Statistical Equilibrium of the Korteweg-de Vries and Benjamin-Ono Unidirectional Soliton Models

TL;DR

This paper investigates the statistical equilibrium of unidirectional soliton models described by Korteweg-de Vries and Benjamin-Ono equations, addressing the challenge of describing thermal equilibrium in a dilute soliton gas.

Contribution

It introduces a stochastic source approach to characterize the equilibrium distribution of soliton momenta in these models.

Findings

Derived the equilibrium distribution of soliton momenta.

Demonstrated the role of stochastic sources in soliton thermalization.

Provided a framework for understanding soliton gas thermodynamics.

Abstract

The Korteweg-de Vries and Benjamin-Ono nonlinear wave equations can describe solitary waves, all of which propagate in the same direction and which emerge from collisions with their shapes unchanged. There are technical challenges to giving a description of the thermal equilibrium of a dilute gas of such solitons. These challenges can be overcome by considering a stochastic source of new solitons located at $x=0$. The total intensity and the momentum distributions of the solitons emitted by such sources determine the equilibrium distribution of soliton momenta for $x>0$.

Equations

$$u_t+6u{u}_x+u_{xxx}=0,$$

$$u(x,t)=\frac{c}{2}{\mathrm{sech}}^2\left[\frac{\sqrt{c}}{2}(x-x_0-ct)\right].$$

$$u_t+u{u}_x+H{u}_{xx}=0,$$

$$Hf(x)=\frac{1}{\pi }\mathcal{P}{\int }_{-\infty }^{\infty }d\xi \frac{f(\xi )}{x-\xi }.$$

$$H\cos (kx+\alpha )=\mathrm{sgn}(k)\cos \left(kx+\alpha -\frac{\pi }{2}\right).$$

$$u(x,t)=\frac{4c}{1+c^2(x-x_0-ct{)}^2},$$

$$H\frac{a}{a^2+x^2}=\frac{x}{a^2+x^2}.$$

$${\mathcal{H}}_{\mathrm{KdV}}$$

$${\mathcal{H}}_{\mathrm{BO}}$$

$$P=\int dx\frac{u^2}{2}.$$

$$-\frac{u_x^2}{2}+u^3=-\frac{c^3}{8}{\mathrm{sech}}^6\left(\frac{\sqrt{c}}{2}\xi \right)\left[{\sinh }^2\left(\frac{\sqrt{c}}{2}\xi \right)-1\right],$$

$$\int dv{\mathrm{sech}}^{6}v({\sinh }^{2}v-1)=-\frac{1}{20}(\cosh 4v+6\cosh 2v+13)\tanh v{\mathrm{sech}}^{4}v+C,$$

$$\cosh Nv$$

$${\mathrm{sech}}^{N}v$$

$$\tanh v$$

$$\int dv{\mathrm{sech}}^{4}v=\frac{1}{3}(\cosh 2v+2)\tanh v{\mathrm{sech}}^{2}v+C,$$

$$H{u}_x=\frac{\partial }{\partial x}(Hu);$$

$$\frac{u\left(H{u}_x\right)}{6}+\frac{u^3}{18}=\frac{(56-24c^2{x}^2)c^3}{9(1+c^2{x}^2{)}^3},$$

$$\int dv\frac{56-24v^2}{9(1+v^2{)}^3}=\frac{2v(19+9v^2)}{9(1+v^2{)}^2}+2{\tan }^{-1}v+C.$$

$$\int dv\frac{1}{{\left(1+v^2\right)}^2}=\frac{1}{2}\left(\frac{v}{1+v^2}+{\tan }^{-1}v\right)+C$$

$$\mathcal{N}=NL=L{\int }_0^{\infty }dPn(P).$$

$$n(P)=\frac{N}{\pi \sqrt{2T}}\exp \left(-\frac{P^2}{8\pi T}\right).$$

$$\mathcal{U}=L{\int }_0^{\infty }dPn(P)E(P)=\frac{\mathcal{N}T}{2},$$

$$S={\int }_0^{\infty }dPs(P).$$

$$\dot{\mathcal{N}}$$

$$0$$

$$s(P)dP=\frac{NP}{4{\pi }^2\sqrt{2T}}\exp \left(-\frac{P^2}{8\pi T}\right)dP.$$

$$S={\int }_0^{\infty }dPs(P)=\frac{N}{\pi }\sqrt{\frac{T}{2}}=\frac{N}{\pi }\sqrt{\frac{\mathcal{U}}{\mathcal{N}}}.$$

$$n(P)=\frac{5^{2/3}N}{\Gamma \left(\frac{3}{5}\right)T^{3/5}}\exp \left(-\frac{3^{5/3}{P}^{5/3}}{5T}\right).$$

