{2}w_{10}w_{20}}\,.$
(5.16) $c^{-}\ =\ 4\eta_{2}^{2}\ -\
\frac{4(\eta_{1}^{2}-\eta_{2}^{2})\alpha_{1}w_{1c}}{1-\alpha_{1}w_{1c}-\alpha_{2}w_{2c}},\quad
c^{+}\ =\ 4\eta_{1}^{2}\ +\
\frac{4(\eta_{1}^{2}-\eta_{2}^{2})\alpha_{2}w_{2c}}{1-\alpha_{1}w_{1c}-\alpha_{2}w_{2c}}\,.$
(5.17)

One can see that velocities $c^{\pm}$ of the edges of the expansing
interaction region coincide with the characteristic velocities (5.10) which is
the basic property of contact discontinuities, cf. (5.4).

The total density of solitons
$\beta=\int_{0}^{1}f(\eta)\mathrm{d}\eta=w_{1}+w_{2}$ satisfies

$\beta(x,t)=\left\\{\begin{array}[]{ll}w_{10},&x\ <\ c^{-}t,\\\\[6.0pt]
w_{1c}+w_{2c},&c^{-}t\ \leq\ x\ <\ c^{+}t,\\\\[6.0pt] w_{20},&x\ \geq\
c^{+}t.\end{array}\right.$ (5.18)

The piece-wise constant solution for $\beta(x,t)$ is shown in Fig. 14 above
the soliton gas plot. It is not difficult to show that the density of the two-
component soliton gas in the interaction region,
$\beta_{c}=w_{1c}+w_{2c}>w_{10},w_{20}$ but $\beta_{c}<w_{10}+w_{20}$. We
emphasize that, although the soliton gas is initially prepared in a rarefied
regime where solitons are spatially well-separated, the total density of
solitons increases in the interaction region, and a more dense soliton gas is
formed for which solitons exhibit significant overlap.

The expanding interaction region in the soliton gas Riemann problem can be
viewed as a stochastic counterpart of the traditional, coherent dispersive
shock wave forming due to a dispersive regularisation of the Riemann initial
data in the KdV equation [144], [18].

Upon using the ansatz (5.7) the values of the two first moments (3.49) in the
interaction region assume the form,

$\langle u\rangle_{c}\ =\ 4(\eta_{1}w_{1c}\ +\
\eta_{2}w_{2c}),\quad\langle{u^{2}}\rangle_{c}\ =\
\frac{16}{3}(\eta_{1}^{3}w_{1c}\ +\ \eta_{2}^{3}w_{2c})\,,$ (5.19)

where $w_{1c}$ and $w_{2c}$ are determined in terms of the initial data by
formulae (5.16).

Comparisons of the theoretical results (5.16), (5.17), (5.19) for the soliton
gas physical ‘observables’ $c^{\pm},\beta,\langle u\rangle,\langle
u^{2}\rangle$ with direct numerical simulations of the shock tube problem for
the KdV soliton gas are presented in Figs. 15, 16.

Solition gases for the defocusing and resonant NLS equations

The spectral Riemann problem (5.1) (5.2) for bidirectional soliton gases for
the defocusing and resonant NLS equations (see Section 2.2) was considered in
[37], where the spectral Riemann problem solution (5.3) – (5.5) was
constructed for the collisions of two- and three-component gas and comparisons
of some analytically obtained ‘observables’ (the speeds of contact
discontinuities and the spatial profiles of the mean density
$\langle\rho\rangle$) with direct numerical simulations of the soliton gas
collision have been made.

Figure 17: (Adapted from [37]). Spatiotemporal plots of the field $\rho(x,t)$
for one realization of the anisotropic soliton gas collision. Trajectories of
the solitons appear in solid lines. Red dash-dotted lines correspond to the
trajectories of the contact discontinuities: $x=z_{1}t$, $x=z_{2}t$, cf.
(5.4), and blue dashed lines to the free soliton trajectories: $x=s_{01}t$,
$x=s_{02}t$. (a) Overtaking collisions $(\zeta_{1},\zeta_{2})=(1.05,1.10)$,
cf. initial condition (i) in Table 1. (b) Head-on collisions
$(\zeta_{1},\zeta_{2})=(-1.05,1.10)$, cf. initial condition (ii) in Table 1.

