Soliton gas of the integrable Boussinesq equation and its generalised
hydrodynamics

T. Bonnemain*, B. Doyon

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

*<EMAIL_ADDRESS>
August 28, 2024

###### Abstract

Generalised hydrodynamics (GHD) is a recent and powerful framework to study
many-body integrable systems, quantum or classical, out of equilibrium. It has
been applied to several models, from the delta Bose gas to the XXZ spin chain,
the KdV soliton gas and many more. Yet it has only been applied to
(1+1)-dimensional systems and generalisation to higher dimensions of space is
non-trivial. We study the Boussinesq equation which, while generally
considered to be less physically relevant than the KdV equation, is
interesting as a stationary reduction of the (boosted) Kadomtsev-Petviashvili
(KP) equation, a prototypical and universal example of a nonlinear integrable
PDE in (2+1) dimensions. We follow a heuristic approach inspired by the
Thermodynamic Bethe Ansatz in order to construct the GHD of the Boussinesq
soliton gas. Such approach allows for a statistical mechanics interpretation
of the Boussinesq soliton gas that comes naturally with the GHD picture. This
is to be seen as a first step in the construction of the KP soliton gas,
yielding insight on some classes of solutions from which we may be able to
build an intuition on how to devise a more general theory. This also offers
another perspective on the construction of anisotropic bidirectional soliton
gases previously introduced phenomenologically by Congy et al (2021).

###### Contents

  1. 1 Introduction
  2. 2 The integrable Boussinesq equation
    1. 2.1 The “good”, the “bad” and the ill-posed
    2. 2.2 Integrability and conserved quantities
    3. 2.3 Soliton solutions
    4. 2.4 Illustration: one- and two-soliton solutions
    5. 2.5 Regularity conditions for the $N-$soliton solution
    6. 2.6 Bound states
  3. 3 Thermodynamics
    1. 3.1 Classical TBA
    2. 3.2 Conventional choice of the momentum function
    3. 3.3 Partition function
    4. 3.4 Large deviation principle and Yang-Yang equation
    5. 3.5 Thermodynamic quantities
  4. 4 Generalised hydrodynamics
    1. 4.1 Dressing operation
    2. 4.2 Effective velocity
    3. 4.3 Thermodynamic averages and correlation functions
    4. 4.4 Euler Hydrodynamics
  5. 5 Conclusion
  6. A GHD of the “bad” Boussinesq soliton gas

##  1 Introduction

The theory of soliton gases, first introduced twenty years ago by Gennady El
[1, 2], has been the subject of rapidly growing interest in the past few
years. This is in part due to the development of the notion of “integrable
turbulence” coined and popularised by Zakharov [3], and to that of generalised
hydrodynamics (GHD) independently introduced in both [4] and [5]. Those two
approaches, though formally different, have a common aim which consists in
investigating the statistical and emergent, hydrodynamical properties of
large-scale, many-body, integrable systems out of equilibrium. As it happens,
soliton gases are physically relevant mathematical objects that can be
naturally interpreted both in terms of a soliton-dominated integrable
turbulence and in terms of GHD.

The prime motivation behind the study of integrable turbulence and GHD lies in
the fact that integrable systems are universal, arising from the asymptotic
expansion of large classes of models, and are known to capture essential
properties of several physical systems [6]. Moreover, a key aspect of
integrable systems is that they feature an infinite number of conserved
quantities (mass, momentum, energy, and their higher-spin versions) that
constrain the dynamics and allow for exact solutions. Many techniques have
been developed to that end, the most notable of which are perhaps the inverse
scattering transform (IST) for classical field theories [7, 8, 9], and the
Bethe Ansatz for quantum systems [10, 11, 12]. But the inherent complexity of
many natural or experimentally observed phenomena, for instance due to initial
or boundary conditions [13, 14, 15], often requires a statistical description
which is beyond well-established mathematical techniques from the theory of
integrable systems.

In particular strongly nonlinear random waves described by one-dimensional
integrable systems, such as the Korteweg–de Vries (KdV) and the nonlinear
Schrödinger (NLS) equations, have recently attracted significant attention
from theoreticians [16, 17, 18, 19, 20] and experimentalists [21, 22, 23, 24,
25, 26, 27] alike. The theory of integrable turbulence then concerns itself
with the statistical properties of the random wave field, for example in terms
of probability density function, power spectrum and correlations, as discussed
in Zakharov’s seminal paper [3].

But those equations notably also feature solutions involving solitary waves
called solitons that exhibit particle-like properties, such as elastic,
pairwise interactions which only result in a simple position shift.
Interacting solitons can then form large, complex and irregular statistical
ensembles, first considered, again, by Zakharov [28]. This provides an
alternative interpretation for the integrable turbulence of a soliton-
dominated wave field in terms of a soliton gas. The focus is then put on the
collective dynamics and kinetics of solitons, which are treated as interacting
(quasi)particles characterised by their velocity (and/or amplitude) and the
associated distribution function.

It is highly non-trivial that such a simple picture of a gas of solitons as a
gas of particles should always hold, particularly for dense gases, as each
soliton continuously overlaps and interacts with many others. But the
structure of the kinetic equations for KdV and NLS soliton gases, rigorously
derived in [1] and [17] from a thermodynamic-type limit of spectral finite gap
solutions and their modulations [29], strongly supports this interpretation.
This in turn allowed for some phenomenological constructions, notably in the
case of bidirectionnal soliton gases [30], based on the so-called collision
rate ansatz. We refer the interested reader to the recent review [31] for an
in depths discussion of soliton gas theory.

Given the aforementioned quasiparticle picture, the soliton gas approach is in
many ways closer in spirit to GHD than it is to integrable turbulence. This
correspondence was noted early [32, 33] and discussed at length in [34] in the
context of the KdV equation. GHD was initially developed to tackle out-of-
equilibrium, integrable, quantum, many-body systems by combining the
(generalised) thermodynamic Bethe Ansatz (TBA) [35, 36, 37] and the
hydrodynamic principles [38] to account for the local relaxation of fluid
cells to generalised Gibbs ensembles (GGE) [39]. The need for such extensions
comes from the fact that, in the case of integrable systems, the infinite
number of conserved quantities is known to impede the process of
“thermalisation” or, at least, it does in the traditional sense (e.g. there is
no equipartition of energy). As it happens, integrable systems _do_
thermalise, yet this notion needs to be generalised: entropy maximisation
occurs, only it does with respect to _all_ the infinitely many conserved
quantities. This recent theory has quickly been applied to quantum gases,
chains and field theories [40, 41, 42], providing exact results regarding
their correlations and fluctuations [43, 44, 45, 46]. It was then extended to
describe classical integrable systems [47, 48, 49, 50, 51], to include higher
order terms in the hydrodynamic expansion like diffusion [52, 53, 54] or
dispersion [55], and to probe integrability breaking [56, 57, 58, 59, 60].
Notably, GHD and some of its extensions have been confirmed in experiments on
cold atomic gases [61, 62, 63]. GHD is a recent but very active field of
research and we refer the interested reader to the lecture notes and reviews
[64, 65, 66, 67, 68].

This paper aims to investigate the soliton gas associated to the integrable
Boussinesq equation

$u_{tt}-u_{xx}=-\alpha\left[6\left(u^{2}\right)_{xx}+u_{xxxx}\right]\;,$ (1)

where $\alpha=\pm 1$. The $\alpha=1$ case is typically refered to as the
“good” Boussinesq equation, while $\alpha=-1$ corresponds to the “bad” one, a
terminology that can be traced back to McKean [69].

It is important to stress that this equation is generally considered to be
less physically relevant than the KdV equation [7]. Although it was
historically derived as a model for propagation of waves in shallow water, it
has since been noted that it is in fact not a valid model for shallow water
hydrodynamics [70]. Moreover it has been argued that the only correct long
shallow water wave model is the decoupled left and right running KdV equations
[71], to which the Boussinesq equation reduces in the (relevant) small
amplitude, weakly nonlinear regime [72, 73].

However, there are several meaningful incentives that motivate our study of
the Boussinesq equation. From the point of view of physics, it arises
naturally (for appropriate initial conditions) as the continuum limit of the
Fermi-Pasta-Ulam-Tsingou chain [74, 73], and characterises stratified fluids
at near (non-semisimple) double criticality [75]. It is arguably more
significant from the point of view of biology, as a simple generalisation of
the Boussinesq equation – seemingly close to integrability, and still
featuring stable solitons that interact almost elastically – has been proposed
as a model for neural activity [76, 77, 78].

But perhaps the biggest incentive, going back to physics, comes from the fact
that it can be seen as a stationary reduction of the (boosted) KP equation [7,
79]

$\left[u_{t}+6(u^{2})_{x}+u_{xxx}\right]_{x}+\alpha u_{yy}=0\;.$ (2)

The case $\alpha=1$ is known as the KP2 equation, associated with the “good”
Boussinesq equation, while $\alpha=-1$ corresponds to KP1 equation, associate
to the “bad” one. The KP equation, in either version, has been used to model
several physical phenomena such as the propagation of ion-accoustic waves in
plasmas [80, 81], of shallow water waves [82, 83], or of a beam of light in
nonlinear media [84, 85]. It is a prototypical and universal example of a
nonlinear integrable PDE in (2+1) dimensions and can be seen as a natural
extension of the KdV equation to two spatial dimensions. As such, constructing
the soliton gas or the GHD of the KP equation would be significant, as both
theories have up to now been developed to deal with, and only applied to,
$(1+1)$-dimensional systems, and generalisation to higher dimensions of space
is non-trivial. We will undertake this endeavour in upcoming works [86]. The
present discussion on the Boussinesq equation is then to be seen as a first
step in the construction of the KP soliton gas, yielding insight on some
classes of solutions from which we may be able to build an intuition on how to
devise a more general theory.

In order to tackle the integrable Boussinesq equation, rather than the
historical and more rigorous approach to soliton gas developed by El [1], we
adopt here a more heuristic approach inspired by [48, 34] and based on the
TBA. This allows for a statistical mechanics interpretation of the Boussinesq
soliton gas that comes naturally with the GHD picture, and thermodynamic
quantities are then defined and evaluated (e.g. free energy, entropy,
temperature). This also offers another perspective on the construction of
anisotropic bidirectional soliton gases introduced phenomenologically in [30].

The paper is organised as follows. In Section 2 we review the Boussinesq
equation and its main properties. In Section 3 we construct its
thermodynamics: we write down integral equations for the exact free energy in
generalised Gibbs ensembles, which take the form of the classical TBA. In
Section 4 we phenomenologically construct its hydrodynamics: we derive its
kinetic equation. Finally, we conclude in Section 5.

##  2 The integrable Boussinesq equation

The Boussinesq equation (1) describes the propagation of waves in (1+1)
dimensional, weakly nonlinear, weakly dispersive systems, an example of which
is shallow water [87, 88]. This is a bidirectional model: contrary to waves
described by the KdV equation, those described by the Boussinesq equation can
move in either direction. This last point will have important consequences on
the GHD of the Boussinesq soliton gas as we shall see in Section 3. Finally,
as mentioned in the previous section, depending on the sign of the parameter
$\alpha$, the Boussinesq equation is refered to as “good” or “bad”. We shall
start our discussion by addressing this dichotomy.

