Dispersive hydrodynamics of soliton condensates for the Korteweg-de Vries equation

Solitons represent the fundamental localised solutions of integrable nonlinear dispersive equations such as the Korteweg-de Vries (KdV), nonlinear Schrödinger (NLS), sine-Gordon, Benjamin-Ono and other equations. Along with the remarkable localisation properties solitons exhibit particle-like elastic pairwise collisions accompanied by definite phase/position shifts. A comprehensive description of solitons and their interactions is achieved within the inverse scattering transform (IST) method framework, where each soliton is characterised by a certain spectral parameter related to the soliton’s amplitude, and the phase related to its position (for the sake of definiteness we refer here to the properties of KdV solitons). Generally, integrable equations support $N$-soliton solutions which can be viewed as nonlinear superpositions of $N$ solitons. Within the IST framework $N$-soliton solution is characterised by a finite set of spectral and phase parameters completely determined by the initial conditions for the integrable PDE.

The particle-like properties of solitons suggest some natural questions pertaining to the realm of statistical mechanics, e.g. one can consider a soliton gas as an infinite ensemble of interacting solitons characterised by random spectral (amplitude) and phase distributions. The key question is to understand the emergent macroscopic dynamics (i.e. hydrodynamics or kinetics) of soliton gas given the properties of the elementary, “microscopic” two-soliton interactions. It is clear that, due to the presence of an infinite number of conserved quantities and the lack of thermalisation in integrable systems the properties of soliton gases will be very different compared to the properties of classical gases whose particle interactions are non-elastic. Invoking the wave aspect of the soliton’s dual identity, soliton gas can be viewed as a prominent example of integrable turbulence [1]. The pertinent questions arising in this connection are related to the determination of the parameters of the random nonlinear wave field in the soliton gas such as probability density function, autocorrelation function, power spectrum etc.

The IST-based phenomenological construction of a rarefied, or diluted, gas of KdV solitons was proposed in 1971 by V. Zakharov [2] who has formulated an approximate spectral kinetic equation for such a gas based on the properties of soliton collisions: the conservation of the soliton spectrum (isospectrality) and the accumulation of phase shifts in pairwise collisions that results in the modification of an effective average soliton’s velocity in the gas. Zakharov’s spectral kinetic equation was generalised in [3] to the case of a dense gas using the finite gap theory and the thermodynamic, infinite-genus, limit of the KdV-Whitham modulation equations [4]. The results of [3] were used in [5] for the formulation of a phenomenological construction of kinetic equations for dense soliton gases for integrable systems describing both unidirectional and bidirectional soliton propagation and including the focusing, defocusing and resonant NLS equations, as well as the Kaup-Boussinesq system for shallow-water waves [6]. The detailed spectral theory of soliton and breather gases for the focusing NLS equation has been developed in [7].

The spectral kinetic equation for a dense soliton gas represents a nonlinear integro-differential equation describing the evolution of the density of states (DOS)—the density function $u(\eta;x,t)$ in the phase space $(\eta,x)\in\Gamma^{+}\times\mathbb{R}$, where $\eta\in\Gamma^{+}\subset\mathbb{R}^{+}$ is the spectral parameter in the Lax pair associated with the nonlinear integrable PDE,

Here $s_{0}(\eta)$ is the velocity of a “free” soliton, and the integral term in the second equation represents the effective modification of the soliton velocity in the gas due to pairwise soliton collisions that are accompanied by the phase-shifts described by the kernel $G(\eta,\mu)$. Both $s_{0}(\eta)$ and $G(\eta,\mu)$ are system specific. In particular, for KdV $s_{0}=4\eta^{2}$ and $G(\eta,\mu)=\tfrac{1}{\eta}\ln\left|\tfrac{\mu+\eta}{\mu-\eta}\right|$. The spectral support $\Gamma^{+}$ of the DOS is determined by initial conditions. We note that, while $\Gamma^{+}\subset\mathbb{R}^{+}$ for the KdV equation, one can have $\Gamma^{+}\subset\mathbb{C}^{+}$ for other equatons, e.g. the focusing NLS equation, see [7] . Equation (1.1) describes the DOS evolution in a dense soliton gas and represents a broad generalisation of Zakharov’s kinetic equation for rarefied gas [2]. The existence, uniqueness and properties of solutions to the “equation of state” (the integral equation in (1.1) for fixed $x,t$) for the focusing NLS and KdV equations were studied in [8].