$${\int }_0^{\infty }dv{v}^{\alpha }{e}^{-A{v}^{5/3}}=\frac{3\Gamma \left[\frac{3}{5}(1+\alpha )\right]}{5A^{\frac{3}{5}(1+\alpha )}},$$

$$\mathcal{U}=\frac{3\mathcal{N}T}{5^{11/15}},$$

$$\Delta =\log \frac{\sqrt{c_1}+\sqrt{c_2}}{\sqrt{c_1}-\sqrt{c_2}}>0,$$

Full text

arXiv:1902.04643

**Statistical Equilibrium of the Korteweg-de Vries and

Benjamin-Ono Unidirectional Soliton Models**

Brett Altschul111baltschu@physics.sc.edu

Department of Physics and Astronomy

University of South Carolina

Columbia, SC 29208

Abstract

The Korteweg-de Vries and Benjamin-Ono nonlinear wave equations can describe solitary waves, all of which propagate in the same direction and which emerge from collisions with their shapes unchanged. This creates a technical challenge to giving a description of the thermal equilibrium for a gas of such solitons, but the challenges can be overcome by considering a stochastic source of new solitons located at $x=0$. The total intensity and the momentum distributions of the solitons emitted by such sources determine the equilibrium distribution of soliton momenta for $x>0$.

1 Introduction

Nonlinear field equations that support solitary waves provide explicit field-theoretic models of extended particles, which propagate without changing their shapes. Through the study of such solitary waves, it is possible to make qualitative and quantitative determinations about how extended but still relatively localized bodies move and interact. These studies have important applications in fluid mechanics, plasma physics, solid state physics, and elementary particle physics.

Perhaps the most remarkable class of solitary waves are the solitons. The term solitons is meant in its oldest and strongest sense, to indicate not merely solitary waves that can propagate in isolation without changing shapes, but localized disturbances that pass through one-another and reemerge with their shapes unchanged. Soliton waveforms will be distorted while they are passing through each another, and although their original shapes are restored after the collision, their positions in space may be still shifted by the interactions. This can be interpreted as forward scattering induced by an effective potential between the soliton excitations. Moreover, some equations that support soliton solutions, such as the sine-Gordon equation $u_{tt}-u_{xx}+\sin u=0$ (using subscript notation for derivatives), actually have solutions in which the attractive potential between two solitons of different moieties may lead bound states—“breather” solutions that expand and contract as the solitons pass back and forth through each other.

However, the present work will be concerned with a simpler class of soliton-bearing theories. The sine-Gordon equation describes a relativistic theory for a scalar field in $1+1$ dimensions, but solitons also occur in nonrelativistic physics, such as in the dynamics of surface waves on a fluid interface. Another much-studied $(1+1)$-dimensional nonlinear wave equation is the Korteweg-de Vries (KdV) equation [1, 2]

[TABLE]

which was discovered—first numerically [3], then analytically [4, 5, 6, 7]—to support soliton solutions. The functional form for a single-soliton field profile is easily found by introducing a traveling ansatz $u=\phi(x-ct)$. The result is

[TABLE]

As is usual for a solitary wave, the fixed shape of the solution represents a balance between the dispersion originating from the third derivative term in the KdV equation and the equation’s nonlinearity.

It is evident from (2) that any speed $c>0$ is possible, with solitary waves of greater amplitudes moving more quickly. However, these solitons can only propagate in one direction—the right-moving direction, toward larger values of $x$. Unlike the sine-Gordon equation, which is second order in time and thus invariant under time reversal, the KdV equation and similar equations are only first order in $t$. An even simpler equation that supports the propagation of solitons, but only unidirectionally, is the first-order wave equation $u_{t}+u_{x}=0$, for which any $u=\phi(x-t)$ is a valid solution.

The conceptual framework that we will utilize here—of a “gas” of multiple solitonic excitations of a KdV system—owes a great deal to the work of Zakharov [8], which extracted the behavior of large numbers solitons simultaneously propagating in parallel. The conception of a soliton gas has proven useful both for analyses of disordered assemblages of many solitons [9], and for correlated ensembles of solitons forming nearly periodic lattices [10, 11]. At sufficient solitary wave densities, these systems can have an effectively hydrodynamic description [12]. Moreover, the statistical methods (using families of random variables [13]) used to study wave turbulence can also be applied to the soliton gas [14]. In such studies of turbulence and other similar contexts, the solutions of the KdV equation can also be used to construct a nonlinear filtering method, expanding random functions in KdV eigenmodes instead of Fourier modes [15].