###  2.1 The “good”, the “bad” and the ill-posed

The “bad” Boussinesq equation is sometimes also called the ill-posed
Boussinesq equation, because of its severe short-wave instability [89]. This
can be made clear by linearising the equation (1) about a constant background
$u_{0}>0$, yielding the linear PDE

$u_{tt}-u_{xx}=-\alpha\left[12u_{0}u_{xx}+u_{xxxx}\right]\;.$ (3)

This equation admits solutions in the form $\exp[\omega t+ikx]$ with
dispersion relation

$\omega^{2}=k^{2}\left[2\Omega_{\alpha}-\alpha k^{2}\right]\;,$ (4)

where $2\Omega_{\alpha}=12\alpha u_{0}-1$. In the case of the “good”
Boussinesq equation $\alpha=+1$, and one can see that perturbations of the
constant solution are stable for $k^{2}>\max[2\Omega_{+},0]$ and unstable
otherwise, with, for $\Omega_{+}>0$, a maximal growth rate $\omega=\Omega_{+}$
attained for $k=\sqrt{\Omega_{+}}$. However, if $\alpha=-1$, perturbations are
unstable to all modes $k^{2}>2|\Omega_{-}|$, with unbounded growth rate, since
$k$ is unbounded and $\omega$ grows monotonically with $k$. That means that
the slightest perturbation of the initial data would instantly and utterly
change the behaviour of the solution up to the smallest scales, and, as such,
solutions of the “bad” Boussinesq equation with arbitrary initial conditions
generically do not exist for positive time.

Fortunately, both for the “good” and “bad” Boussinesq equations, certain
classes of solutions remain stable at all time. In particular, that is the
case of the soliton solutions we will be considering in the remainder of this
paper. For the sake of simplicity and legibility we will now drop the
parameter $\alpha$ and focus exclusively on the “good” Boussinesq equation.
All the upcoming results can be straightforwardly extended to the “bad”
Boussinesq equation as is discussed in Appendix A.

###  2.2 Integrability and conserved quantities

In 1973, Zakharov showed that the Boussinesq equation (1) was integrable, and
solvable via IST [90], by introducing the Lax pair

$\left\\{\begin{aligned} &\hat{L}=i\left[\frac{{\rm d}^{3}}{{\rm
d}x^{3}}+u\frac{{\rm d}}{{\rm d}x}+\frac{{\rm d}}{{\rm
d}x}u\right]-\sqrt{\frac{4}{3}}\ w_{x}\\\
&\hat{A}=\sqrt{\frac{3}{4}}\frac{{\rm d}^{2}}{{\rm
d}x^{2}}+\sqrt{\frac{4}{3}}\ u\end{aligned}\right.\;,$ (5)

where the function $w(x,t)$ is defined by $u_{t}=w_{xx}$. Indeed, one can see
by direct substitution that the Lax equation $\hat{L}_{t}=i[\hat{L},\hat{A}]$
is equivalent to the original equation (1).

In the same paper, Zakharov also devised a recurrence formula to write an
infinite set of integrals of motion in involution. We shall not discuss this
recurrence of formula or how it came to be, however, as an illustration, we
provide here the expressions of the first four conserved charges:

$\begin{aligned} &Q_{1}=\int_{\mathbb{R}}{\rm d}x\ u\\\
&Q_{2}=\int_{\mathbb{R}}{\rm d}x\ w_{x}\\\ &Q_{3}=\int_{\mathbb{R}}{\rm d}x\
uw_{x}\\\ &Q_{4}=\int_{\mathbb{R}}{\rm
d}x\left[\frac{u^{2}}{2}+\frac{(w_{x})^{2}}{2}-2u^{3}+\frac{(u_{x})^{2}}{2}\right]\end{aligned}\;,$
(6)

which all commute with respect to the canonical Poisson bracket defined as the
bilinear map

$\\{F;G\\}=\int_{\mathbb{R}}\left(\frac{\delta F}{\delta u(x)}\frac{\delta
G}{\delta w(x)}-\frac{\delta F}{\delta w(x)}\frac{\delta G}{\delta
u(x)}\right)\ ,$ (7)

for real-analytic functionals in the sense of [9]. In particular, the fourth
charge $Q_{4}$ plays the role of a Hamiltonian, and one can alternatively
write the Boussinesq equation (1) in the Hamiltonian form

$\left\\{\begin{aligned} &u_{t}=w_{xx}=\\{Q_{4};u(x)\\}=-\frac{\delta
Q_{4}}{\delta w}\\\ &w_{t}=u-6u^{2}-u_{xx}=\\{Q_{4};w(x)\\}=\frac{\delta
Q_{4}}{\delta u}\end{aligned}\right.\;.$ (8)

Finally, given the definition of $w$, the current associated to the conserved
density $u$ (for $Q_{1}$) is $-w_{x}$, but we see that $w_{x}$ is also the
conserved density for $Q_{2}$. Thus $w_{x}$ is a self-conserved current, in
particular by defining

$J_{1}\equiv\int_{\mathbb{R}}{\rm d}t\ w_{x}\ ,$ (9)

one has

$\partial_{x}J_{1}=\partial_{t}J_{1}=0\;,$ (10)

if we assume that $u$, $w$ and their derivatives vanish at $|t|=\infty$ and/or
$|x|=\infty$, which we will be doing since we are only interested in
reflectionless (i.e., pure soliton) solutions [72]. The presence of a self-
conserved current justifies the use of the collision rate ansatz further down
the line [50, 91].

###  2.3 Soliton solutions

Although very powerful, the IST formalism is not best suited for what we aim
to do here. It will instead be more convenient to use Hirota’s bilinear method
[72] to construct the soliton solutions and extract their 2-body scattering
shift. Indeed, Hirota showed in 1973 that

$u(x,t)=\log\left[\tau(x,t)\right]_{xx}\;,$ (11)

is a solution of the Boussinesq equation (1) provided the $\tau-$function
solves

$\left[\left(\partial_{t}-\partial_{t^{\prime}}\right)^{2}-\left(\partial_{x}-\partial_{x^{\prime}}\right)^{2}-\left(\partial_{x}-\partial_{x^{\prime}}\right)^{4}\right]\tau(x,t)\tau(x^{\prime},t^{\prime})=0\;.$
(12)

Notably, $N-$soliton solutions are obtained from $\tau-$functions of the form

$\tau(x,t)=1+\sum^{N}_{n=1}\sum_{{}_{N}C_{n}}a(i_{1},i_{2},\cdots,i_{n})\exp(\theta_{i_{1}}(x,t)+\theta_{i_{2}}(x,t)+\cdots+\theta_{i_{n}}(x,t))\;,$
(13)

where ${}_{N}C_{n}$ indicates summation over all combinations of $n$ elements
taken from $N$. Both the weights $a(i_{1},i_{2},\cdots,i_{n})$ and the
arguments of the exponential $\theta_{i}$ in (13) need to be discussed: they
can be made explicit by simply plugging the ansatz (13) in Eq. (12) and can be
interpreted in terms of the properties of the solitons.

An $N-$soliton solution is parameterised by three sets of $N$ parameters:
$0<\eta_{i}<1$, $\epsilon_{i}=\pm 1$ and $x_{i}^{0}\in\mathbb{R}$, for
$i=1,\ldots,N$. The function $\theta_{i}(x,t)$ can be seen as the “phase”
associated with a soliton indexed by the spectral doublet
$(\eta_{i},\epsilon_{i})$ and an “initial position” $x_{i}^{0}$

$\theta_{i}(x,t)=\eta_{i}\left(x-\epsilon_{i}t\sqrt{1-\eta_{i}^{2}}-x_{i}^{0}\right)\;,$
(14)

where $\epsilon_{i}=+1,-1$ distinguishes between right- and left-moving
solitons, respectively, which have speeds
$\mathsf{v}_{i}=\sqrt{1-\eta_{i}^{2}}$ and velocities (that is, including the
direction) $v_{i}=\epsilon_{i}\mathsf{v}_{i}$. Additionally, the coefficient
$a(i_{1},i_{2},\cdots,i_{n})$, in front of the exponential in Eq. (13), can be
expressed in terms of the 2-body phase shift $\varphi_{ij}$

$a(i_{1},i_{2},\cdots,i_{n})=\prod^{n}_{k<l}\exp\varphi_{i_{k}i_{l}}\;,$ (15)

where

$\varphi_{ij}=\log\frac{\left(\epsilon_{i}\sqrt{1-\eta_{i}^{2}}-\epsilon_{j}\sqrt{1-\eta_{j}^{2}}\right)^{2}-3(\eta_{i}-\eta_{j})^{2}}{\left(\epsilon_{i}\sqrt{1-\eta_{i}^{2}}-\epsilon_{j}\sqrt{1-\eta_{j}^{2}}\right)^{2}-3(\eta_{i}+\eta_{j})^{2}}\;.$
(16)

Note that for small values of $\eta_{i}$ and $\eta_{j}$, we recover the KdV
phase shift in the case of overtaking collisions.

###  2.4 Illustration: one- and two-soliton solutions

To ensure these interpretations are correct, let us first look at the
1-soliton solution

$\displaystyle u_{1}(x,t)$
$\displaystyle=\log\left[1+e^{\theta_{1}}\right]_{xx}$ (17)
$\displaystyle=\left(\frac{\eta_{1}}{2}\right)^{2}\operatorname{sech}^{2}\left[\frac{\eta_{1}}{2}\left(x-\epsilon_{1}t\sqrt{1-\eta_{1}^{2}}-x_{1}^{0}\right)\right]\;,$

which confirms $\theta_{1}$ can be seen as a phase. Note how, contrary to the
phenomenology of the KdV equation, the speed of the soliton
$\mathsf{v}_{i}=\sqrt{1-\eta_{i}^{2}}$ decreases as its amplitude grows. Given
the 1-soliton solution (17), we can define the set of $\\{h_{n}\\}$’s, namely
the amout of charge $Q_{n}$ carried by a single soliton of spectral doublet
$(\eta,\epsilon)$. Here are the first four, up to a multiplicative constant
(which will be irrelevant for us), given definitions (6)

$\begin{aligned} &h_{1}(\eta,\epsilon)\propto\eta\\\
&h_{2}(\eta,\epsilon)\propto\epsilon\eta\sqrt{1-\eta^{2}}\\\
&h_{3}(\eta,\epsilon)\propto\epsilon\eta^{3}\sqrt{1-\eta^{2}}\\\
&h_{4}(\eta,\epsilon)\propto 5\eta^{3}-4\eta^{5}\end{aligned}\;.$ (18)

Now, let us focus on the 2-soliton solution

$u_{2}(x,t)=\log\left[1+e^{\theta_{1}}+e^{\theta_{2}}+a_{12}e^{\theta_{1}+\theta_{2}}\right]_{xx}\;.$
(19)

As an example assuming that $\eta_{1}<\eta_{2}$, $\epsilon_{1}=+1$ and
$\epsilon_{2}=\pm 1$, so that $v_{1}>v_{2}$, let us look in the vicinity of
$x\approx v_{1}t$. If we take the limit $t\to-\infty$, so that $\theta_{1}$
stays finite but $\theta_{2}\to-\infty$, we directly recover the one soliton
solution (17)

$u_{2}(x,t)\approx u_{1}(x,t)\;.$ (20)

However, still looking in the vicinity of $x\approx v_{1}t$, if this time we
take the limit $t\to\infty$, then $\theta_{2}\to\infty$ and we obtain

$\begin{aligned}
u_{2}(x,t)&\approx\log\left[e^{\theta_{2}}(1+a_{12}e^{\theta_{1}})\right]_{xx}\\\
&=\left(\frac{\eta_{1}}{2}\right)^{2}\operatorname{sech}^{2}\left[\frac{\eta_{1}}{2}\left(x-\epsilon_{1}t\sqrt{1-\eta_{1}^{2}}-x_{1}^{0}\right)+\frac{\varphi_{12}}{2}\right]\end{aligned}\;,$
(21)

showing that $\varphi_{ij}/\eta_{i}$ indeed plays the role of the phase shift
of soliton 1 as it crosses soliton 2. Here, soliton 1 is right-moving,
$\epsilon_{1}=+$, and we have considered simultaneously the overtaking,
$\epsilon_{2}=+$, and the head-on, $\epsilon_{2}=-$, collisions.