The original spectral theory of the KdV soliton gas [3] has been developed under the assumption that the spectral support $\Gamma^{+}$ of the DOS is a fixed, simply-connected interval of $\mathbb{R}^{+}$; without loss of generality one can assume $\Gamma^{+}=[0,1]$. In [7], [8] this restriction has been removed by allowing the spectral support $\Gamma^{+}$ to be a union of $N+1$ disjoint intervals $\gamma_{j}=[\lambda_{2j-1},\lambda_{2j}]$, termed here s-bands: $\Gamma^{+}=\cup_{j=0}^{N}\gamma_{j}$, ($\gamma_{i}\cap\gamma_{j}=\emptyset$, $i\neq j$). In this paper we introduce a further generalization of the existing theory by allowing the endpoints $\lambda_{i}$ of the s-bands be functions of $x,t$. We show that this generalization has profound implications for soliton gas dynamics, in particular, the kinetic equation implies certain nonlinear evolution of the endpoints $\lambda_{j}(x,t)$ of the s-bands. We determine this evolution for a special type of soliton gases, termed in [7] soliton condensates. Soliton condensate represents the “densest possible” gas whose DOS is uniquely defined by a given spectral support $\Gamma^{+}$. The number $N$ of disjoint s-bands in $\Gamma^{+}$ determines the genus $g=N-1$ of the soliton condensate. We show that the evolution of $\lambda_{j}$’s in a soliton condensate is governed by the $g$-phase averaged KdV-Whitham modulation equations [4], also derived in the context of the semi-classical, zero-dispersion limit of the KdV equation [9].

We then consider the soliton condensate dynamics arising in the Riemann problem initiated by a rapid jump in the DOS. Our results suggest that in the condensate limit the KdV dynamics of soliton gas is almost everywhere equivalent to the (deterministic) generalised rarefaction waves (RWs) and generalized dispersive shock waves (DSWs) of the KdV equation. We prove this statement for the genus zero case and present a strong numerical evidence for genus one. Our results also suggest direct connection of the “deterministic KdV soliton gases” constructed in the recent paper [10] with modulated soliton condensates.

Our work puts classical results of integrable dispersive hydrodynamics (Flaschka-Forest-McLaughlin [4], Lax-Levermore [9], Gurevich-Pitaevskii [11]) in a broader context of the soliton gas theory. Namely, we show that the KdV-Whitham modulation equations describe the emergent hydrodynamic motion of a special soliton gas—a condensate—resulting from the accumulated effect of “microscopic” two-soliton interactions. This new interpretation of the Whitham equations is particularly pertinent in the context of generalized hydrodynamics, the emergent hydrodynamics of quantum and classical many-body systems [52]. The direct connection between the kinetic theory of KdV soliton gas and generalized hydrodynamics was established recently in [53] (see also [54] where the Whitham equations for the defocusing NLS equation were shown to arise in the semi-classical limit of the generalised hydrodynamics of the quantum Lieb-Liniger model).

Our work also paves the way to a major extension of the existing dispersive hydrodynamic theory by including the random aspect of soliton gases. To this end we consider “diluted” soliton condensates whose DOS has the same spectral distribution as in genuine condensates but allows for a wider spacing between solitons giving rise to rich incoherent dynamics associated with “integrable turbulence” [1]. In particular, we show numerically that evolution of initial discontinuities in diluted soliton condensates results in the development of incoherent oscillating rarefaction and dispersive shock waves.

An important aspect of our work is the numerical modelling of soliton condensates using $n$-soliton KdV solutions with large $n$ configured according to the condensate density of states. The challenges of the numerical implementation of standard $n$-soliton formulae for sufficiently large $n$ due to rapid accumulation of roundoff errors are known very well. Here we use the efficient algorithm developed in [45], which relies on the Darboux transformation. We improve this algorithm following the recent methodology developed in [46] for the focusing NLS equation with the implementation of high precision arithmetic routine. Our numerical simulations show excellent agreement with analytical predictions for the solutions of soliton condensate Riemann problems and provide a strong support to the basic conjecture about the connection of KdV soliton condensates with finite-gap potentials.