Figure 18: (Adapted from [37]). Comparison between the ensemble average
$\langle\rho(x,t)\rangle$ obtained by averaging over 100 realisations of the
numerical solution of the three-component soliton gas shock tube problem
(solid line) for bidirectional dispersive hydrodynamics and the analytical
solution obtained via the corresponding spectral Riemann problem (dash-dotted
line). The dashed line corresponds to the average field density in the
artificial soliton gas composed of non-interacting solitons with the spectral
distribution given by (5.6). (a) Defocusing NLS soliton gas at $t=2000$, case
(iii) Table 1. (b) Resonant NLS soliton gas at $t=5000$, case (iv) in Table 1.

Fig. 17 displays the numerically obtained $x-t$ diagrams extracted from a
single realisation of the soliton gas collision and illustrating the
difference between overtaking and head-on interactions in the anisotropic
soliton gas of the resonant NLS equation.

In Fig. 18 the comparisons are shown for the profiles of ensemble average
$\langle\rho\rangle$ of the wave field $\rho(x,t)$ associated with the
spectral Riemann problem solution for the collision of three-component soliton
gases for the defocusing and resonant NLS equations. The spectral data used
for the numerical realisation of the soliton gases are collected in Table 1.
The numerical evaluation of the ensemble average $\langle\rho\rangle$ involves
averaging over 100 realisations. As expected, the solution is composed of 4
plateaus, where regions II and III contain at least two distinct soliton
components and are region of interactions. The comparison shows an excellent
agreement confirming the efficacy of the developed spectral kinetic theory.
Note that in Fig. 18 the discontinuities in $\langle\rho(x,t)\rangle$ have a
finite slope which is an artefact of the statistical averaging procedure
detailed in [37].

|  spectral parameters |  left boundary condition |  right boundary condition
---|---|---|---
(i) |  $(\zeta_{1}=1.05,\zeta_{2}\in[1.06,1.10])$ |  $\boldsymbol{w}^{\rm L}=(0,6.6)\times 10^{-2}$ |  $\boldsymbol{w}^{\rm R}=(6.6,0)\times 10^{-2}$
(ii) |  $(\zeta_{1}=-1.05,\zeta_{2}\in[1.06,1.1])$ |  $\boldsymbol{w}^{\rm L}=(0,6.6)\times 10^{-2}$ |  $\boldsymbol{w}^{\rm R}=(6.6,0)\times 10^{-2}$
(iii) |  $(\zeta_{1},\zeta_{2},\zeta_{3})=(-0.2,0.1,0.4)$ |  $\boldsymbol{w}^{\rm L}=(2.5,0,7.5)\times 10^{-2}$ |  $\boldsymbol{w}^{\rm R}=(5,5,0)\times 10^{-2}$
(iv) |  $(\zeta_{1},\zeta_{2},\zeta_{3})=(-1.1,1.05,1.1)$ |  $\boldsymbol{w}^{\rm L}=(1.6,0,5)\times 10^{-2}$ |  $\boldsymbol{w}^{\rm R}=(3.3,3.3,0)\times 10^{-2}$
Table 1: Spectral Riemann problem data used in the numerical simulations for
Figs. 17, 18.