###  2.5 Regularity conditions for the $N-$soliton solution

In full generality, given a $\tau-$function of form (13), we may
asymptotically write the $N-$soliton solution as

$u_{N}(x,t)\approx\sum^{N}_{i=1}\left(\frac{\eta_{i}}{2}\right)^{2}\operatorname{sech}^{2}\left[\frac{\eta_{i}}{2}\left(x-\epsilon_{i}t\sqrt{1-\eta_{i}^{2}}-x_{i}^{\pm}\right)\right]\;,\quad\text{as
}t\to\pm\infty\;,$ (22)

where $x_{i}^{-}$ and $x_{i}^{+}$ are respectively called the “in” and “out”
impact parameters, and are related by

$x_{i}^{+}-x_{i}^{-}=\frac{1}{\eta_{i}}\sum_{j\neq i}\varphi_{ij}\;.$ (23)

In effect, $N-$soliton solutions are entirely characterised by the set of
triplets $\\{\eta_{i};\ x_{i}^{-};\ \epsilon_{i}\\}_{i=1}^{N}$.

Let us now consider a $N-$soliton solution, $M$ of which are left-moving and
$(N-M)$ of which are right-moving. By convention, we shall order the spectral
parameters according to the velocities of the solitons
$v_{i}=\epsilon_{i}\sqrt{1-\eta_{i}^{2}}$, such that

$v_{1}<\cdots<v_{M}<0<v_{M+1}<\cdots<v_{N}\;,$ (24)

or, equivalently

$\eta_{1}<\cdots<\eta_{M}\quad\text{and}\quad\eta_{N}<\cdots<\eta_{M+1}\;.$
(25)

Without loss of generality, because of Galilean invariance, we can assume that
the slowest (largest) soliton is right-moving, $\eta_{M+1}>\eta_{M}$.

Noting that the expression of the 2-body phase shift (16), in terms of the
spectral parameters of the interacting solitons, only depends on the product
$\epsilon_{i}\epsilon_{j}=\pm 1$, and not on the direction in which they are
moving, we define the overtaking position shift

$\Delta_{{\rm
O}}(\eta_{i},\eta_{j})\equiv\operatorname{sgn}(v_{j}-v_{i})\frac{\varphi_{{\rm
O}}(\eta_{i},\eta_{j})}{\eta_{i}}=\frac{\operatorname{sgn}(v_{j}-v_{i})}{\eta_{i}}\log\frac{\left(\sqrt{1-\eta_{i}^{2}}-\sqrt{1-\eta_{j}^{2}}\right)^{2}-3(\eta_{i}-\eta_{j})^{2}}{\left(\sqrt{1-\eta_{i}^{2}}-\sqrt{1-\eta_{j}^{2}}\right)^{2}-3(\eta_{i}+\eta_{j})^{2}}\;,$
(26)

and the head-on position shift

$\Delta_{{\rm H}}(\eta_{i},\eta_{j})\equiv-\frac{\varphi_{{\rm
H}}(\eta_{i},\eta_{j})}{\epsilon_{i}\eta_{i}}=\frac{-1}{\epsilon_{i}\eta_{i}}\log\frac{\left(\sqrt{1-\eta_{i}^{2}}+\sqrt{1-\eta_{j}^{2}}\right)^{2}-3(\eta_{i}-\eta_{j})^{2}}{\left(\sqrt{1-\eta_{i}^{2}}+\sqrt{1-\eta_{j}^{2}}\right)^{2}-3(\eta_{i}+\eta_{j})^{2}}\;.$
(27)

An important aspect of the system we are considering is that the argument of
the $\log$ in the position shifts (26) and (27) is not always positive for any
couple $(\eta_{i},\eta_{j})$. This means the $N-$soliton solution does not
necessarily remain regular for all time. Given the ordering (25), the
$N-$soliton solution is always regular for either [92]

$\left\\{\begin{aligned} &\eta_{M+1}<\frac{\sqrt{3}}{2}\\\
&\eta_{M}<\frac{1}{2}\left|\eta_{M+1}-\sqrt{3\left(1-\eta_{M+1}^{2}\right)}\right|\end{aligned}\right.\quad\text{or}\quad\left\\{\begin{aligned}
&\frac{\sqrt{3}}{2}<\eta_{M+1}<1\\\
&\eta_{M+2}<\frac{1}{2}\left[\eta_{M+1}+\sqrt{3\left(1-\eta_{M+1}^{2}\right)}\right]\\\
&\eta_{M}<\frac{1}{2}\left[\eta_{M+1}-\sqrt{3\left(1-\eta_{M+1}^{2}\right)}\right]\end{aligned}\right.\;,$
(28)

if $\eta_{M+2}>\eta_{M}$, or

$\left\\{\begin{aligned} &\eta_{M+1}<\frac{\sqrt{3}}{2}\\\
&\eta_{M+2}<\frac{1}{2}\left|\eta_{M+1}-\sqrt{3\left(1-\eta_{M+1}^{2}\right)}\right|\end{aligned}\right.\quad\text{or}\quad\left\\{\begin{aligned}
&\frac{\sqrt{3}}{2}<\eta_{M+1}<1\\\
&\eta_{M}<\frac{1}{2}\left[\eta_{M+1}-\sqrt{3\left(1-\eta_{M+1}^{2}\right)}\right]\end{aligned}\right.\;,$
(29)

if $\eta_{M}>\eta_{M+2}$. Recall that, because of ordering (24) and (25), the
highest value of the spectral parameters is $\eta_{M+1}$ followed either by
$\eta_{M}$ or $\eta_{M+2}$ 111Note that this is slightly different from the
description in [92].. Note that allowing for complex phases $\theta_{i}$, a
complex phase shift would result in a soliton transforming into an antisoliton
and vice-versa [93], however this is not something we will be considering
here.

From the expressions of the position shifts (26) and (27), one may see that
solitons with greater velocity receive positive overtaking shifts222Note that
$\varphi_{{\rm O}}(\eta_{i},\eta_{j})<0\ \forall\ (i,j)$ given the constraint
(28)., while solitons with smaller velocities receive negative shifts. In
other words, smaller solitons are shifted forward with respect to their
direction of propagation, and larger solitons are shifted backward as a result
of overtaking interactions. In case of head-on collisions, solitons are
shifted in the same direction, regardless of their amplitude: if
$\varphi_{{\rm H}}>0$ right-moving solitons are shifted to the left and left-
moving to the right; if $\varphi_{{\rm H}}<0$ right-movers are shifted to the
right and left-movers to the left. While the position shift induced by
overtaking collisions can always be interpreted as solitons repelling each
other, $\varphi_{{\rm O}}<0$, head-on interactions can be seen as attractive
for $\varphi_{{\rm H}}>0$ and bound states may appear.

###  2.6 Bound states

On top of all this, in [92] Lambert et al. identified what they call
degenerate and resonant soliton solutions of the Boussinesq equation (1),
which are related to the Miles resonnant solutions of the KP equqtion [94, 95,
96]. Those correspond to regular solutions for which the phase shift between
two solitons may become infinite, $\varphi_{ij}\to-\infty$ or
$\varphi_{ij}\to\infty$ respectively for what they call degenerate or resonant
solutions. Those can only occur under specific conditions and, given the
ordering (24) and (25), one can identify three cases:

  * -

Overtaking degeneracy for

$\left\\{\begin{aligned} &\frac{\sqrt{3}}{2}<\eta_{M+1}<1\\\
&\eta_{M+2}=\frac{1}{2}\left[\eta_{M+1}+\sqrt{3(1-\eta_{M+1}^{2})}\right]\\\
&\eta_{M}<\frac{1}{2}\left[\eta_{M+1}-\sqrt{3(1-\eta_{M+1}^{2})}\right]\end{aligned}\right.\;,$
(30)

  * -

Head-on degeneracy for

$\left\\{\begin{aligned} &\frac{\sqrt{3}}{2}<\eta_{M+1}<1\\\
&\eta_{M}=\frac{1}{2}\left[\eta_{M+1}-\sqrt{3(1-\eta_{M+1}^{2})}\right]\\\
&\eta_{M+2}<\frac{1}{2}\left[\eta_{M+1}+\sqrt{3(1-\eta_{M+1}^{2})}\right]\end{aligned}\right.\;,$
(31)

  * -

Head-on resonance for

$\left\\{\begin{aligned} &\frac{1}{2}<\eta_{M+1}<\frac{\sqrt{3}}{2}\\\
&\eta_{M-1}=\frac{1}{2}\left[\eta_{M+1}-\sqrt{3(1-\eta_{M+1}^{2})}\right]\end{aligned}\right.\;.$
(32)

Phenomenologically those three cases are similar and, in fact, any one can be
made into another through time and/or space reversal symmetry. They correspond
to inelastic processes: a soliton of large amplitude $\eta_{M+1}$ decays in
two smaller ones, of amplitudes $\eta_{M}$ and $(\eta_{M+1}-\eta_{M})$, moving
in opposite direction (head-on degeneracy); a right-moving soliton of large
amplitude $(\eta_{M+1}+\eta_{M})$ decays in two smaller ones, $\eta_{M+1}$ and
$\eta_{M}$ (head-on resonance); two smaller solitons, $\eta_{M+2}$ and
$(\eta_{M+1}-\eta_{M+2})$, moving in opposite direction merge into a larger
one $\eta_{M+1}$ (overtaking degeneracy). Those degenerate/resonant solutions
asymptotically appear either as $(N-1)-$soliton solutions (at $t\to-\infty$
for head-on degeneracy or resonance, at $t\to\infty$ for overtaking
degeneracy) or as $N-$soliton solutions (at $t\to\infty$ for head-on
degeneracy or resonance, at $t\to-\infty$ for overtaking degeneracy), while
the other $(N-2)$ “spectator” solitons remain unaffected. Lastly, one may
consider a combination of the two degenerate cases in which
$\eta_{M+2}=\frac{1}{2}\left[\eta_{M+1}+\sqrt{3(1-\eta_{M+1}^{2})}\right]$ and
$\eta_{M}=\frac{1}{2}\left[\eta_{M+1}-\sqrt{3(1-\eta_{M+1}^{2})}\right]$
(which implies $\eta_{M+1}=\eta_{M+2}+\eta_{M}$) appearing as a degenerate
$(N-1)-$soliton solution with $(N-3)$ spectator solitons.

Because resonances happen at the boundaries of the regularity region, only the
cases discussed in this section should remain regular at all times, which has
been confirmed in [97], and no four soliton processes are possible (contrary
to KP which feature resonant tetrads [92]). As such this is not an effect that
we expect to be thermodynamically relevant.