It should be noted that soliton condensates have been recently studied for the focusing NLS equation, where they represent incoherent wave fields exhibiting distinct statistical properties. In particular, it was shown in [55] that the so-called bound state soliton condensate dynamics underies the long-term behavior of spontaneous modulational instability, the fundamental physical phenomenon that gives rise to the statistically stationary integrable turbulence [56, 57].

The paper is organised as follows. In Section 2 we present a brief outline of the spectral theory of soliton gas for the KdV equation and introduce the notion of soliton condensate for the simplest genus zero case. In Section 3, following [8], we generalize the spectral definition of soliton condensate to an arbitrary genus case and prove the main Theorem 3.2 connecting spectral dynamics of non-uniform soliton condensates with multiphase Whitham modulation theory [4] describing slow deformations of the spectrum of periodic and quasiperiodic KdV solutions. Section 3 is concerned with properties of KdV solutions corresponding to the condensate spectral DOS, i.e. the soliton condensate realizations. We formulate Conjecture 4.1 that any realization of an equilibrium soliton condensate almost surely coincides with a finite-gap potential defined on the condensate’s hyperelliptic spectral curve. This proposition is proved for genus zero condensates and a strong numerical evidence is provided for genus one and two. In Section 5 we construct solutions to Riemann problems for the soliton gas kinetic equation subject to discontinuous condensate initial data. These solutions describe evolution of generalized rarefaction and dispersive shock waves. In Section 6 we present numerical simulations of the Riemann problem for the KdV soliton condensates and compare them with analytical solutions from Section 5. Finally, in Section 7 we consider basic properties of “diluted” condensates having a scaled condensate DOS and exhibiting rich incoherent behaviors. In particular, we present numerical solutions to Riemann problems for such diluted condensates. Appendix A contains details of the numerical implementation of dense soliton gases. In Appendix B we present results of the numerical realization of the genus 2 soliton condensate and its comparison with two-phase solution of the KdV equation.

Here we present an outline of the spectral theory of KdV soliton gas developed in [3, 12]. We consider the KdV equation in the form

The inverse scattering theory associates soliton of the KdV equation (2.1) with a point $z=z_{1}=-\eta_{1}^{2}$, $\eta_{1}>0$ of the discrete spectrum of the Lax operator

with sufficiently rapidly decaying potential $\varphi(x,t)$: $\varphi(x,t)\to 0$ as $|x|\to\infty$. The corresponding KdV soliton solution is given by

where the soliton amplitude $a_{1}=2\eta_{1}^{2}$, the speed $s_{1}=4\eta_{1}^{2}$, and $x_{1}^{0}$ is the initial position or ‘phase’. Along with the simplest single-soliton solution (2.3) the KdV equation supports $N$-soliton solutions $\varphi_{n}(x,t)$ characterized by $n$ discrete spectral parameters $0<\eta_{1}<\eta_{2}<\dots<\eta_{n}$ and the set of initial positions $\{x_{i}^{0}|i=1,\dots,n\}$ associated with the phases of the so-called norming constants [13]. It is also known that $n$-soliton solutions can be realized as special limits of more general $n$-gap solutions, whose Lax spectrum $\mathcal{S}_{n}$ consists of $N$ finite and one semi-infinite bands separated by $n$ gaps [13],

The $n$-gap solution of the KdV equation (2.1) represents a multiphase quasiperiodic function

where $k_{j}$ and $\omega_{j}$ are the wavenumber and frequency associated with the $j$-th phase $\theta_{j}$, and $\theta_{j}^{0}$ are the initial phases. Details on the explicit representation of the solution (2.5) in terms of Riemann theta-functions can be found in classical papers and monographs on finite-gap theory, see [59] and references therein.