##  6 Summary and Outlook

In this article we have reviewed the results on the spectral theory of soliton
gases obtained by the author and his collaborators over the last two decades,
although many of the developments are relatively recent. The theory is
inspired by Zakharov’s 1971 paper [28] where the ideas of the spectral kinetic
description of a weakly interacting, rarefied soliton gas for the KdV equation
were introduced for the first time. It has transpired later that the
perspective of dispersive hydrodynamics, particularly the Whitham theory of
modulated spectral finite-gap solutions provides an appealing mathematical
framework for the generalisation of Zakharov’s kinetic equation to the case of
strongly interacting, dense soliton gas [32], [36], [38]. It has also been
realised that soliton gas represents a ubiquitous physical phenomenon, a kind
of strongly nonlinear turbulent wave motion, that can observed in the
environmental conditions [21], [39] and realised in laboratory experiments
[23], [24]. The recently discovered connections of the soliton gas dynamics
and statistics with the topical areas of modulational instability and rogue
wave formation [25], [26], [10] have provided the soliton gas research with
further relevance.

One can envisage many important and interesting mathematical and physical
problems arising in connection with the developed spectral theory of soliton
gas. Probably the most pertinent one is the determination of the statistical
parameters of the integrable turbulence associated with a given spectral DOS
of a soliton gas, in particular, obtaining the probability density function,
the power spectrum and the autocorrelation function of the nonlinear wave
field. This would provide the theoretical explanation of the intriguing
statistical properties of integrable turbulence observed in the numerical
simulations and physical experiments [8], [10], [25], [27], [145]. Further,
the spectral hydrodynamics/kinetics of non-equilibrium soliton gas could
provide the means for the manipulation of the integrable turbulence
statistics, and therefore, control of the rogue wave emergence in physical
systems described at leading order by integrable equations.

Another pertinent question is the generation of soliton gas out of ‘non-gas’
initial or boundary conditions. This problem is important from both the
theoretical and experimental points of view. One possibility is to use the
spontaneous ‘soliton fission’ mechanism [64], [31] inspired by the original
Zabusky-Kruskal work [12] and employed in the experiments on the creation of a
shallow water bidirectional anisotropic soliton gas in [23]. In modulationally
unstable media a semiclassical scenario of the transition to integrable
soliton turbulence via a chain of topological bifurcations of the finite-gap
spectra was theoretically proposed in [63] and to some extent realised in an
optics experiment reported in [54]. Of particular relevance is the IST based
approach to the controlled (as opposed to the spontaneous) experimental
realisation of soliton gas developed in [24]. This approach enables the
generation of a soliton gas with a prescribed DOS, which is crucial for the
verification and utilisation of the spectral kinetic theory.

Yet another prospective area of the further development of the soliton gas
theory is the interaction of soliton gas with external potentials, either
‘static’ as in the soliton gas in a trapped BECs [48], [146] or dynamic, as in
the recently proposed hydrodynamic soliton tunnelling framework [17], [147],
[110]. Related to this, the development of the theory of soliton gas in
perturbed integrable systems is of particular importance for applications
where the higher order, non-integrable corrections to the core integrable
dynamics are always present and generally result in the slow evolution of the
DOS, that is distinct from the spectral kinetic transport by the continuity
equation, see [24].

Finally, the recently emerged intriguing parallels between the spectral theory
of soliton gas and generalised hydrodynamics [57], [78] provide yet another
avenue for the novel cross-disciplinary research that could be of significant
benefit for both fields of research.

## Acknowledgments

This work was partially supported by EPSRC grant EP/R00515X/2.

I would like to express my gratitude to the late A.L. Krylov, who had
suggested this topic to me many years ago. I am grateful to M. Hoefer, A.
Kamchatnov, M. Pavlov, S. Randoux, P. Suret and A. Tovbis for the
collaboration and many stimulating discussions over the years. I would like to
thank T. Congy and G. Roberti for their more recent contributions and for the
help with preparing the figures. Special thanks to T. Bonnemain for reading
the manuscript and providing a number of helpful comments.