The KdV equation and other closely related dynamical equations (such as the modified KdV equation, which has a $u^{2}$ in the nonlinear term [16]; or the periodic KdV equation, with periodic boundary conditions limiting long-wavelength behavior) continue to arise in many different ($1+1$)-dimensional systems—for example, in magnetized multicomponent plasmas [17, 18]. Because of the ongoing interest in such systems, analyses of the KdV equation and its relatives remain important areas of research.

Less well known than the KdV equation is the Benjamin-Ono (BO) equation [19, 20],

[TABLE]

where $H$ is the Hilbert transform operator, defined by the principal value integral

[TABLE]

The Hilbert transform is clearly nonlocal in coordinate space, but it has a particularly simple representation in Fourier space, commuting with the spatial derivative $\partial/\partial x$ and shifting the phases of waves by $\frac{\pi}{2}$, as in

[TABLE]

Note that this phase shift is similar to the the effect of spatial differentiation, except that it is not accompanied by multiplicative factor of $-|k|$. The phase shift due to the Hilbert transform is what makes the BO equation conservative, in contrast to the better-known Burgers’ equation [21, 22] $u_{t}+uu_{x}-u_{xx}=0$, which is a nonlinear generalization of the heat equation ($u_{t}-u_{xx}=0$) and thus entails dissipation.

The single-soliton solution of the BO equation is a Lorentzian

[TABLE]

for $c>0$. The key relations needed to demonstrate that (6) is a solution of the BO equation are that the Hilbert transform is translation invariant [as is evident from the definition (4)], as well as the specific Hilbert transform

[TABLE]

The formula (7), as well as many other Hilbert transforms, are tabulated in Ref. [23].

As with the KdV solitons, the field profiles (6) advance more rapidly if they have greater amplitudes. However, the character of soliton collisions are different between the two equations. The KdV solitons emerge from collisions with phase shifts, but the BO solitons do not [24]; although the BO solitons’ profiles are distorted while they overlap in space, their positions after the interaction are just what they would have been had the interactions not occurred at all. This will make the statistical analysis of a system of BO solitons relatively simple.

The KdV and BO equations are the simplest nontrivial examples of a large class of unidirectional wave equations that support soliton solutions. More complicated examples involve additional spatial derivatives and higher degrees of nonlinearity. As occurs in the BO equation, any term with an even number of derivatives will be accompanied by a Hilbert transform. (The first-order equation $u_{t}+u_{x}=0$ can be seen as a degenerate case at the bottom of the whole soliton-bearing equation hierarchy.) Moreover, aside from their simplicity, the KdV and BO equations are notable in that they were first studied not for their remarkable mathematical properties but as models for the behavior of real fluidic systems. The nonlinear $uu_{x}$ terms arise from the appearance of the convective fluid derivative $D\vec{u}/Dt=(\partial/\partial{t}+\vec{u}\cdot\vec{\nabla})\vec{u}$—where $\vec{u}$ is the fluid velocity field—in the Navier-Stokes equation for the momentum. In contrast, only once the importance of solitons was coming to be appreciated were the higher-order soliton-supporting equations discovered as theoretical generalizations of the KdV and BO equations.

Of course, the KdV and BO equations, as well as their more elaborate generalizations, are conventionally expressed in a partially nondimensionalized form. In a fluid mechanics context the dynamical field $u$ is ultimately proportional to the fluid velocity, and for both the KdV and BO solitons, the field profile $u(x,t)$ is proportional to the soliton speed $c$, times a dimensionless function. However, the partially nondimensionalized units of the field $u$ and speed $c$ are different between the equations. In the KdV equation, $u$ and $c$ have units of $($length$)^{-2}$. However, in the BO equation, the units of these quantities are $($length$)^{-1}$.

One general characteristic of the field equations that support solitonic solutions is that they are fully integrable and support infinite numbers of conservation laws. (See, for instance, Refs. [6, 25, 26] for the KdV equation and Ref. [27] for the BO equation.) That there can be no finite limit on the number of conserved quantities in these theories is evident from the following simple, nonrigorous argument. Suppose that a finite number of solitons are moving in the $x$-direction. In general, they will all have different amplitudes, and thus speeds. Given sufficient time, the separations between the solitons may be made arbitrarily large, and therefore the portion of the total energy that is tied up in the interactions between the solitons may be made arbitrarily small. Each soliton essentially has a conserved energy associated solely with its own structure and motion, and since there is no limit on the total number of solitons, this means that there cannot be any finite limit to the number of conservation laws either. (In contrast, for a system with solitary waves that are not true solitons, the solitary waves are never truly isolated from one-another, even at large physical separations. Interactions between such solitary waves are always accompanied by the emission of radiation, and even when they are arbitrarily far apart, the solitary waves will be slowly bleeding away their kinetic energies into finite-wavelength periodic modes. Thus—and unlike in the soliton case—the energies of such individual solitary waves are never perfectly conserved, unless there is just a single wave with absolutely nothing else for it to interact with.) The localized single-soliton energies are not the usual way in which the infinite families of conservations laws are normally expressed; but the point of this argument was merely to demonstrate the that existence of an unlimited number of conserved quantities is intimately tied to the occurrence of soliton behavior.