The $n$-phase ($n$-gap) KdV solution (2.5) is parametrized by $2n+1$ spectral parameters—the endpoints $\{\zeta_{j}\}_{j=1}^{2n+1}$ of the spectral bands. The nonlinear dispersion relations (NDRs) for finite gap potentials can be represented in the general form, see [4] for the concrete expressions,

—and connect the wavenumber-frequency set $\{k_{j},\omega_{j}\}_{j=1}^{n}$ of (2.5) with the spectral set $\mathcal{S}_{n}$ (2.4). These are complemented by the relation $\langle{\varphi}\rangle=\Phi(\zeta_{1},\dots,\zeta_{2n+1})$, where $\langle\varphi\rangle=\int F_{n}d\theta_{1}\dots d\theta_{n}$ is the mean obtained by averaging of $F_{n}$ over the phase $n$-torus $\mathbb{T}^{n}=[0,2\pi)\times\dots\times[0,2\pi)$, assuming respective non-commensurability of $k_{j}$’s and $\omega_{j}$’s and, consequently, ergodicity of the KdV flow on the torus.

The $n$-soliton limit of an $n$-gap solution is achieved by collapsing all the finite bands $[\zeta_{2j-1},\zeta_{2j}]$ into double points corresponding to the soliton discrete spectral values,

It was proposed in [3] that the special infinite-soliton limit of the spectral $n$-gap KdV solutions, termed the thermodynamic limit, provides spectral description the KdV soliton gas. The thermodynamic limit is achieved by assuming a special band-gap distribution (scaling) of the spectral set $\mathcal{S}_{n}$ for $n\to\infty$ on a fixed interval $[\zeta_{1},\zeta_{2n+1}]$ (e.g. $[-1,0]$). Specifically, we set the spectral bands to be exponentially narrow compared to the gaps so that $\mathcal{S}_{n}$ is asymptotically characterized by two continuous nonnegative functions on some fixed interval $\Gamma^{+}\subset\mathbb{R}^{+}$: the density $\phi(\eta)$ of the lattice points $\eta_{j}\in\Gamma^{+}$ defining the band centers via $-\eta_{j}^{2}=(\zeta_{2j}+\zeta_{2j-1})/2$, and the logarithmic bandwidth distribution $\tau(\eta)$ defined for $n\to\infty$ by

The scaling (2.8) was originally introduced by Venakides [14] in the context of the continuum limit of theta functions.

Complementing the spectral distributions (2.8) with the uniform distribution of the initial phase vector ${\boldsymbol{\theta}}^{0}$ on the torus $\mathbb{T}^{n}$ we say that the resulting random finite gap solution $\varphi(x,t)$ approximates soliton gas as $n\to\infty$. An important consequence of this definition of soliton gas is ergodicity, implying that spatial averages of the KdV field in a soliton gas are equivalent to the ensemble averages, i.e. the averages over $\mathbb{T}^{n}$ in the thermodynamic limit $n\to\infty$. We shall use the notation $\langle F[\varphi]\rangle$ for ensemble averages and $\overline{F[\varphi]}$ for spatial averages.

From now on we shall refer to $\eta$ as the spectral parameter and $\Gamma^{+}$–the spectral support. The density of states (DOS) $u(\eta)$ of a spatially homogeneous (equilibrium) soliton gas is phenomenologically introduced in such a way that $u(\eta_{0})d\eta dx$ gives the number of solitons with the spectral parameter $\eta\in[\eta_{0};\eta_{0}+d\eta]$ contained in the portion of soliton gas over a macroscopic (i.e. containing sufficiently many solitons) spatial interval $x\in[x_{0},x_{0}+dx]\subset\mathbb{R}$ for any $x_{0}$ (the individual solitons can be counted by cutting out the relevant portion of the gas and letting them separate with time). The corresponding spectral flux density $v(\eta)$ represents the temporal counterpart of the DOS i.e. $v(\eta_{0})d\eta$ is the number of solitons with the spectral parameter $\eta\in[\eta_{0};\eta_{0}+d\eta]$ crossing any given point $x=x_{0}$ per unit interval of time. These definitions are physically suggestive in the context of rarefied soliton gas where solitons are identifiable as individual localized wave structures. The general mathematical definitions of $u(\eta)$ and $v(\eta)$ applicable to dense soliton gases are introduced by applying the thermodynamic limit to the finite-gap NDRs (2.6), leading to the integral equations [3, 12]:

for all $\eta\in\Gamma^{+}$. Here the spectral scaling function $\sigma:\Gamma^{+}\to[0,\infty)$ is a continuous non-negative function that encodes the Lax spectrum of soliton gas via $\sigma(\eta)=\phi(\eta)/\tau(\eta)$. Equations (2.9), (2.10) represent the soliton gas NDRs.

