Physics Letters A 372 (2008) 1024–1033

www.elsevier.com/locate/pla

Vlasov moments, integrable systems and singular solutions

John Gibbons a, Darryl D. Holm a,b,∗, Cesare Tronci a,c

a Department of Mathematics, Imperial College London, London SW7 2AZ, UKb Computer and Computational Science Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

c TERA Foundation for Oncological Hadrontherapy, 11 V. Puccini, Novara 28100, Italy

Received 30 May 2007; received in revised form 14 August 2007; accepted 15 August 2007

Available online 8 September 2007

Communicated by C.R. Doering

Abstract

The Vlasov equation governs the evolution of the single-particle probability distribution function (PDF) for a system of particles interactingwithout dissipation. Its singular solutions correspond to the individual particle motions. The operation of taking the moments of the Vlasovequation is a Poisson map. The resulting Lie–Poisson Hamiltonian dynamics of the Vlasov moments is found to be integrable is several cases. Forexample, the dynamics for coasting beams in particle accelerators is associated by a hodograph transformation to the known integrable Benneyshallow-water equation. After setting the context, the Letter focuses on geodesic Vlasov moment equations. Continuum closures of these equationsat two different orders are found to be integrable systems whose singular solutions characterize the geodesic motion of the individual particles.© 2007 Elsevier B.V. All rights reserved.

1. Introduction

1.1. The Vlasov equation

The evolution of N identical particles in phase space T ∗Mwith coordinates (qi ,pi), i = 1, 2, . . . ,N , may be described byan evolution equation for their joint probability distributionfunction. Integrating over all but one of the particle phase-space coordinates yields an evolution equation for the single-particle probability distribution function (PDF). This is theVlasov equation, which may be expressed as an advection equa-tion for the phase-space density f along the Hamiltonian vectorfield XH corresponding to single-particle motion with Hamil-tonian H(q,p):

(1)∂f

∂t= {f ,H} = −div(q,p)(f XH ) = −LXH

f

(2)with XH (q,p) =(

∂H

∂p,−∂H

∂q

),

* Corresponding author at: Department of Mathematics, Imperial CollegeLondon, London SW7 2AZ, UK.

E-mail address: d.holm@ic.ac.uk (D.D. Holm).

0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2007.08.054

where LX denotes Lie derivative with respect to the vectorfield X.

The solutions of the Vlasov equation reflect its heritagein particle dynamics, which may be reclaimed by writing itsmany-particle PDF as a product of delta functions in phasespace. Any number of these delta functions may be integratedout until all that remains is the dynamics of a single particle inthe collective field of the others.

In the mean-field approximation of plasma dynamics, thiscollective field generates the total electromagnetic propertiesand the self-consistent equations obeyed by the single par-ticle PDF are the Vlasov–Maxwell equations. In the elec-trostatic approximation, these reduce to the Vlasov–Poisson(VP) equations, which govern the statistical distributions ofparticle systems ranging from integrated circuits (MOSFETS,metal-oxide semiconductor field-effect transistors), to charged-particle beams, to the distribution of stars in a galaxy.

1.2. Vlasov moments

The Euler fluid equations arise by imposing a closure rela-tion on the first three momentum moments, or p-moments ofthe Vlasov PDF f (p,q, t). The zeroth p-moment is the spatialmass density of particles as a function of space and time. The

J. Gibbons et al. / Physics Letters A 372 (2008) 1024–1033 1025

first p-moment is the mean fluid momentum density. Introduc-ing an expression for the fluid pressure in terms of the massdensity and momentum density closes the system of p-momentequations, which otherwise would possess a countably infinitenumber of dependent variables.

The operation of taking p-moments preserves the geomet-ric nature of Vlasov dynamics. In particular, this operation isa Poisson map. That is, it takes the Lie–Poisson structure forVlasov dynamics into another Lie–Poisson system. However,strictly speaking, the solutions for the p-moments cannot yetbe claimed to undergo coadjoint motion, as in the case of theVlasov PDF solutions, because the group action underlying theLie–Poisson structure found for the p-moments is not yet un-derstood.

Given the beauty and utility of the solution behavior forEuler’s equation for the first p-moment, one is intrigued toknow more about the dynamics of the other moments ofVlasov’s equation. Of course, the dynamics of the p-momentsof the Vlasov–Poisson equation is one of the mainstream sub-jects of plasma physics and space physics.

1.3. Summary

This Letter formulates the dynamics of Vlasov p-momentsgoverned by quadratic Hamiltonians. This dynamics is a certaintype of geodesic motion on the symplectic diffeomorphismsThese symplectomorphisms are smooth invertible symplecticmaps acting on the phase space and possessing smooth inverses.The theory of moment dynamics for the Vlasov equation turnsout to be equivalent to the theory of shallow water equations,and a particular example is the one-dimensional system of Ben-ney long wave equations, which is integrable [1,2].

Here we shall consider the singular solutions of the geo-desic dynamics of the Vlasov p-moments. Remarkably, theseequations turn out to be related to other integrable systemsgoverning shallow water wave theory. For example, when theVlasov p-moment equations for geodesic motion on the sym-plectomorphisms are closed at the level of the first p-moment,their singular solutions are found to recover the peaked soli-ton of the integrable Camassa–Holm (CH) equation for shallowwater waves [3]. These singular Vlasov moment solutions alsocorrespond to individual particle motion.