##  3 Thermodynamics

As we have seen in the previous section, $N-$soliton solutions of the
Boussinesq equation (1) are entirely specified by the triplets $\\{\eta_{i};\
x_{i}^{-};\ \epsilon_{i}\\}_{i=1}^{N}$ characterising each individual soliton.
This, along with the asymptotic form (22), naturally suggests an
interpretation of solitons in terms of quasi-particles, and of a _soliton gas_
as a litteral gas of solitons. Naturally, this interpretation is only to be
taken as a useful tool for reasoning but not as a rigorous description:
solitons are completely unrecognisable once in the gas and constantly overlap
with one another. Nevertheless, one expects that it be possible to determine
the soliton content of any large enough region of space by a “time-of-flight”
thought experiment: one extracts the Boussisnesq field from this region and
puts it in the vacuum (i.e. the field is to zero away from the chosen region),
and then lets the full, new field configuration on the line evolve in time
until separate solitons are seen. In this way, one may determine the
approximate spatially resolved soliton content on the line, and, it turns out,
a heuristic argument taking its root in the “gas of solitons” interpretation
is rather accurate to predict both its dynamics and fluctuations. In this
section, from such an heuristic argument, we shall develop the thermodynamics
of the Boussinesq soliton gas characterised by the associated Generalised
Gibbs Ensemble (GGE). Not only will we derive the bidirectional kinetic
equations phenomenologically proposed in [30], but we will also describe the
gas in the language of statistical mechanics, introducing an entropy, free
energy, temperature, etc.

###  3.1 Classical TBA

Following the heuristic procedure previously applied to the classical Toda
system in [48] and to the KdV equation in [34], we may construct the classical
TBA of the Boussinesq soliton gas, given either condition (28) or (29) are
satisfied. To that end, we shall consider solitons as quasi-particles indexed
by $i$, of position $x_{i}^{t}$ at time $t$, of positive or negative velocity,
and spectral parameter $\eta_{i}$. We will assume that, at all times, the only
effect of the interactions between solitons is the shift in position
introduced in Section 2.5. This is motivated by the fact that solitons
asymptotically behave as free particles

$x_{i}^{t}=x_{i}^{\pm}+v_{i}t\ ,\quad\text{for }t\to\pm\infty\ ,$ (33)

that are spatially ordered according to their velocity.

Now, let us consider a $N-$soliton solution initially supported on a finite
interval $[0,L]$ in the sense that $u_{N}(x,t=0)$ decays exponentially fast if
$x\notin[0,L]$ or, in terms of the quasi-particle picture
$\\{x_{i}^{0}\in[0,L],\ \forall i=1,2\cdots,N\\}$. Recall that we take the
labelling ordered as per velocities, (24). If we assume that the right-moving
quasi-particle $i$ is initially the left-most particle, $x_{i}^{0}=0$, then,
between $t\to-\infty$ and $t=0$, it almost surely only incurred shifts from
faster right-moving solitons, such that we can evaluate its (in) impact
parameter $x_{i}^{{\rm left}}$ as

$0=x_{i}^{{\rm
left}}+\frac{1}{\eta_{i}}\sum_{j=i+1}^{N}\log\frac{\left(\sqrt{1-\eta_{i}^{2}}-\sqrt{1-\eta_{j}^{2}}\right)^{2}-3(\eta_{i}-\eta_{j})^{2}}{\left(\sqrt{1-\eta_{i}^{2}}-\sqrt{1-\eta_{j}^{2}}\right)^{2}-3(\eta_{i}+\eta_{j})^{2}}\;.$
(34)

Similarly, if we assume that the right-moving quasi-particle $i$ is initially
the right-most particle, $x_{i}^{0}=L$, then, between $t\to-\infty$ and $t=0$,
it almost surely only incurred shifts from slower right-moving solitons and
from all left-moving ones, and its impact parameter $x_{i}^{{\rm right}}$
takes the form

$\displaystyle L=x_{i}^{{\rm right}}-\frac{1}{\eta_{i}}$
$\displaystyle\left[\sum_{j=M+1}^{i-1}\log\frac{\left(\sqrt{1-\eta_{i}^{2}}-\sqrt{1-\eta_{j}^{2}}\right)^{2}-3(\eta_{i}-\eta_{j})^{2}}{\left(\sqrt{1-\eta_{i}^{2}}-\sqrt{1-\eta_{j}^{2}}\right)^{2}-3(\eta_{i}+\eta_{j})^{2}}\right.$
(35)
$\displaystyle\left.+\sum_{j=1}^{M}\log\frac{\left(\sqrt{1-\eta_{i}^{2}}+\sqrt{1-\eta_{j}^{2}}\right)^{2}-3(\eta_{i}-\eta_{j})^{2}}{\left(\sqrt{1-\eta_{i}^{2}}+\sqrt{1-\eta_{j}^{2}}\right)^{2}-3(\eta_{i}+\eta_{j})^{2}}\right]\;.$

In other words, if a quasiparticle $i$ is initially located in a box $[0,L]$,
its impact parameter must satisfy $x_{i}^{-}\in[x_{i}^{{\rm left}},x_{i}^{{\rm
right}}]$. By substracting Eq (34) to Eq (35) we may compute the length of the
asymptotic space for right-movers $L_{i}^{{\rm r}}=x_{i}^{{\rm
right}}-x_{i}^{{\rm left}}$

$\displaystyle L_{i}^{{\rm r}}=L+\frac{1}{\eta_{i}}$
$\displaystyle\left[\sum_{j=M+1,\ j\neq
i}^{N}\log\frac{\left(\sqrt{1-\eta_{i}^{2}}-\sqrt{1-\eta_{j}^{2}}\right)^{2}-3(\eta_{i}-\eta_{j})^{2}}{\left(\sqrt{1-\eta_{i}^{2}}-\sqrt{1-\eta_{j}^{2}}\right)^{2}-3(\eta_{i}+\eta_{j})^{2}}\right.$
(36)
$\displaystyle\left.+\sum_{j=1}^{M}\log\frac{\left(\sqrt{1-\eta_{i}^{2}}+\sqrt{1-\eta_{j}^{2}}\right)^{2}-3(\eta_{i}-\eta_{j})^{2}}{\left(\sqrt{1-\eta_{i}^{2}}+\sqrt{1-\eta_{j}^{2}}\right)^{2}-3(\eta_{i}+\eta_{j})^{2}}\right]\;.$

A similar reasonning can be carried out for left-moving solitons and one may
find that

$L_{i}^{{\rm l}}=L+\frac{1}{\eta_{i}}\left[\sum_{j=1,\ j\neq
i}^{M}\varphi_{{\rm O}}(\eta_{i},\eta_{j})+\sum_{j=M+1}^{N}\varphi_{{\rm
H}}(\eta_{i},\eta_{j})\right]\;.$ (37)

Introducing two functions $\mathcal{K}_{N,M}^{{\rm r}}(\eta)$ and
$\mathcal{K}_{N,M}^{{\rm l}}(\eta)$ that respectively interpolate between the
$(L_{i}^{{\rm r}}/L)$’s and the $(L_{i}^{{\rm l}}/L)$’s for fixed couples of
$(N,M)$, we may then define the right- and left-asymptotic space densities,
$\mathcal{K}_{{\rm r}}(\eta)$ and $\mathcal{K}_{{\rm l}}(\eta)$, by taking the
thermodynamic limit: $L\to\infty$, $N\to\infty$, $M\to\infty$, while keeping
the ratios $N/L=\varkappa$ and $M/N=\gamma$ finite. Then, from Eqs (36) and
(37) one may write

$\left\\{\begin{aligned} &\mathcal{K}_{{\rm
r}}(\eta)=1+\frac{1}{\eta}\left[\int_{\Gamma_{{\rm r}}}{\rm d}\mu\
\varphi_{{\rm O}}(\eta,\mu)\rho_{{\rm r}}(\mu)+\int_{\Gamma_{{\rm l}}}{\rm
d}\mu\ \varphi_{{\rm H}}(\eta,\mu)\rho_{{\rm l}}(\mu)\right]\\\
&\mathcal{K}_{{\rm l}}(\eta)=1+\frac{1}{\eta}\left[\int_{\Gamma_{{\rm l}}}{\rm
d}\mu\ \varphi_{{\rm O}}(\eta,\mu)\rho_{{\rm l}}(\mu)+\int_{\Gamma_{{\rm
r}}}{\rm d}\mu\ \varphi_{{\rm H}}(\eta,\mu)\rho_{{\rm
r}}(\mu)\right]\end{aligned}\right.\;,$ (38)

where $\Gamma_{{\rm r}}\subset[0,1]$ and $\Gamma_{{\rm l}}\subset[0,1]$
represent the spectral support of right- and left-moving solitons,
respectively, and where

$\left\\{\begin{aligned} &\rho_{{\rm
r}}(\eta)=\frac{\varkappa(1-\gamma)}{N-M}\sum_{i=M+1}^{N}\delta(\eta-\eta_{i})\\\
&\rho_{{\rm
l}}(\eta)=\frac{\varkappa\gamma}{M}\sum_{i=1}^{M}\delta(\eta-\eta_{i})\end{aligned}\right.\;,$
(39)

are the right- and left-densities of states (DOS) expressed in terms of the
empirical densities of spectral points and of the spatial densities of
solitons. Equations (38) is akin to the nonlinear dispersion relation of
soliton gases (see [31] for a comprehensive review on the topic), this time
for two types of quasi-particles, in which

$\sigma_{{\rm l}}(\eta)\equiv\frac{\eta\mathcal{K}_{{\rm l}}(\eta)}{\rho_{{\rm
l}}(\eta)}\ ,\quad\text{and}\quad\sigma_{{\rm
r}}(\eta)\equiv\frac{\eta\mathcal{K}_{{\rm r}}(\eta)}{\rho_{{\rm
r}}(\eta)}\;,$ (40)

would play the role of the left- and right- spectral scaling functions.

Before we continue, some remarks are in order. First, one should note that, in
effect, left- and right- moving solitons are to be considered as two different
particle types with their distinct DOS’s. That is because the shifts for head-
on and overtaking collision are different and one must keep track of each type
of collisions. Second, echoing the discussion carried out in Section 4.5 of
[34] regarding the ambiguity associated with the choice of the GHD momentum
function, we should clarify which convention we shall adopt in the rest of the
paper.

###  3.2 Conventional choice of the momentum function

As argued in [34] the parametrisation of solitons as quasi-particles in terms
of the spectral and directional parameters $\eta$ and $\epsilon$ is somewhat
arbitrary. When defining the momentum function $P(\eta,\epsilon)$ of the
quasi-particles associated with solitons in our GHD construction, there are, a
priori, two natural choices: the physical momentum of the soliton

$P_{{\rm phys}}(\eta,\epsilon)\equiv
h_{2}(\eta,\epsilon)=\epsilon\eta\sqrt{1-\eta^{2}}\ ,$ (41)

or its velocity

$P_{{\rm vel}}(\eta,\epsilon)\equiv
v(\eta,\epsilon)=\epsilon\sqrt{1-\eta^{2}}\ .$ (42)

For the purpose of constructing the GHD of the Boussinesq soliton gas, we
could use the same convention as [34] and express all the upcoming results in
terms of the momentum of the quasi-particle $v$, and not in terms of the
“physical” momentum of the soliton $h_{2}$. However there is a third
alternative which is less physically meaningful but formally more practical.