Eliminating $\sigma(\eta)$ from the NDRs (2.9), (2.10) yields the equation of state for KdV soliton gas:

where $s(\eta)=v(\eta)/u(\eta)$ can be interpreted as the velocity of a tracer soliton in the gas. It was shown in [3] that for a weakly non-uniform (non-equilibrium) soliton gas, for which $u(\eta)\equiv u(\eta;x,t)$, $s(\eta)\equiv s(\eta;x,t)$, the DOS satisfies the continuity equation

so that $s(\eta;x,t)$ acquires the natural meaning of the transport velocity in the soliton gas. Equations (2.12), (2.11) form the spectral kinetic equation for soliton gas. One should note that the typical scales of spatio-temporal variations in the kinetic equation (2.12) are much larger than in the KdV equation (2.1), i.e. the kinetic equation describes macroscopic evolution, or hydrodynamics, of soliton gases.

Let the spectral support $\Gamma^{+}$ be fixed. Then, differentiating equation (2.9) with respect to $t$, equation (2.10) with respect to $x$, and using the continuity equation (2.12) we obtain the evolution equation for the spectral scaling function

which shows that $\sigma(\eta;x,t)$ plays the role of the Riemann invariant for the spectral kinetic equation.

Finally, the ensemble averages of the conserved densities of the KdV wave field in the soliton gas (the Kruskal integrals) are evaluated in terms of the DOS as $\langle{\cal P}_{n}[\varphi]\rangle=C_{n}\int_{\Gamma^{+}}\eta^{2n-1}u(\eta)d\eta$, where ${\cal P}_{n}[\varphi]$ are conserved quantities of the KdV equation and $C_{n}$ constants [3, 12] (see also [15] for rigorous derivation in the NLS context). In particular, for the two first moments we have, on dropping the $x,t$-dependence [3, 12],

We note that in the original works on KdV soliton gas it was assumed (explicitly or implicitly) that the spectral support $\Gamma^{+}$ of the KdV soliton gas is a fixed, simply connected interval (without loss of generality one can assume that in this case $\Gamma^{+}=[0,1]$). In what follows we will be considering a more general configuration where $\Gamma^{+}$ represents a union of $N+1$ disjoint intervals.

A special kind of soliton gas, termed soliton condensate, is realized spectrally by letting $\sigma\to 0$ in the NDRs (2.9), (2.10). This limit was first considered in [7] for the soliton gas in the focusing NLS equation and then in [8] for KdV. Loosely speaking, soliton condensate can be viewed as the “densest possible” gas (for a given spectral support $\Gamma^{+}$) whose properties are fully determined by the interaction (integral) terms in the NDRs (2.9), (2.10).

For the KdV equation, setting $\sigma=0$ and, considering the simplest case $\Gamma^{+}=[0,1]$ in (2.9), (2.10), we obtain the soliton condensate NDRs [12]:

These are solved by

as verified by direct substitution (it is advantageous to first differentiate equations (2.15) with respect to $\eta$). The formula (2.16) for $u(\eta)$ is sometimes called the Weyl distribution, following the terminology from the semiclassical theory of linear differential operators [9].

We now consider the general case of the soliton gas NDRs (2.9), (2.10) by letting the support $\Gamma^{+}\subset\mathbb{R}^{+}$ of $u(\eta),v(\eta)$ to be a union of disjoint intervals $\gamma_{k}\subset{\mathbb{R}}^{+}$ with endpoints $\lambda_{j}>0$, $j=1,2,\dots,2N+1$, where $\gamma_{0}=[0,\lambda_{1}]$ and $\gamma_{k}=[\lambda_{2k},\lambda_{2k+1}]$, $k=1,\dots,N$, i.e.