The object of this paper is to study systems with multiple solitons in thermal equilibrium. When the density of the solitons is low, they present an example of a gas of particles in one spatial dimension, with exactly solvable interactions. However, two facts make this analysis somewhat trickier than it might first appear: firstly, that the solitons are all moving in the same overall rightward direction; and secondly, that the collisions between solitons never actually alter their speeds. To deal with these complications, it will be useful to consider solutions of the field equations within a bounded region, with a stochastic source introducing new excitations at the left-hand end of the region of interest. The equilibrium distribution will then represent a balance between new solitons being generated by this stochastic source on the left, versus the solitons that are escaping away to the right. With this formalism, it will be possible to describe the statistical equilibrium of KdV and BO soliton systems quantitatively.

Moreover, throughout our analysis, it will be worthwhile to keep in mind the fact that nothing about the methodology depends specifically on the soliton distribution being thermal. The calculations could proceed in the same manner with a different momentum distribution, such as a power law distribution. We have selected the thermalized Boltzmann distribution partially just for the sake of definiteness, although the Boltzmann distribution is special in the sense that it represents the uniquely likeliest distribution of energies.

However, this work will not be a complete statistical mechanical treatment of either the KdV or BO equation, because (as alluded to above), in addition to their solitary waves, both equations support periodic wave train solutions. The periodic solutions of the KdV equation are known as cnoidal waves, because they involve the Jacobi elliptic function $\operatorname{cn}(z,m)$. Just as for the solitary waves, the phase speeds of the periodic waves depend on the wave amplitudes, because of the nonlinearity. In fact, the soliton solutions of both the KdV and BO equations may be found by taking the wavelengths of the associated periodic solutions to infinity.

The paper is organized as follows. In section 2, we present the energies and momenta of single-soliton solutions. Section 3 then introduces the formalism that will be needed for the analysis of theories that contain only right-moving solitonic particles. The final results for the BO equation are quite simple, because it turns out that the BO solitons have a conventional energy-momentum relation, and because the collisions between solitons do not actually modify their trajectories. Section 4 applies the same methodology to a gas of KdV solitons, which is a more complicated problem. The relationship between energy and momentum is more complicated for a KdV soliton, and pairs of KdV solitons can undergo nontrivial forward scattering processes. Finally, the conclusions and outlook for future work are summarized in section 5.

2 Soliton Energies and Momenta

For each equation that supports soliton solutions, there exists a infinite hierarchy of conserved integrals of the motion, involving progressively higher powers of the field $u$ and its spatial derivatives (as well as Hilbert transforms in the case of the BO equation and its generalizations). The two lowest-order conserved quantities represent the total momentum and total energy of the system. These first two conservation laws also exist for other conservative wave equations, but the higher-order laws that ensure solitonic behavior are typically absent. Interactions between solitary waves in more general theories are accompanied by the emission of radiation; energy is lost from the initial kinetic energies of the solitary waves into propagating wave trains. Moreover, while individual solitary waves may be topologically protected kinks (interpolating between different vacua), a collision between a kink and an anti-kink may result in complete annihilation into radiation. Nevertheless, because other theories that support solitary waves but not true solitons do still typically possess the first two conserved energy integrals, the techniques developed in this paper might also be useful for studying such theories, at least when the density of kinks (and thus the frequency of collisions) is low.

The energy integrals for the KdV and BO theories are

[TABLE]

From these, the energies of single-soliton configurations may be determined. (For the KdV case at least, ${\cal H}_{{\rm KdV}}$—treated as a functional of the field $u$—also provides a full Hamiltonian formulation of the system [28].)

There is also a momentum invariant, the same for each equation,

[TABLE]

For the KdV equation, the energy of the solitary wave may be found by direct integration. The integrand takes the form

[TABLE]

where $\xi=x-x_{0}-ct$. The indefinite integral

[TABLE]

along with the asymptotic forms, as $v\rightarrow\pm\infty$,

[TABLE]

give the energy of the KdV soliton as $E_{{\rm KdV}}=\frac{1}{5}c^{5/2}$. Since the overall energy functional ${\cal H}_{{\rm KdV}}$ is really only defined up to an overall multiplicative constant (related to—among other things—the mass density in a fluid flow problem), the key feature is the $c^{5/2}$ dependence of the energy on the velocity, which is determined by just the dimensions of the formula (8) for ${\cal H}_{{\rm KdV}}$.