This comes from the fact that GHD was originally developed to study quantum
integrable systems, in which the momentum function $P_{{\rm quant}}$ is
unambiguous. In these systems the scattering phase $\varphi_{{\rm quant}}$ is
related to the scattering shift $\Delta_{{\rm quant}}$ by the relation

$\varphi_{{\rm quant}}(\eta)=P^{\prime}_{{\rm quant}}(\eta)\Delta_{{\rm
quant}}(\eta)\ ,$ (43)

and comparing with Eqs. (26)-(27), we construct

$P(\eta,\epsilon)\equiv P_{{\rm
GHD}}(\eta,\epsilon)=\epsilon\frac{\eta^{2}}{2}\ .$ (44)

In the upcoming sections, this parametrisation is the one that will yield the
most compact and easiest to handle expressions. In effect, when it comes to
the KdV equation discussed in [34], the conventions (42) and (44) are
equivalent up to a multiplication by 8. The fact that the Boussinesq equation
actually differentiates between these two conventions extends the previous
discussion of [34] and provides a better way of selecting the arbitrary
momentum function through Eq (43).

###  3.3 Partition function

In this section we will build up on the previous TBA-inspired approach to
construct the thermodynamics of our soliton gas and the associated GGE. This
can be done by writing the generalised Gibbs measure for an ensemble of
$N-$soliton solutions and the corresponding partition fucntion which, on a
formal level, takes the form

$\begin{aligned}
\mathcal{Z}_{N}=\int\mathcal{D}[u_{N}]\exp\big{(}S[u_{N}]-W[u_{N}]\big{)}\end{aligned}\;,$
(45)

where $S$ is the entropy of the gas to be defined, and $W$ a Lagrange
parameter, the Gibbs weight, to account for the infinite set of conservation
laws

$W=\sum_{n=1}^{\infty}\beta_{n}Q_{n}\ ,$ (46)

associated with the infinite set of inverse temperatures
$\\{\beta_{n}\\}_{n=1}^{\infty}$. At this point, expression (45) is admitedly
rather unwieldy. However, since the $N-$soliton solution is entirely specified
by the triplets $\\{\eta_{i};\ x_{i}^{-};\ \epsilon_{i}\\}_{i=1}^{N}$ (with
$\epsilon_{i}=-1$ for $i=1,2,\ldots M$, and $\epsilon_{i}=1$ for
$i=M+1,M+2,\ldots N$), one may instead write its partition function in terms
of the asymptotic coordinates

$\displaystyle\mathcal{Z}_{N}=\sum_{M=0}^{N-1}\frac{M!(N-M)!}{(N!)^{2}}$
$\displaystyle\int_{\Gamma_{{\rm l}}^{M}\times\Gamma_{{\rm
r}}^{N-M}\times\mathbb{R}^{N}}\prod^{N}_{i=1}\frac{{\rm
d}P(\eta_{i})}{2\pi}{\rm d}x_{i}^{-}$ (47)
$\displaystyle\exp\left[-\sum^{N}_{i=1}w(\eta_{i},\epsilon_{i})\right]\chi\left(u_{N}(x,t=0)<\varepsilon_{x},\
x\notin[0,L]\right)$

where $\varepsilon_{x}\to 0$ fast enough as ${\rm max}(-x,x-L)\to\infty$ and

${\rm d}P(\eta_{i})=\eta_{i}{\rm d}\eta_{i}\;.$ (48)

This amounts to counting all possible realisations of a $N$-soliton solution
$u_{N}$, with solitonic Gibbs weights $w$, and given the constraint that, at
$t=0$, it is supported on a finite interval $[0,L]$, represented by the
indicator function $\chi$. Note that, in summing/integrating over the full
spectral space of each solitons $(\eta_{i},\epsilon_{i})$, we only explicitly
integrate over the $\eta_{i}$’s, as the sum over the $\epsilon_{i}$’s is
already taking into account by the combinatoric factor. The solitonic weights
$w(\eta,\epsilon)$ are defined as the individual contribution of a single
soliton to the weight $W$ so that

$w(\eta,\epsilon)\equiv\sum_{n}^{\infty}\beta_{n}h_{n}(\eta,\epsilon)\
,\quad\quad W=\sum_{i=1}^{N}w(\eta_{i},\epsilon_{i})\;.$ (49)

An important point is that there is in fact no need to write
$w(\eta,\epsilon)$ as an expansion in the functions $h_{n}(\eta,\epsilon)$ –
any function $w(\eta,\epsilon)$ which is bounded from below will do. Of
course, it is not clear that an arbitrary function of $\eta,\epsilon$ can be
written as an expansion of the $h_{n}(\eta,\epsilon)$’s, whose first few are
written in (18). In view of Eq. (46), we assume that there should exist a
possibly wider family of conserved charges, including the $Q_{n}$’s whose
first few members are written in (6), such that the corresponding one-soliton
values $h_{n}(\eta,\epsilon)$ form a complete basis for the space of functions
of $\eta,\epsilon$. Then, an arbitrary choice of the corresponding Lagrange
parameters gives rise to an arbitrary choice of $w(\eta,\epsilon)$ as a
function of both arguments. In order to make this statement precise, one would
have to specify the space of functions of $\eta$ and $\epsilon$ more
accurately, and possibly to construct explicitly the required, extra conserved
quantities; these may be expected to be quasi-local, following the notion
developed in quantum many-body systems [98]. We leave this for future
research, and simply assume $w(\eta,\epsilon)$ to be an arbitrary function of
both arguments.

For large values of $N$, the constraint can be expressed as bounds for the
integration of the impact parameters ${x_{i}^{-}}$ through the TBA approach
discussed in the previous section 333This essentially amounts to assuming that
varying the impact parameter of a single soliton has only negligible effects
on the overall gas.

$\displaystyle\int_{\mathbb{R}^{N}}\prod^{N}_{i=1}{\rm
d}x_{i}^{-}\chi\left(u_{N}(x,t=0)<\varepsilon_{x},x\notin[0,L]\right)$
$\displaystyle\approx\prod^{N}_{i=1}\left(\int^{x_{i}^{\rm right}}_{x_{i}^{\rm
left}}{\rm d}x^{-}\right)$ (50) $\displaystyle=\prod^{N}_{i=M+1}L_{i}^{{\rm
r}}\prod^{M}_{i=1}L_{i}^{{\rm l}}$
$\displaystyle=L^{N}\prod^{N}_{i=M+1}\mathcal{K}_{N,M}^{{\rm
r}}(\eta_{i})\prod^{M}_{i=1}\mathcal{K}_{N,M}^{{\rm l}}(\eta_{i})\;.$

In a way, the TBA provides the Jacobian of the transformation from expression
(45), in terms of the field, to expression (47), in terms of (asymptotically)
free quasi-particles. Noting that we can estimate the prefactor through
Sterling’s formula for large values of $N$ and $M$

$L^{N}\frac{M!(N-M)!}{(N!)^{2}}\approx\exp\left\\{N+N\log\left[\gamma^{\gamma}(1-\gamma)^{1-\gamma}\right]-N\log\varkappa\right\\}\;,$
(51)

we may write the partition function (47) as

$\begin{aligned} \mathcal{Z}_{N}=\sum_{M=0}^{N-1}&\int_{\Gamma_{{\rm
l}}^{M}\times\Gamma_{{\rm r}}^{N-M}}\prod^{N}_{i=1}{\rm
d}\eta_{i}\exp\left\\{\vphantom{\log\frac{\eta_{i}}{\sqrt{1-\eta_{i}}}}\right.\\\
&-\sum^{M}_{i=1}\left[w_{{\rm
l}}(\eta_{i})-1+\log\varkappa-\log\left[\gamma^{\gamma}(1-\gamma)^{1-\gamma}\right]-\log\frac{\eta_{i}}{2\pi}-\log\mathcal{K}_{N,M}^{{\rm
l}}(\eta_{i})\right]\\\ &\left.-\sum^{N}_{i=M+1}\left[w_{{\rm
r}}(\eta_{i})-1+\log\varkappa-\log\left[\gamma^{\gamma}(1-\gamma)^{1-\gamma}\right]-\log\frac{\eta_{i}}{2\pi}-\log\mathcal{K}_{N,M}^{{\rm
r}}(\eta_{i})\right]\right\\}\end{aligned}\;,$ (52)

where we decomposed the Gibbs weight into left- and right-contributions
$w_{{\rm l}}(\eta)=w(\eta,-1)$, $w_{{\rm r}}(\eta)=w(\eta,+1)$.

###  3.4 Large deviation principle and Yang-Yang equation

In the thermodynamic limit $N\to\infty$, we can evaluate the partition
function through the large deviation principle

$\mathcal{Z}_{N}\asymp\exp\left(-N\mathcal{F}^{\rm MF}[\rho_{{\rm
l}}^{*}(\eta),\rho_{{\rm r}}^{*}(\eta)]\right)\,,$ (53)

where $\rho_{{\rm l}}^{*}$ and $\rho_{{\rm r}}^{*}$ is the (assumed unique)
couple of minimizers of the mean-field free energy functional

$\begin{aligned} \mathcal{F}^{\rm MF}[\rho_{{\rm l}}(\eta),\rho_{{\rm
r}}(\eta)]=&\int_{\Gamma_{{\rm l}}}{\rm d}\eta\rho_{{\rm
l}}(\eta)\left[w_{{\rm l}}(\eta)-1+\nu-\log\frac{\eta\mathcal{K}_{{\rm
l}}(\eta)}{2\pi}+\log\rho_{{\rm l}}(\eta)\right]\\\ &+\int_{\Gamma_{{\rm
r}}}{\rm d}\eta\rho_{{\rm r}}(\eta)\left[w_{{\rm
r}}(\eta)-1+\nu-\log\frac{\eta\mathcal{K}_{{\rm
r}}(\eta)}{2\pi}+\log\rho_{{\rm r}}(\eta)\right]\end{aligned}\;.$ (54)

This amounts to applying Laplace method on Eq. (52), in which we introduced
$\nu=\log\left[\gamma^{\gamma}(1-\gamma)^{1-\gamma}\right]$, and where the
configuration entropy term $\log\rho$ comes from Sanov’s theorem in regards to
the empirical density [99]. Recalling that $v(\eta)=\pm\sqrt{1-\eta^{2}}$ is
the momentum of the quasi-particle and introducing the right/left-occupation
functions

$n_{{\rm r}}(\eta)=\frac{2\pi}{\eta}\frac{\rho_{{\rm
r}}(\eta)}{\mathcal{K}_{{\rm r}}(\eta)}\ ,\quad n_{{\rm
l}}(\eta)=\frac{2\pi}{\eta}\frac{\rho_{{\rm l}}(\eta)}{\mathcal{K}_{{\rm
l}}(\eta)}\;,$ (55)

as well as the pseudo-energies

$\varepsilon_{{\rm r}}=-\log n_{{\rm r}}\ ,\quad\varepsilon_{{\rm l}}=-\log
n_{{\rm l}}\;,$ (56)

minimising the mean-field free energy functional (54) amounts to solving the
system