We shall call the intervals $\gamma_{k}$ the s-bands, and the soliton gas spectrally supported on $\Gamma^{+}$ (3.1)— the genus $N$ soliton gas. Correspondingly, we refer to the intervals $c_{j}=(\lambda_{2j-1},\lambda_{2j})$ separating the s-bands as to s-gaps. Note that the s-bands and s-gaps are different from the original bands and gaps in the spectrum $\mathcal{S}_{n}$ of finite-gap potential (cf. (2.4)) as they emerge after the passage to the thermodynamic limit: loosely speaking, one can view the s-bands as a continuum limit of the “thermodynamic band clusters”, each representing an isolated dense subset of $\mathcal{S}_{\infty}$ consisting of the collapsing original bands. The existence and uniqueness of solutions $u(\eta)$, $v(\eta)$ for (2.9), (2.10) respectively, as well as the fact that $u(\eta)\geq 0$ on $\Gamma^{+}$ with some mild constraints, was established in [8]. Our goal here is to find explicit expressions for $u,v$ for the genus $N$ soliton condensate, that is, solutions of (2.9), (2.10) for the particular case ${\sigma}\equiv 0$ on $\Gamma^{+}$.

Denote by $\Gamma^{-}$ the symmetric image of $\Gamma^{+}$ with respect to the origin, i.e., $\Gamma^{-}=-\Gamma^{+}$. If we take the odd continuation of $u,v$ to $\Gamma^{-}$ (preserving the same notations), we observe that equations (2.9), (2.10) become

where $\Gamma:=\Gamma^{+}\cup\Gamma^{-}$, for all $\eta\in\Gamma^{+}$. In fact, if we symmetrically extend ${\sigma}(\eta)$ from $\Gamma^{+}$ to $\Gamma$, equations (3.2), (3.3) should be valid on $\Gamma$ since every term in these equations is odd. The expressions (2.14) for the first two moments (ensemble averages) of the KdV wave field in the soliton gas become

We now consider soliton condensate of genus $N$ by setting ${\sigma}\equiv 0$ in (3.2), (3.3). Then, differentiating in $\eta$ we obtain

where $H$ denotes the Finite Hilbert Transform (FHT) on $\Gamma$, see for example [17], [18],

Equations (3.5) are the (transformed) NDRs for the KdV soliton condensate.

To find $u,v$ for the soliton condensate, it is sufficient to invert the FHT $H$ on $\Gamma$. Denote by $\mathcal{R}_{2N}$ the hyperelliptic Riemann surface of the genus $2N$, defined by the branchcuts (s-bands) $\gamma_{k}$, $k=0,\pm 1,\dots,\pm N$, where $\gamma_{-k}=-\gamma_{k}$. Define two meromorphic differentials of second kind, $dp$ and $dq$ on $\mathcal{R}_{2N}$ by

where

and $P,Q$ are odd monic polynomials of degree $2N+1$ and $2N+3$ respectively that are chosen so that all their s-gap integrals are zero, i.e.

Equivalently, one can say that $dp,dq$ are real normalized differentials. Note that Equations (3.7), (3.9) uniquely define $dp,dq$.

Theorem 3.1 for $u$ was proven in [8], Section 6, for the so-called bound state soliton condensate. The proof for KdV is analogous. The proof for $v$ goes along the same lines, except $v(\eta)$ attains different signs.

Thus, for the soliton condensate of genus $N$, we obtain, on using Theorem 3.1 and Equation (3.7),

The velocity of a tracer soliton with the spectral parameter $\eta\in\Gamma^{+}$ propagating in the soliton condensate with DOS $u(\eta)$ is then found as

As an illustrative example we present in Fig. 1 the plots of the DOS and tracer velocity for the genus 2 soliton condensate.