The integral for the KdV soliton’s $P$ is simpler. Using the indefinite integral

[TABLE]

the result is $P_{{\rm KdV}}=\frac{1}{3}c^{3/2}$. Once again, the key fact is that, based on dimensional considerations, $P_{{\rm KdV}}\propto c^{3/2}$, so that $E_{{\rm KdV}}\propto(P_{{\rm KdV}})^{5/3}$. The normalizations of the field energy and momentum integrals are not entirely standardized in the literature, but the normalizations here have been selected so that they obey the correct velocity relation $c=\partial E/\partial P$ for particle-like excitations.

The same energy and momentum determination procedures may be carried out for the BO equation. For $H\,u_{x}$, we use the fact that the the Hilbert transform commutes with spatial differentiation:

[TABLE]

combined with the equations (6) and (7). Then the integrand in the energy $E_{{\rm BO}}$ is

[TABLE]

and the indefinite integral of this rational function gives

[TABLE]

So the result for the energy is $E_{{\rm BO}}=2\pi c^{2}$. For the momentum $P_{{\rm BO}}$, only the straightforward integral

[TABLE]

is required, giving $P_{{\rm BO}}=4\pi c$. Once again, the soliton velocity $c$ is equal to $\partial E/\partial P$. However, the BO case is also further simplified by that fact that $E_{{\rm BO}}\propto(P_{{\rm BO}})^{2}$, which is the usual relation for a nonrelativistic particle.

3 BO Statistics in the Steady State

Because of this greater simplicity, and also because the solitons of the BO equation do not experience any phase delays as a result of passing through each other, it will be easier to begin the statistical analysis by looking at the BO case. The theory of the exclusively solitonic BO excitations is almost equivalent to that of a gas of noninteracting particles in one space dimension. However, unlike in the standard kinetic theory of gasses, where taking the particles to be (effectively) noninteracting is an idealization that simplifies the analysis of the system, in this case, the solitons truly do not interact, except for the way they distort each other when they are physically overlapping. That actually complicates the formulation and interpretation of the theory, because the gas will never thermalize on its own. The initial distribution of soliton energies will remain unchanged in time; there are not even the instantaneous collisions that reassign momentum and energy in an ideal gas—which quickly move the ideal gas toward the overwhelmingly likely Maxwell-Boltzmann velocity distribution. A separate complication arises from the fact that the KdV and BO equations only support soliton propagation in one direction.

Both of these difficulties can be overcome by considering solitons not along the full $x$-axis, but only within a finite length $L$. Since the solitons are all moving rightward with different velocities, they will all eventually leave this finite interval. The solitary waves cannot be reflected, since the BO equation does not support leftward-moving solutions. Periodic boundary conditions could be imposed, but then the problem that the soliton energy and momentum distribution functions will never thermalize becomes acute. Instead, we shall therefore envision the interval of length $L$ being put in contact, at its left end ($x=0$), with a “thermal” reservoir of solitons, which enter the interval at random times, traverse the full interval with individual velocities $c_{i}$, and then leave the interval at $x=L$ after times $L/c_{i}$. In doing this, we will effectively be treating the solitary waves as pointlike particles. Although this idealization is not crucial for the BO equation, the assumption that the soliton gas is sufficiently dilute will be of greater practical importance in our subsequent analysis of the KdV soliton gas.

Understanding the necessary behavior of the thermal source at $x=0$ will be the main goal of this section. With a satisfactory model of this source, the full behavior of the system follows straightforwardly.

Before proceeding further, a few notation conventions are required. Since we shall be dealing with densities in the full two-dimensional position-momentum phase space, we adopt the following terminology. A total extrinsic quantity will be denoted by a capital script letter; the corresponding spatial density will be denoted by a capital italic letter, as will any other thermodynamically intrinsic quantity; and the density in momentum space will be lower case. For example, the total number of solitonic particles ${\cal N}$ contained in the interval $0\leq x\leq L$ may be expressed as

[TABLE]

The basic problem will be to determine what conditions must be satisfied in order for the distribution of solitons in the $0\leq x\leq L$ region to remain thermal. In a situation where the reservoir at $x=0$ is releasing a thermalized stream of soliton excitations, the momentum distribution of the solitons must have a Maxwellian form with temperature $T$. Using the BO relation $E=P^{2}/8\pi$ and normalizing the distribution function appropriately, this yields,

[TABLE]

Direct calculation of the total internal energy

[TABLE]

produces the expected result that (since $E$ is a quadratic function of $P$) the average particle energy is $T/2$.