$\left\\{\begin{aligned} &0=\frac{\delta\mathcal{F}^{\rm MF}[\rho_{{\rm
l}},\rho_{{\rm r}}]}{\delta\rho_{{\rm l}}}\\\ &0=\frac{\delta\mathcal{F}^{\rm
MF}[\rho_{{\rm l}},\rho_{{\rm r}}]}{\delta\rho_{{\rm
r}}}\end{aligned}\right.\;,$ (57)

or, more explicitly, one must solve a system of Yang-Yang type equations

$\left\\{\begin{aligned} &\varepsilon_{{\rm l}}(\eta)=w_{{\rm
l}}(\eta)+\nu-\int_{\Gamma_{{\rm l}}}\frac{{\rm d}\mu}{2\pi}\ \varphi_{{\rm
O}}(\eta,\mu)e^{-\varepsilon_{{\rm l}}(\mu)}-\int_{\Gamma_{{\rm r}}}\frac{{\rm
d}\mu}{2\pi}\varphi_{{\rm H}}(\eta,\mu)e^{-\varepsilon_{{\rm r}}(\mu)}\\\
&\varepsilon_{{\rm r}}(\eta)=w_{{\rm r}}(\eta)+\nu-\int_{\Gamma_{{\rm
r}}}\frac{{\rm d}\mu}{2\pi}\ \varphi_{{\rm O}}(\eta,\mu)e^{-\varepsilon_{{\rm
r}}(\mu)}-\int_{\Gamma_{{\rm l}}}\frac{{\rm d}\mu}{2\pi}\varphi_{{\rm
H}}(\eta,\mu)e^{-\varepsilon_{{\rm l}}(\mu)}\end{aligned}\right.\;.$ (58)

This generalises to bidirectional equations (involving two different soliton
types) the Yang-Yang type equation introduced in [34] for the KdV soliton gas.
These equations define a one to one map between the set of inverse
temperatures $\\{\beta_{n}\\}_{n=1}^{\infty}$ defining the GGE and the
occupation functions, $n_{l}$ and $n_{r}$, which can then be related to the
DOS’s, $\rho_{{\rm l}}$ and $\rho_{{\rm r}}$, through the NDR’s (38) and the
definitions (55).

###  3.5 Thermodynamic quantities

Using the previous relations (58) in Eq. (54), along with the dispersion
relations (38), we may define the free energy density $\mathcal{F}$ as the
minimised mean-field free energy functional

$\mathcal{F}=-\int_{\Gamma_{{\rm l}}}\frac{{\rm d}P(\mu)}{2\pi}\ n_{{\rm
l}}(\mu)-\int_{\Gamma_{{\rm r}}}\frac{{\rm d}P(\mu)}{2\pi}\ n_{{\rm
r}}(\mu)\;,$ (59)

or, using the equivalence (40)

$\mathcal{F}=-\int_{\Gamma_{{\rm l}}}\frac{{\rm d}P(\mu)}{\sigma_{{\rm
l}}(\mu)}-\int_{\Gamma_{{\rm r}}}\frac{{\rm d}P(\mu)}{\sigma_{{\rm
r}}(\mu)}\;,$ (60)

which is a straightforward generalisation to two quasi-particle types of the
KdV free energy found in [34]. This then provides a straightforward way to
compute the gas’ entropy density $\mathcal{S}$ using the usual definition from
statistical mechanics

$\mathcal{F}=\mathcal{W}-\mathcal{S}\;,$ (61)

where $\mathcal{W}=\int_{\Gamma_{{\rm l}}}{\rm d}\eta\ w_{{\rm
l}}(\eta)\rho_{{\rm l}}(\eta)+\int_{\Gamma_{{\rm r}}}{\rm d}\eta\ w_{{\rm
r}}(\eta)\rho_{{\rm l}}(\eta)$. Using both the Yang-Yang system (58) to
eliminate both $w_{{\rm l}}$ and $w_{{\rm r}}$, and again the dispersion
relations (38) to simplify the expression, we end up with

$\mathcal{S}=\int_{\Gamma_{{\rm l}}}{\rm d}\eta\ \rho_{{\rm
l}}(\eta)\left[1-\log n_{{\rm l}}(\eta)-\nu\right]+\int_{\Gamma_{{\rm r}}}{\rm
d}\eta\ \rho_{{\rm r}}(\eta)\left[1-\log n_{{\rm r}}(\eta)-\nu\right]\;.$ (62)

Note that at condensation $\mathcal{K}_{{\rm r}}=\mathcal{K}_{{\rm l}}=0$, so
that $n_{{\rm r}}\to\infty$ and $n_{{\rm l}}\to\infty$, meaning that
$\mathcal{S}$ is indeed minimal.

##  4 Generalised hydrodynamics

###  4.1 Dressing operation

Going back to the Gibbs weights (49), we can be more specific and define it in
terms of the conserved quantities carried by a single soliton (18) and some
Lagrange parameters $\\{\beta_{n}\\}_{n=1}^{\infty}$

$w(\eta)=\sum_{n=1}^{\infty}\beta_{n}h_{n}(\eta,\epsilon)\;,$ (63)

where $h_{n}(\eta,-1)=h^{{\rm l}}_{n}(\eta)$ if the soliton is left-moving and
$h_{n}(\eta,+1)=h^{{\rm r}}_{n}(\eta)$ if it is right-moving. With this
definition we can now ask ourselves how the pseudo-energies vary with respect
to one of those Lagrange parameters. Given Yang-Yang equations (58) we expect

$\left\\{\begin{aligned} &\partial_{\beta_{n}}\varepsilon_{{\rm
l}}(\eta)=h^{{\rm l}}_{n}(\eta)+\int_{\Gamma_{{\rm l}}}\frac{{\rm
d}\mu}{2\pi}\ \varphi_{{\rm O}}(\eta,\mu)\partial_{\beta_{n}}\varepsilon_{{\rm
l}}(\mu)n_{{\rm l}}(\mu)+\int_{\Gamma_{{\rm r}}}\frac{{\rm
d}\mu}{2\pi}\varphi_{{\rm H}}(\eta,\mu)\partial_{\beta_{n}}\varepsilon_{{\rm
r}}(\mu)n_{{\rm r}}(\mu)\\\ &\partial_{\beta_{n}}\varepsilon_{{\rm
r}}(\eta)=h^{{\rm r}}_{n}(\eta)+\int_{\Gamma_{{\rm r}}}\frac{{\rm
d}\mu}{2\pi}\ \varphi_{{\rm O}}(\eta,\mu)\partial_{\beta_{n}}\varepsilon_{{\rm
r}}(\mu)n_{{\rm r}}(\mu)+\int_{\Gamma_{{\rm l}}}\frac{{\rm
d}\mu}{2\pi}\varphi_{{\rm H}}(\eta,\mu)\partial_{\beta_{n}}\varepsilon_{{\rm
l}}(\mu)n_{{\rm l}}(\mu)\end{aligned}\right.\;,$ (64)

recalling that $n_{{\rm l}}=e^{-\varepsilon_{{\rm l}}}$ and $n_{{\rm
r}}=e^{-\varepsilon_{{\rm r}}}$. By writing
$\partial_{\beta_{n}}\varepsilon_{{\rm l}}(\eta)\equiv(h^{{\rm
l}}{}_{n})^{\rm{l,dr}}(\eta)$ and $\partial_{\beta_{n}}\varepsilon_{{\rm
r}}(\eta)\equiv(h^{{\rm r}}{}_{n})^{\rm{r,dr}}(\eta)$ we define the left- and
right- dressing operation

$\left\\{\begin{aligned} &h^{\rm{l,dr}}(\eta)=h^{{\rm
l}}(\eta)+\int_{\Gamma_{{\rm l}}}\frac{{\rm d}\mu}{2\pi}\ \varphi_{{\rm
O}}(\eta,\mu)n_{{\rm l}}(\mu)h^{\rm{l,dr}}(\mu)+\int_{\Gamma_{{\rm
r}}}\frac{{\rm d}\mu}{2\pi}\varphi_{{\rm H}}(\eta,\mu)n_{{\rm
r}}(\mu)h^{\rm{r,dr}}(\mu)\\\ &h^{\rm{r,dr}}(\eta)=h^{{\rm
r}}(\eta)+\int_{\Gamma_{{\rm r}}}\frac{{\rm d}\mu}{2\pi}\ \varphi_{{\rm
O}}(\eta,\mu)n_{{\rm r}}(\mu)h^{\rm{r,dr}}(\mu)+\int_{\Gamma_{{\rm
l}}}\frac{{\rm d}\mu}{2\pi}\varphi_{{\rm H}}(\eta,\mu)n_{{\rm
l}}(\mu)h^{\rm{l,dr}}(\mu)\end{aligned}\right.\;,$ (65)

so that the NDRs (38) can be reinterpreted as a dressing relation. Indeed we
may define

$\left\\{\begin{aligned} &\rho_{{\rm s,l}}(\eta)\equiv\eta\mathcal{K}_{{\rm
l}}(\eta)=\sigma_{{\rm l}}(\eta)\rho_{{\rm
l}}(\eta)=\left|P^{\prime}(\eta,-1)\right|^{\rm{l,dr}}(\eta)=(\eta)^{\rm{l,dr}}(\eta)\\\
&\rho_{{\rm s,r}}(\eta)\equiv\eta\mathcal{K}_{{\rm r}}(\eta)=\sigma_{{\rm
r}}(\eta)\rho_{{\rm
r}}(\eta)=\left|P^{\prime}(\eta,1)\right|^{\rm{r,dr}}(\eta)=(\eta)^{\rm{r,dr}}(\eta)\end{aligned}\right.\,$
(66)

by recalling the relations (55).

In view of the remark made after Eq. (49), we note how we assume here that
$h_{n}(\eta,\epsilon)$ may be chosen as an arbitrary function of
$\eta,\epsilon$: as is usual in many-body integrability and especially in GHD
[64], the general form of the dressing operation is obtained by perturbing the
“source term” $w(\eta,\epsilon)$ by an arbitrary function, thus assuming that
we have access to this large space.