We now consider slow modulations of non-equilibrium (non-uniform) soliton condensates by assuming $u\equiv u(\eta;x,t)$, $v\equiv v(\eta;x,t)$, $\Gamma\equiv\Gamma(x,t)$. Equations (2.12), (3.10) then yield the kinetic equation for genus $N$ soliton condensate:

that is valid for $\eta\in\Gamma=\cup_{k=-N}^{N}(\gamma_{k})$. The velocity (3.11) then assumes the meaning of the tracer, or transport, velocity in a non-uniform genus $N$ soliton condensate.

Proof. (See [19]) Multiplying (3.12) by $(\eta^{2}-\lambda_{j}^{2})^{3/2}$ and passing to the limit $\eta\to\lambda_{j}$ we obtain equations (3.13), (3.14) for the evolution of the spectral $s$-bands (i.e. the evolution of $\Gamma(x,t)$). These are the KdV-Whitham modulation equations [4], [9] (see also Remark 3.2 below).

∎

The finite-genus Whitham modulation system (3.13), (3.14) can be viewed as an exact hydrodynamic reduction of the full kinetic equation (2.12), (2.11) under the ansatz (3.10), (3.11). Recalling the origin of the soliton gas kinetic equation as a singular, thermodynamic limit of the Whitham equations [3] the recovery of the finite-genus Whitham dynamics in the condensate limit might not look surprising. On the other hand, viewed from the general soliton gas perspective the condensate reduction notably shows that the highly nontrivial nonlinear modulation (hydro)dynamics emerges as a collective effect of the elementary two-soliton scattering events. This understanding is in line with ideas of generalised hydrodynamics, a powerful theoretical framework for the description of non-equilibrium macroscopic dynamics in many-body quantum and classical integrable systems [21]. The connection of the KdV soliton gas theory with generalised hydrodynamics has been recently established in [22]. Relevant to the above, it was shown in [23] that the semiclassical limit of the generalised hydrodynamics for the Lieb-Liniger model of Bose gases yields the Whitham modulation system for the defocusing NLS equation.

A different type of hydrodynamic reductions of the soliton gas kinetic equation defined by the multi-component delta-function ansatz $u(\eta,x,t)=\sum_{i=1}^{m}w_{i}(x,t)\delta(\eta-\eta_{j})$ for the DOS has been studied in [24] for $\eta_{j}=const$ and in [25, 26] for $\eta_{j}=\eta_{j}(x,t)$. One of the definitive properties of the multicomponent hydrodynamic reductions of this type is their linear degeneracy which, in particular, implies the absence of the wavebreaking and the occurrence of contact discontinuities in the solutions of Riemann problems [27]. In contrast, the condensate (Whitham) system (3.13), (3.14) obtained under the condition $\sigma\equiv 0$ is known to be genuinely nonlinear, $\partial V_{j}/\partial\lambda_{j}\neq 0$, $j=1,\dots 2N+1$ [28] implying the inevitability of wavebreaking for general initial data, which is in stark contrast with linear degeneracy of the multicomponent “cold-gas” hydrodynamic reductions. Reconciling the genuine nonlinearity property of soliton condensates with linearly degenerate non-condensate multicomponent cold-gas dynamics is an interesting problem which will be considered in future publications.

Thus, we have shown that the spectral dynamics of soliton condensates are equivalent to those of finite gap potentials, which naturally suggests a close connection (or even equivalence) of these two objects at the level of realizations, i.e. the corresponding solutions $\varphi(x,t)$ of the KdV equation. This connection will be explored in the next section using a combination of analytical results and numerical simulations for genus 0 and genus 1 soliton condensates.

Having developed the spectral description of KdV soliton condensates, we now look closer at the two simplest representatives: genus 0 and genus 1 condensates. In particular, we shall be interested in the characterization of the realizations of soliton condensates, i.e. the KdV solutions, denoted $\varphi_{\rm c}^{(N)}(x,t)$, corresponding to the condensate spectral DOS $u^{(N)}(\eta)$ for $N=0,1$. We do not attempt here to construct the soliton gas realizations explicitly via the thermodynamic limit of finite gap potentials (see Section 2), instead, we infer some of their key properties from the expressions (3.4) for the ensemble averages as integrals over the spectral DOS. We then conjecture the exact form of soliton condensate realizations and support our conjecture by detailed numerical simulations.