Each soliton remains in the $0\leq x\leq L$ domain only for a time $\tau(P)=L/c(P)=4\pi L/P$, after which it exits at $x=L$. These departures must be replaced by the stochastic generation of new solitons at $x=0$ at a rate $S$, where

[TABLE]

The quantity $s(P)$ is an abstraction of whatever underlying physics is responsible for the thermalization of the momentum distribution and whatever the solitons are doing before they reach the boundary at $x=0$.

$\dot{{\cal N}}$, the total average rate of change of the number of solitons, should vanish, so

[TABLE]

In fact, in order that we should not just have $\dot{{\cal N}}$ vanish, but actually have $\dot{n}=0$, so that the distribution function is in equilibrium, the integrand in (26) must vanish. Thus the rate at which new solitons are emitted with momenta in the range from $P$ to $P+dP$ is

[TABLE]

This means that the total emission rate (solitons per unit time) from the thermal source is

[TABLE]

Note that because the more energetic solitons spend less time in the $0\leq x\leq L$, they must be emitted more rapidly than might be guessed just from the equilibrium density distribution $n(P)$. This is the reason for the extra factor of momentum in (27) relative to (22). The total emission rate $S$ also increases with the average energy ${\cal U}/{\cal N}$ for this reason; the more energetic the mean particle, the faster it reaches $x=L$ and needs to be replaced at $x=0$.

The key ideas of this section can be summarized as follows. The BO theory can essentially be characterized as a theory for a gas of particles with only rightward momentum and no collisional interactions at all. The lack of interactions makes things tricky, since there are no collisions to drive the approach to equilibrium. If the BO equation is satisfied over the entire real line $-\infty<x<\infty$, then then the density in momentum space $n(P)$ is constant in time, because the momentum of any given soliton never changes. Thus, in order to grapple with this lack of nontrivial interactions, it is necessary to introduce a stochastic source. The source distribution $s(P)$ provides the key characterization of the system; it describes the number of solitons per unit time, per unit momentum, that are emerging into the $x\geq 0$ region. This source of solitons at $x=0$ provides the requisite randomness, and the system takes on a (one-sided) Maxwellian velocity distribution. As would be expected for a momentum-dependent quantity arising in a free theory, the source intensity $s(P)$ is simply proportional to a power of the momentum, times a Boltzmann factor. [If we had used a different equilibrium distribution $n(P)$, it would have automatically given rise to a different $s(P)$.]

4 KdV Statistics in the Steady State

The analysis for the KdV solitons will follow along much the same lines as the BO analysis from section 3. However, the KdV solitons have both a nonstandard energy-momentum relation and nontrivial forward scattering interactions. As a result, to obtain formulas analogous to all those obtained in section 3 will require additional nontrivial integrations. There is actually quite a bit of interesting structure to three-body KdV collisions, while they are going on [29, 30, 31]. However, there are no intrinsic three-body phase shifts [8]. The asymptotic position of a soliton after it has passed through two others does not depend on the relative or absolute positions of those other two; the net shift for the first soliton is the same whether the other two are widely separated or right on top of each other at the time of interaction. This means that the approximation of a dilute gas is not necessary for the determination of the average soliton traversal times. There does remain, however, an implicit assumption that the gas is sufficiently rarefied that it is possible to speak meaningfully about the locations of individual solitary waves; for this reason, as well as for computational reasons, we shall treat the interactions perturbatively, up to ${\cal O}(N^{2})$.

The normalization of the thermal density $n(P)$ with the KdV dispersion relation $E=(3^{5/3}/5)P^{5/3}$ involves a Gamma function,

[TABLE]

The numerical constants appearing in this equilibrium density are $3^{5/3}/5\approx 1.248$ and $5^{2/3}/\Gamma(3/5)\approx 1.963$. Using

[TABLE]

the total energy associated with this momentum distribution is

[TABLE]

where $3/5^{11/15}\approx 0.922$.

The time required for noninteracting solitons to cross $0\leq x\leq L$ was examined in section 3. In the KdV theory, the key to determining the required source function $s(P)$ that will maintain the thermal $n(P)$ is determining how many other solitons a given soliton will collide with during its traversal from $x=0$ to $x=L$. A soliton will manage to overtake some of the slower solitons in front of it, and it will correspondingly be overtaken by some faster-moving ones that start out behind it. The computation of this effect was not needed for the BO case, because the BO solitons are not displaced during collisions.

When a larger, faster KdV soliton overtakes a smaller, slower one, the faster one emerges from the interaction with the argument of the $\operatorname{sech}^{2}$ in (2) advanced by an amount [32, 33]

[TABLE]

where the solitons have the asymptotic speeds $c_{1}>c_{2}$. The slower soliton emerges with its phase retarded by an amount $-\Delta$. This effect is often characterized as describing an effective attractive force between the two bodies. The attraction accelerates the faster, overtaking soliton and decelerates the slower soliton as the collision approaches. After the collision, the force reverses directions, and its action eventually returns the two solitons to their original speeds. The net result is that the faster soliton ends up ahead of where it would have been in the absence of interaction, while the slower one ends up behind where it would have been.