###  4.2 Effective velocity

Recall the nonlinear dispersion relation (66) characterising the DOS in terms
of the dressing operation. Similarly it is possible to define a left- and
right- flux density $f_{\cdot}(\eta)$. In GHD, this flux density is generally
defined in terms of the dressing of the energy function $E(\eta)$

$\left\\{\begin{aligned} &\sigma_{{\rm l}}(\eta)f_{{\rm
l}}(\eta)\equiv-\left[E^{\prime}(\eta)\right]^{\rm{l,dr}}(\eta)\\\
&\sigma_{{\rm r}}(\eta)f_{{\rm
r}}(\eta)\equiv\left[E^{\prime}(\eta)\right]^{\rm{r,dr}}(\eta)\end{aligned}\right.\;.$
(67)

Given our chosen convention (44) regarding the momentum function, the
corresponding energy function is defined in terms of the bare velocity of the
quasi-particle

$v(\eta,\epsilon)=\frac{E^{\prime}(\eta)}{P^{\prime}(\eta,\epsilon)}\quad\Rightarrow\quad
E=-\frac{1}{3}\left[1-\eta^{2}\right]^{3/2}\ ,$ (68)

the usual expression of the group velocity. This identification of the flux
density with a dressed quantity provides the second set of NDRs for our
soliton gas

$\left\\{\begin{aligned} &\sigma_{{\rm l}}(\eta)f_{{\rm
l}}(\eta)=-\eta\sqrt{1-\eta^{2}}+\int_{\Gamma_{{\rm l}}}{\rm d}\mu\
\varphi_{{\rm O}}(\eta,\mu)f_{{\rm l}}(\mu)+\int_{\Gamma_{{\rm r}}}{\rm d}\mu\
\varphi_{{\rm H}}(\eta,\mu)f_{{\rm r}}(\mu)\\\ &\sigma_{{\rm r}}(\eta)f_{{\rm
r}}(\eta)=\eta\sqrt{1-\eta^{2}}+\int_{\Gamma_{{\rm r}}}{\rm d}\mu\
\varphi_{{\rm O}}(\eta,\mu)f_{{\rm r}}(\mu)+\int_{\Gamma_{{\rm l}}}{\rm d}\mu\
\varphi_{{\rm H}}(\eta,\mu)f_{{\rm l}}(\mu)\end{aligned}\right.\;,$ (69)

using the definition of the dressing operation (65) along with the fact that,
from Eqs (40) and (55), $\sigma_{{\rm l}}=2\pi/n_{{\rm l}}$ and $\sigma_{{\rm
r}}=2\pi/n_{{\rm r}}$. From this we may introduce the left- and right-
effective velocity

$v^{{\rm eff}}_{{\rm l}}\equiv\frac{f_{{\rm l}}}{\rho_{{\rm
l}}}\quad\text{and}\quad v^{{\rm eff}}_{{\rm r}}\equiv\frac{f_{{\rm
r}}}{\rho_{{\rm r}}}$ (70)

of solitons moving through the gas. These effective velocities can be computed
by multiplying the left- and right- NDR equations (38) respectively by
$v^{{\rm eff}}_{{\rm l}}$ and $v^{{\rm eff}}_{{\rm r}}$, then subtracting the
second NDR relations (69), identifying both $f_{{\rm l}}=\rho_{{\rm l}}v^{{\rm
eff}}_{{\rm l}}$ and $f_{{\rm r}}=\rho_{{\rm r}}v^{{\rm eff}}_{{\rm r}}$ and,
ultimately, dividing by $\eta$. This yields

$\left\\{\begin{aligned} &v^{{\rm eff}}_{{\rm
l}}(\eta)=v(\eta,-1)-\frac{1}{\eta}\left[\int_{\Gamma_{{\rm l}}}{\rm d}\mu\
\varphi_{{\rm O}}(\eta,\mu)\rho_{{\rm l}}(\mu)[v^{{\rm eff}}_{{\rm
l}}(\eta)-v^{{\rm eff}}_{{\rm l}}(\mu)]+\int_{\Gamma_{{\rm r}}}{\rm d}\mu\
\varphi_{{\rm H}}(\eta,\mu)\rho_{{\rm r}}(\mu)[v^{{\rm eff}}_{{\rm
l}}(\eta)-v^{{\rm eff}}_{{\rm r}}(\mu)]\right]\\\ &v^{{\rm eff}}_{{\rm
r}}(\eta)=v(\eta,+1)-\frac{1}{\eta}\left[\int_{\Gamma_{{\rm r}}}{\rm d}\mu\
\varphi_{{\rm O}}(\eta,\mu)\rho_{{\rm r}}(\mu)[v^{{\rm eff}}_{{\rm
r}}(\eta)-v^{{\rm eff}}_{{\rm r}}(\mu)]+\int_{\Gamma_{{\rm l}}}{\rm d}\mu\
\varphi_{{\rm H}}(\eta,\mu)\rho_{{\rm l}}(\mu)[v^{{\rm eff}}_{{\rm
r}}(\eta)-v^{{\rm eff}}_{{\rm l}}(\mu)]\right]\end{aligned}\right.\;,$ (71)

recovering the results presented in [30] in which the authors
phenomenologically proposed a general expression for the effective velocity of
a bidrectional soliton gas.

On top of that, it is well-known that the Boussinesq equation can be reduced
to the KdV equation is the long long-wave (or equivalently small $\eta$)
limit. As an additional check, Eqs. (71) should be consistent with the
effective velocity of the KdV soliton gas. Indeed, in the small $\eta$ limit
the head-on an overtaking phase shifts simplify in such fashion

$\varphi_{{\rm H}}\approx 0\ ,\quad\quad\varphi_{{\rm O}}\approx
2\log\left|\frac{\eta-\mu}{\eta+\mu}\right|\ .$ (72)

Left- and right-moving solitons essentially decouple, while the overtaking
phase shift reduces to the KdV phase shift. Similarly the velocity of solitons
simplifies to

$v(\eta,\epsilon)=\epsilon\left[1-\frac{\eta^{2}}{2}\right]\ .$ (73)

Because of Galilean invariance, the constant part of the velocity can be
absorbed through a boost, similarly, and time-reversal takes care of the
factor $\epsilon$. Finally, choosing a different normalisation for the
amplitude of the wave field and for the space and time variables

$u\ \to\ 8u\ ,\quad x\ \to\ \sqrt{2}x\ ,\quad t\ \to\ \sqrt{2}t\ ,$ (74)

yields

$\frac{\eta^{2}}{2}\ \to\ 4\eta^{2}\quad\text{and}\quad\Delta(\eta,\mu)\ \to\
\frac{1}{\eta}\log\left|\frac{\eta-\mu}{\eta+\mu}\right|\ .$ (75)

Putting all of this together, equations (71) eventually reduce to the single
equation

$v^{{\rm eff}}(\eta)=4\eta^{2}-\frac{1}{\eta}\int_{\Gamma}{\rm
d}\mu\log\left|\frac{\eta-\mu}{\eta+\mu}\right|\rho(\mu)\left[v^{{\rm
eff}}(\eta)-v^{{\rm eff}}(\mu)\right]\ ,$ (76)

which is precisely the one satisfied by the KdV effective velocity first
derived in [1].

###  4.3 Thermodynamic averages and correlation functions

Putting together the statistical mechanics picture we developed in section 3
with the notion of dressing introduced in section 4.1, we can now borrow
general results from [43, 45] pertaining to thermodynamic quantities in GHD.

What we present in this section can be seen as a straightforward
generalisation of usual results from statistical mechanics. Provided our
soliton gas is accurately described by the GGE defined by (47), associated
with the free energy density (59), we can conjecture expressions of
thermodynamic averages and correlations in our soliton gas. Indeed, given
time-conserved Poisson commuting charges of type (e.g. Eqs. (6) )

$Q_{n}=\int_{\mathbb{R}}\ {\rm d}x\ q_{n}(x)\ ,$ (77)

and the associated space conserved currents

$J_{n}=\int_{\mathbb{R}}\ {\rm d}x\ j_{n}(x)\ ,$ (78)

we can easily write the ensemble averages of the densities $q_{n}$ and
$j_{n}$. As per standard statistical mechanics we have

$\begin{aligned} \langle
q_{i}\rangle&=\frac{\partial\mathcal{F}}{\partial\beta_{n}}\\\
&=\int_{\Gamma_{{\rm l}}}\frac{{\rm
d}P(\mu)}{2\pi}n_{l}(\mu)h^{\rm{l,dr}}(\mu)+\int_{\Gamma_{{\rm r}}}\frac{{\rm
d}P(\mu)}{2\pi}n_{r}(\mu)h^{\rm{r,dr}}(\mu)\\\ &=\int_{\Gamma_{{\rm l}}}{\rm
d}\mu\ \rho_{{\rm l}}(\mu)h^{{\rm l}}(\mu)+\int_{\Gamma_{{\rm r}}}{\rm d}\mu\
\rho_{{\rm r}}(\mu)h^{{\rm r}}(\mu)\end{aligned}\ ,$ (79)

where the relation between the first and second lines comes from using the
definition the dressing operation (64)-(65), along identity (56), in the
definition of the free energy density (59). The third line, which is more
natural, is obtained thanks to the symmetry of the dressing operation [64]

$\sum_{a=\textrm{l},\textrm{r}}\int_{\Gamma_{a}}{\rm d}\mu\
g^{a}(\mu)n_{a}(\mu)h^{a,{\rm
dr}}(\mu)=\sum_{a=\textrm{l},\textrm{r}}\int_{\Gamma_{a}}{\rm d}\mu\ g^{a,{\rm
dr}}(\mu)n_{a}(\mu)h^{a}(\mu)\ ,$ (80)

while recalling the NDR’s (66) and definitions (55). Similarly, space-
integrated connected correlations are obtained as

$\begin{aligned} \mathsf{C}_{ab}&\equiv\int{\rm d}x\,\left(\left\langle
q_{a}(x)q_{b}(0)\right\rangle-\left\langle q_{a}(x)\right\rangle\left\langle
q_{b}(0)\right\rangle\right)=-\frac{\partial^{2}\mathcal{F}}{\partial\beta_{a}\partial\beta_{b}}\\\
&=\int_{\Gamma_{{\rm l}}}{\rm d}\eta\,\rho_{{\rm l}}(\eta)h_{a}^{\rm
l,dr}(\eta)h_{b}^{\rm l,dr}(\eta)+\int_{\Gamma_{{\rm r}}}{\rm
d}\eta\,\rho_{{\rm r}}(\eta)h_{a}^{\rm r,dr}(\eta)h_{b}^{\rm
r,dr}(\eta)\end{aligned}\ .$ (81)

The averages of currents can then be obtained by differentiating the free
energy flux introduced in [4]

$\mathcal{G}=-\int_{\Gamma_{{\rm l}}}\frac{{\rm d}E(\mu)}{2\pi}\ n_{{\rm
l}}(\mu)-\int_{\Gamma_{{\rm r}}}\frac{{\rm d}E(\mu)}{2\pi}\ n_{{\rm r}}(\mu)\
.$ (82)

This expression is similar to that of the free energy density (59) in which
the momentum measure has been replaced by the energy measure. It yields fairly
natural results regarding the average of the currents

$\begin{aligned} \langle
j_{i}\rangle&=\frac{\partial\mathcal{G}}{\partial\beta_{n}}\\\
&=\int_{\Gamma_{{\rm l}}}\frac{{\rm
d}E(\mu)}{2\pi}n_{l}(\mu)h^{\rm{l,dr}}(\mu)+\int_{\Gamma_{{\rm r}}}\frac{{\rm
d}E(\mu)}{2\pi}n_{r}(\mu)h^{\rm{r,dr}}(\mu)\\\ &=\int_{\Gamma_{{\rm l}}}{\rm
d}\mu\ v^{{\rm eff}}_{{\rm l}}(\mu)\rho_{{\rm l}}(\mu)h^{{\rm
l}}(\mu)+\int_{\Gamma_{{\rm r}}}{\rm d}\mu\ v^{{\rm eff}}_{{\rm
r}}(\mu)\rho_{{\rm r}}(\mu)h^{{\rm r}}(\mu)\end{aligned}\ ,$ (83)

and the field-current correlator

$\begin{aligned} \mathsf{B}_{ab}&\equiv\int{\rm d}x\,\left(\left\langle
q_{a}(x)j_{b}(0)\right\rangle-\left\langle q_{a}(x)\right\rangle\left\langle
j_{b}(0)\right\rangle\right)=-\frac{\partial^{2}\mathcal{G}}{\partial\beta_{a}\partial\beta_{b}}\\\
&=\int_{\Gamma_{{\rm l}}}{\rm d}\eta\,v^{{\rm eff}}_{{\rm l}}(\eta)\rho_{{\rm
l}}(\eta)h_{a}^{\rm l,dr}(\eta)h_{b}^{\rm l,dr}(\eta)+\int_{\Gamma_{{\rm
r}}}{\rm d}\eta\,v^{{\rm eff}}_{{\rm r}}(\eta)\rho_{{\rm r}}(\eta)h_{a}^{\rm
r,dr}(\eta)h_{b}^{\rm r,dr}(\eta)\end{aligned}\ .$ (84)