##  Appendix A Nonlinear dispersion relations for the finite-gap solutions of
the focusing NLS equation

We introduce the fundamental wavenumber and frequiency vectors:
${\boldsymbol{k}}=(k_{1},\dots,k_{N},\tilde{k}_{1},\dots,\tilde{k}_{N})$ and
${\boldsymbol{\omega}}=(\omega_{1},\dots,\omega_{N},\tilde{\omega}_{1},\dots,\tilde{\omega}_{N})$
respectively, whose components are defined as follows

$\displaystyle k_{j}$ $\displaystyle=$
$\displaystyle-\oint_{\alpha_{j}}\mathrm{d}p,\quad\omega_{j}=\oint_{\alpha_{j}}\mathrm{d}q,\quad
j=1,\dots,N,$ (A.1) $\displaystyle\tilde{k}_{j}$ $\displaystyle=$
$\displaystyle\oint_{\beta_{j}}\mathrm{d}p,\quad\tilde{\omega}_{j}=-\oint_{\beta_{j}}\mathrm{d}q,\quad
j=1,\dots,N.$ (A.2)

Here $\mathrm{d}p(z)$ and $\mathrm{d}q(z)$ are the meromorphic quasimomentum
and quasienergy differentials with the only poles at $z=\infty$ on both sheets
of the Riemann surface (3.57) defined by (see e.g. [122], [124])

$\mathrm{d}p=1+\mathcal{O}(z^{-2}),\qquad\mathrm{d}q=4z+\mathcal{O}(z^{-2})$
(A.3)

near $z=\infty$ on the main sheet respectively, and the normalisation
conditions requiring that all the periods of $\mathrm{d}p,\mathrm{d}q$ are
real (real normalised differentials). The homology basis $\alpha_{j}$,
$\beta_{j}$ is defined in Section 3.3.1, see Fig. 11. The signs of integrals
in (A.1), (A.2) will be opposite if we replace $j$ by $-j$. The signs of the
integrals in (A.1), (A.2) will be opposite if we replace $j$ by $-j$.

The wavenumbers and frequencies are symmetrically extended to negative indices
by $k_{-j}=k_{j},\omega_{-j}=\omega_{j}$, $j=1,\dots,N,$ and similar equations
for the ‘tilded’ quantities. They also satisfy the corresponding equations
(A.1), (A.2), but the signs of integrals in (A.1), (A.2) will be opposite when
we replace $j$ by $-j$.

The proof that $k_{j}$, $\tilde{k}_{j}$, $\omega_{j}$, $\tilde{\omega}_{j}$
defined by (A.1), (A.2) are indeed the wavenumbers and frequencies of the
finite-gap focusing NLS solution associated with the spectral surface
$\mathcal{R}$ of (3.57) can be found in [38]. It was then shown in [38] that
for the wavenumbers and frequencies (A.1), (A.2) associated with the finite-
gap focusing NLS solution the nonlinear dispersion relations are given by

$\displaystyle\tilde{k}_{j}+\sum_{|m|=1}^{N}k_{m}\oint_{\beta_{m}}{{P_{j}(\zeta)\mathrm{d}\zeta}\over{R(\zeta)}}=-\frac{1}{2}\oint_{\tilde{\gamma}}{{\zeta
P_{j}(\zeta)\mathrm{d}\zeta}\over{R(\zeta)}},$
$\displaystyle\tilde{\omega}_{j}+\sum_{|m|=1}^{N}\omega_{m}\oint_{\beta_{m}}{{P_{j}(\zeta)\mathrm{d}\zeta}\over{R(\zeta)}}=-\oint_{\tilde{\gamma}}{{\zeta^{2}P_{j}(\zeta)\mathrm{d}\zeta}\over{R(\zeta)}},$
$\displaystyle|j|=1,\dots,N,$ (A.4)

where $\tilde{\gamma}$ is a large clockwise oriented contour containing
$\Gamma$, and $P_{j}(z)$ and $w_{j}$ are defined in (3.62), (3.63) Taking real
and imaginary parts of (A.4) and using the residues in the right hand side, we
obtain separate systems for the solitonic components $k_{m}$, $\omega_{m}$ and
the carrier components $\tilde{k}_{m}$, $\tilde{\omega}_{m}$.

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