The shift $\Delta$ is largest when the two speed are very close together. Collisions of this type are uncommon, because the relative velocity between the two solitons is small. However, such collisions, when they do occur, are quite drawn out, giving the effective force between the solitons a long time to act. This explains why the position shifts are so large in these situations. Moreover, when $c_{1}$ and $c_{2}$ are close, the actual spatial displacements of the two solitons are also nearly equal. $\Delta$ is the dimensionless magnitude by which the argument of each soliton’s $\operatorname{sech}^{2}$ changes; however, these correspond to different position shifts $\pm 2\Delta/\sqrt{c_{i}}$, according to (2).

There is also an alternative interpretation, in which the slower soliton leeches energy from the faster one as the two get close together [33]. The amplitude of the front soliton grows, at the expense of the one behind, until they have interchanged amplitudes. Then the soliton in front speeds away at its new, faster velocity, and the apparent position shift for the fast soliton just derives from the fact that the outgoing fast soliton actually started out well in front of the original fast soliton, before they switched roles.

To determine what $s(P)$ is required to maintain the thermal distribution (29), it is necessary to know the rate at which solitons reach the right-hand boundary at $x=L$. A soliton that experienced no interactions would take a time $L/c(P)$ to cross to $x=L$. However, the presence of other solitons in its path can change the time required. Treating the average speed for a soliton of momentum $P$ moving through a thermal background as an unknown function of $P$ produces, in the most general case, an intractable nonlinear integral equation in momentum space. However, matters are simplified greatly by approximation that the soliton gas is sufficiently dilute. Then the predominant contribution to how far a given soliton travels in a time $\tau$ is the free propagation distance $(3P)^{2/3}\tau$. The distance the the soliton is displaced by any collisions is a small correction on top of this.

Schematically, we may write that if a soliton (the “primary” soliton, which has momentum $P$) crosses the distance from $0\leq x\leq L$ in a time $\tau$, we must have

[TABLE]

The initial term on the right-hand-side of (33) is just the free travel distance $c(P)\tau$. Then the first sum represents a sum over all the solitons that are overtaken as the primary soliton traverses the interval. Each of these solitons has a momentum $P^{\prime}<P$, so that they are all moving more slowly than the primary soliton. If a soliton is less than a distance $L[1-c(P^{\prime})/c(P)]$ ahead of the primary soliton when the primary soliton enters the interval at $x=0$, it will be overtaken. So the number density in momentum space of the solitons that will be overtaken is $L[1-c(P^{\prime})/c(P)]n(P^{\prime})$. By an analogous argument, the momentum space density of $P^{\prime}>P$ solitons that will overtake the primary soliton is $L[c(P^{\prime})/c(P)-1]n(P^{\prime})$. The position shifts due to these are collected in the second sum on the right-hand side of (33). Note that both the $P^{\prime}<P$ and $P^{\prime}>P$ densities are positive.

The summands in each of the sums are just $2\Delta/\sqrt{c(P)}$. These are the forward or backward displacements corresponding to the phase differences $\Delta$, according to (2) and (32).

The $P^{\prime}<P$ sum in (33) is therefore

[TABLE]

while the $P^{\prime}>P$ sum is correspondingly

[TABLE]

These two integrals can evidently be combined to give a single expression,

[TABLE]

using the $n(P^{\prime})$ from (29).

Following the same reasoning as with (25) in section 3, it follows that the source of solitons at $x=0$ must be emitting them with a rate density in momentum space of

[TABLE]

Treating the collision tern in (36) as a small correction, this involves

[TABLE]

where the function $f(u)$ contains the complicated integral,

[TABLE]

For low temperatures (meaning large $u$), the integral for $f(u)$ is tractable, since it will be dominated by the small-$v$ integration region. Using

[TABLE]

the source strength in this limit is

[TABLE]

The additional constants appearing in this expression are $15^{2/3}/\Gamma(3/5)\approx 4.084$ and $4(5^{7/15})\Gamma(4/5)/3^{2/3}\Gamma(3/5)\approx 3.186$.

It makes sense that, when the temperature is sufficiently small compared with the scales set by the soliton density $N$ and the primary soliton momentum $P$, the collision term is small. The reason lies in the formula (32) for $\Delta$. When a fast-moving, energetic soliton overtakes a much slower one, the displacements caused by $\Delta$ are small, because the argument of the logarithm in (32) is nearly unity. In short, solitons with wildly different momenta barely affect each other, because they pass through each other so quickly.