Finally, the Drude weight characterising ballistic transport can be written,
via the Kubo formula, in termsof a time integral of correlations

$\begin{aligned}
\mathsf{D}_{ab}&\equiv\lim_{t\to\infty}\frac{1}{2t}\int_{-t}^{t}{\rm
d}\tau\int{\rm d}x\ \left(\left\langle
j_{a}(x,\tau)j_{b}(0,0)\right\rangle-\left\langle
j_{a}(x,\tau)\right\rangle\left\langle j_{b}(0,0)\right\rangle\right)\\\
&=\int_{\Gamma_{{\rm l}}}{\rm d}\eta\,\left(v^{{\rm eff}}_{{\rm
l}}(\eta)\right)^{2}\rho_{{\rm l}}(\eta)h_{a}^{\rm l,dr}(\eta)h_{b}^{\rm
l,dr}(\eta)+\int_{\Gamma_{{\rm r}}}{\rm d}\eta\,\left(v^{{\rm eff}}_{{\rm
r}}(\eta)\right)^{2}\rho_{{\rm r}}(\eta)h_{a}^{\rm r,dr}(\eta)h_{b}^{\rm
r,dr}(\eta)\end{aligned}\ .$ (85)

###  4.4 Euler Hydrodynamics

We shall now consider a weakly non-homogeneous, out of equilibrium gas in
which space-time variations of the DOS’s and of the effective velocities occur
over macroscopic scales that are much larger than the typical width of a
soliton. In that case we may, as is standard in hydrodynamic theories [38],
assume local entropy maximisation and propagation of such local equilibria.

The main ingredient in our hydrodynamic construction is the infinite set of
local conservation laws

$\partial_{t}q_{n}+\partial_{x}j_{n}=0\ ,$ (86)

induced by the integrability structure of the Boussinesq equation. As an
illustration, recalling equations (6), we present here the first three charge
and current densities

$\displaystyle q_{1}=u\ ,\ \ \ \ \quad\quad j_{1}=w_{x}\ ,$ (87)
$\displaystyle q_{2}=w_{x}\ ,\ \ \quad\quad j_{2}=-u+6u^{2}+u_{xx}\ ,$
$\displaystyle q_{3}=uw_{x}\ ,\quad\quad
j_{3}=\frac{(w_{x})^{2}+(u_{x})^{2}}{2}-\frac{u^{2}}{2}-4u^{3}-uu_{xx}\ .$

We then assume our system can be divided into mesoscopic, thermodynamically
large fluid cells, that are still small compared to the scales associated with
the variations of the statistical properties of the soliton gas. In other
words, we work under the hydrodynamic approximation: we assume that the
averages of local observables can be well approximated, at large times, by
averages evaluated in local GGEs

$\langle o(x,t)\rangle\approx\langle
o\rangle_{w(\eta,\epsilon;x,t)}\equiv\bar{o}_{n}(x,t)\;.$ (88)

We may now write mesoscopic conservation equations by averaging the
microscopic laws (86) over such a fluid cell444Formally we integrate equations
(86) over a contour $[0,X]\times[0,T]$, make the substitutions
$\frac{1}{T}\int_{0}^{T}q_{n}{\rm d}t\approx\bar{q}_{n}$ and
$\frac{1}{X}\int_{0}^{X}q_{n}{\rm d}t\approx\bar{q}_{n}$ according to the law
of large numbers, and end up with equations (89) provided $\bar{q}_{n}$ and
$\bar{j}_{n}$ are differentiable.

$\partial_{t}\bar{q}_{n}(x,t)+\partial_{x}\bar{j}_{n}(x,t)=0\ .$ (89)

Since $\bar{q}_{n}(x,t)$ and $\bar{j}_{n}(x,t)$ are related to the DOS’s and
effective velocities through equations (79)-(83), provided the set of
$\\{h_{n}\\}$’s is complete (which should be the case since the scattering of
solitons is 2-body reducible [3]) one eventually obtains the fundamental
equation of (Euler scale) GHD

$\partial_{t}\rho_{\cdot}(\eta;x,t)+\partial_{x}\left[v^{{\rm
eff}}_{\cdot}(\eta;x,t)\rho_{\cdot}(\eta;x,t)\right]=0\ .$ (90)

##  5 Conclusion

In this paper, we have constructed the generalised hydrodynamics of the
soliton gas associated with the Boussinesq integrable PDE, a theory for a
field lying in one dimension of space and one of time. We have constructed its
hydrodynamics – the kinetic equation – and thermodynamics – the free energy.
We have done this by simply following the general intuitive rules from soliton
gases [2] and generalised hydrodynamics [4, 5] according to which both the
hydrodynamics and thermodynamics can be obtained from the known two-soliton
scattering shifts. This follows, and in some sense generalises, the analysis
done for the soliton gas of the KdV equation in [34] to bidirectional models.
In particular, it was important to correctly account for the choice of the
“momentum function” in order to construct the thermodynamics. By contrast to
the KdV case, in the Boussinesq case the most natural momentum function, with
respect to the known soliton constructions, is not the physical momentum.

We note that the soliton phenomenology of the Boussinesq equation is
significantly more complex than that of the KdV equation. But in some regimes
of parameters, on which we have concentrated, it is nevertheless well
understood, and, interestingly, includes both head-on and overtaking
interactions. In certain cases, non-soliton-conserving, yet integrable,
interactions can occur, but, in the chosen regime of parameters, they cannot
occur in macroscopic number, hence to not affect the hydrodynamics or
thermodynamics. A natural extension of the present work would be to analyse
other regimes of parameters, where it may be possible to include a macroscopic
number of non-soliton-conserving interactions. However, under different
regimes of parameters, the $N-$soliton solution is known to develop
singularities in finite time [97] that may only remain for a finite duration.
One would then need to develop a better understanding of the physical meaning
of such singularities, as well as carefully study their impact on the
thermodynamics of the gas. Alternatively, one may have to work with non-zero
background, which seems to make the phenomenology even richer as observed in
[100], a priori allowing for more than two resonances while keeping the
solution regular at all time. Going beyond the simple Boussinesq equation,
this could prove significant for GHD in general as a prototypical example of
an integrable model in which the quasi-particles are unstable and in which
their number is not conserved.

On top of all that, it should be possible to extend the construction presented
in this paper by straightforwardly adapting known GHD results. GHD, just like
conventional hydrodynamics, is fundamentally a derivative expansion. While the
equation (90) generalises the Euler equations to account for integrable
systems, it is possible to compute an additional diffusion operator [52, 53]
(or even dispersion [55]), which would transform equation (90) into the GHD
equivalent of the Navier-Stokes equation. It is also possible to probe
integrability breaking, either by adding an external potential [101] or
through space-time inhomogeneous interactions [47], which would result in the
addition of an “effective acceleration” term in equation (90). This suggests
it should be possible to ultimately adapt the soliton gas approach developed
in this paper to study the generalisation of the Boussinesq equation
introduced in [76] as a model for neural activity.

Beyond that the primary interest of developing the soliton gas theory for the
Boussinesq equation is its eventual application to the KP equation, a PDE for
a field in two dimensions of space. Indeed, it is known, as we have recalled,
that certain “essentially stationary” solutions of the KP equation (stationary
in Galilean frames with non-zero velocities) in fact are simply space-time
solutions of the Boussinesq equation, where the Boussinesq time is identified
with one of the KP space directions. This relation, which we study in more
details in [86], will lay the ground for developing the soliton gas theory for
the KP equation, and will help reveal the structure of the theory beyond
models in one dimension of space.

##  Appendix A GHD of the “bad” Boussinesq soliton gas

Generalisation of the previous construction to the bad Boussinesq equation is
fairly straightforward once the main building blocks – namely the scattering
shifts, the momentum and energy functions, and the dressing operations – have
been provided.

First of all, the bad Boussinesq equation admits $N-$soliton $\tau-$functions
in the form (13) except, this time, the phase is given by

$\theta_{i}(x,t)=\eta_{i}\left(x-\epsilon_{i}t\sqrt{1+\eta_{i}^{2}}-x_{i}^{0}\right)\;,$
(91)

and the phase shift by

$\varphi_{ij}=\log\frac{\left(\epsilon_{i}\sqrt{1+\eta_{i}^{2}}-\epsilon_{j}\sqrt{1+\eta_{j}^{2}}\right)^{2}+3(\eta_{i}-\eta_{j})^{2}}{\left(\epsilon_{i}\sqrt{1+\eta_{i}^{2}}-\epsilon_{j}\sqrt{1+\eta_{j}^{2}}\right)^{2}+3(\eta_{i}+\eta_{j})^{2}}\;.$
(92)

The main difference here is that the argument of the square root, in both the
expression of the soliton velocity and of the phase shift, has changed from
$(1-\eta^{2})$ to $(1+\eta^{2})$. This, along the fact that the argument of
the $\log$, in the expression of the phase shift, is now a ratio of sums of
squares, and not of differences, means that the $N-$soliton solution remains
regular at all time for any set of real spectral parameters. Despite the bad
Boussinesq being ill-posed for generic arbitrary initial data, conditions for
the regularity of its $N-$soliton solution are less restrictive than they are
for the good Boussinesq equation. Additionally, the phenomenology is not as
rich, the phase shift is always negative and there are no bound states.

Then, by convention we choose the same momentum function we did earlier for
the good Boussinesq equation which, given the velocity of solitons, also fixes
the energy function

$P(\eta,\epsilon)=\epsilon\frac{\eta^{2}}{2}\ ,\quad
E(\eta)=\frac{1}{3}\left[1+\eta^{2}\right]^{3/2}\ .$ (93)

The dressing operation takes again the form (65), changing the phase shift
accordingly, for which infer the NDRs

$\left\\{\begin{aligned} &\sigma_{{\rm l}}(\eta)\rho_{{\rm
l}}(\eta)=(\eta)^{\rm{l,dr}}(\eta)\\\ &\sigma_{{\rm r}}(\eta)\rho_{{\rm
r}}(\eta)=(\eta)^{\rm{r,dr}}(\eta)\end{aligned}\right.\quad\text{and}\quad\left\\{\begin{aligned}
&\sigma_{{\rm l}}(\eta)f_{{\rm
l}}(\eta)=-\left(\eta\sqrt{1+\eta^{2}}\right)^{\rm{l,dr}}(\eta)\\\
&\sigma_{{\rm r}}(\eta)f_{{\rm
r}}(\eta)=\left(\eta\sqrt{1+\eta^{2}}\right)^{\rm{r,dr}}(\eta)\end{aligned}\right.\
,$ (94)

which yields again an effective velocity of form (71) by taking the ratio
$v^{{\rm eff}}_{\cdot}\equiv f_{\cdot}/\rho_{\cdot}$. Generalisation of any
other identity constructed for the good Boussinesq equation is then immediate.

## Acknowledgments

This work has benefited from discussions with Gino Biondini, Gennady El and
Giacomo Roberti. We are especially thankful for their comments regarding early
versions of this manuscript. The authors were supported by the Engineering and
Physical Sciences Research Council (EPSRC) under grant EP/W010194/1.

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