If the ${\cal O}(N^{2})$ collision term is entirely neglected, we may relate the source strength $S$ to the total energy ${\cal U}$, as in section 3. For such free flowing solitons, using (30) again,

[TABLE]

The power of ${\cal U}/{\cal N}$ involved is determined by dimensional considerations alone, but the overall constant $5^{24/25}/3^{2/5}\Gamma(3/5)\approx 2.203$ is nontrivial.

Ultimately, we see that the statistical KdV theory possesses all the complications of the analogous BO theory, plus some additional ones. Even though the BO soliton gas represents an essentially free theory, the fact that the solitons only move unidirectionally complicates affairs, and in order to have steady-state thermal behavior, it is necessary to have an input of new solitons entering the region of interest from the left. This feature obviously continues to be important when the solitons are governed by the KdV equation. Moreover, the solitons in the KdV theory do have nontrivial pairwise interactions. All collisions are completely elastic, with no momentum redistribution. However, the solitons are attracted toward one-another, and each two-body collision alters the positions of the solitons involved. In this section, we have generalized the statistically modeling techniques used to study the BO system to address the KdV equation, with its nonstandard energy-momentum relation and nontrivial interactions. To get tractable results, the interactions were treated perturbatively, up to second order in the soliton density $N$. In order to get a closed form expression (41) without the complicated integral function $f(u)$, it was also necessary to look at the limit of high temperatures.

5 Conclusions and Outlook

We have seen how to construct a theory of statistical equilibrium for gasses of KdV and BO solitons. These two theories share a number of complications; in particular, they only support right-moving solitary waves, and the solitary waves are true solitons—meaning that there is no redistribution of momentum during collisions between two or more solitons. For the solitons of the BO theory, these are essentially the only complications of note. The BO solitons have a conventional dispersion relation, with $E_{{\rm BO}}\propto(P_{{\rm BO}})^{2}$, just like for ordinary nonrelativistic particles. The KdV system, in contrast, is more subtle.

Throughout our analysis, we have assumed the existence of thermalized Boltzmann equilibrium distributions. However, as previously noted, it would be entirely possible to carry through the same kind of analysis with a momentum distribution function $n(P)$ that had a different functional form. (In fact, some detailed simulations of the periodic KdV equation have indeed used entirely different soliton energy distributions [34].) In practice, our calculations took the thermal distribution of soliton momenta as an assumption and worked backwards from there to determine the necessary source function $s(P)$. Starting with a different $n(P)$, it would be possible to get a different $s(P)$. This raises the natural question of whether a $s(P)$ function like (27) or (41) is really special in any way. In fact, these particular forms for $s(P)$ do have special significance, precisely because they gives rise to a thermal distributions of soliton momenta. When the number of solitons is thermodynamically large, the conventionally structured thermal distribution is unique in that it represents the most likely way of dividing up the total energy among the many solitonic particles. Perturbations to the KdV and similar equation have been shown to have a thermalizing effect [35, 36]. If the interactions between solitons are perturbed by weak soliton-soliton momentum exchange interactions, the state of the gas will evolve toward a thermal distribution. In our model, the soliton source at $x=0$ is a “black box”; it may be that for $x<0$, the field equation governing the solitons is slightly modified, so that they do interact in that domain, and the soliton gas in the $x<0$ region thermalizes before the solitons are injected into the $0\leq x\leq L$ region of interest.

An alternative viewpoint would be to start with a known distribution of injected solitons $s(P)$ and work in the other direction—deriving the equilibrium $n(P)$ that corresponds to such a $s(P)$. This is a straightforward generalization of the approach described here. However, other generalizations of our approach may be trickier to implement. For instance, the fact that a number of simplifying approximations were used in section 4 raises the obvious possibility that it might be possible to further improve the results—to have closed form expressions that would be valid beyond the small $N$, small $T$ limit. An even more ambitious, and perhaps more important, generalization would be to consider a statistical treatment that includes the full spectrum of solutions of the BO or KdV equations. These nonlinear wave equations have periodic wave train solutions, with the soliton solutions representing only the infinite wavelength limits. Nontrivial interactions between finite-wavelength waves will make matters significantly more complicated, in part because there is no notion of the excitations being far apart. A further adjustment to how the thermal theory is formulated may be necessary in order to obtain a tractable description.

However, even with just the soliton results presented in this paper, progress has been made, both conceptually and computationally, toward a complete theory of statistical equilibrium for fully integrable systems. The resulting picture of thermal equilibrium may be useful for characterizing plasmas, turbulently interacting surface waves, or other systems with weakly interacting nonlinear dynamics.