Thus, geodesic symplectic dynamics of the Vlasov p-mo-ments is found to possess singular solutions whose closure atthe fluid level for the CH equation recovers the peakon solu-tions of shallow water theory. Being solitons, the CH peakonssuperpose and undergo elastic collisions in fully nonlinear in-teractions. The singular solutions for Vlasov p-moments pre-sented here also superpose and interact nonlinearly as coherentstructures.

The plan of the Letter follows:Section 2 reviews the Vlasov p-moment equations and

recounts their Lie–Poisson Hamiltonian structure using theKupershmidt–Manin Lie–Poisson (KMLP) bracket. Variationalformulations of the p-moment dynamics are also provided.

Section 3 shows how the Lagrangian framework for fluiddynamics is recovered from the Vlasov p-moments and es-

tablishes connections with some equations for shallow waterwaves. In this case, the p-moment equations are shown to pos-sess singular solutions.

Section 4 establishes the connections between the integrableBenney equations and the physics of charged-particle accelera-tor beams. To our knowledge, these connections are noted herefor the first time. We also point out how the experimental real-ization of solitary waves in coasting particle beams has its rootsin the integrability of the Benney system.

Section 5 formulates the problem of geodesic motion on thesymplectomorphisms in terms of the Vlasov p-moments andidentifies the singular solutions of this problem. This geodesicmotion is related to a geodesic form of the Vlasov equation.Thus the singular solutions are found to originate in the sin-gle particle dynamics on phase space. In a special case, thetruncation of geodesic symplectic motion to geodesic diffeo-morphic motion for the first p-moment recovers the singular so-lutions of the Camassa–Holm equation, and thereby correspondto single particle dynamics. Other truncations of the geodesicp-moment equations are also found that extend the Camassa–Holm equation to include multiple components and orientationdependence.

The geodesic form of the Vlasov equation and its extensionto higher dimensions were introduced in [4].

2. Review of Vlasov moment dynamics

The Vlasov equation may be expressed as

(3)

∂f

∂t= −

[f ,

δh

δf

]=

∂f

∂p

∂

∂q

δh

δf− ∂f

∂q

∂

∂p

δh

δf=: − ad∗

δh/δf f.

Here the canonical Poisson bracket [·,·] is defined for smoothfunctions on phase space with coordinates (q,p) and f (q,p, t)is the evolving Vlasov single-particle distribution function. Thevariational derivative δh/δf regulates the particle motion andthe quantity ad∗

δh/δf f is explained as follows.A functional g[f ] of the Vlasov distribution f evolves ac-

cording to

dg

dt=

∫ ∫δg

δf

∂f

∂tdq dp = −

∫ ∫δg

δf

[f ,

δh

δf

]dq dp

(4)

=∫ ∫

f

[δg

δf,δh

δf

]dq dp =:

⟨⟨f ,

[δg

δf,δh

δf

]⟩⟩=: {g,h}.

In this calculation boundary terms were neglected upon inte-grating by parts in the third step and the notation 〈〈·,·〉〉 is in-troduced for the L2 pairing in phase space. The quantity {g,h}defined in terms of this pairing is the Lie–Poisson Vlasov (LPV)bracket [5]. This Hamiltonian evolution equation may also beexpressed as

(5)dg

dt= {g,h} = −

⟨⟨f , adδh/δf

δg

δf

⟩⟩=: −

⟨⟨ad∗

δh/δf f ,δg

δf

⟩⟩

which defines the Lie-algebraic operations ad and ad∗ in thiscase in terms of the L2 pairing on phase space 〈〈·,·〉〉 : s∗ ×s �→ R. The notation ad∗

δh/δf f in (3) expresses coadjoint ac-

1026 J. Gibbons et al. / Physics Letters A 372 (2008) 1024–1033

tion of δh/δf ∈ s on f ∈ s∗, where s is the Lie algebra of singleparticle Hamiltonian vector fields and s∗ is its dual under L2

pairing in phase space. This is the sense in which the Vlasovequation represents coadjoint motion on the group of symplec-tic diffeomorphisms (symplectomorphisms). This Lie–Poissonstructure was extended to include Yang–Mills theories in [6]and [7].

In higher dimensions, particularly n = 3, we may take the di-rect sum of the Vlasov Lie–Poisson bracket, together the Pois-son bracket for an electromagnetic field (in the Coulomb gauge)where the electric field E and magnetic vector potential A arecanonically conjugate. For discussions of the Vlasov–Maxwellequations from a geometric viewpoint in the same spirit as thepresent approach, see [5,8–10].

2.1. Dynamics of Vlasov q,p-moments

The phase space of q,p-moments of the Vlasov distributionfunction are defined by

(6)gmm(t) =∫ ∫

f (q,p, t)qmpm dq dp.

The q,p-moments gmm are often used in treating the colli-sionless dynamics of plasmas and particle beams [11]. This isusually done by considering low-order truncations of the poten-tially infinite sum over phase space moments,

(7)g(t) =∞∑

m,m=0

ammgm,m(t), h(t) =∞∑

n,n=0

bnngn,n(t),

with constants amm and bnn, with m,m, n,n = 0, 1, . . . . If h isthe Hamiltonian, the sum over q,p-moments g evolves underthe Vlasov dynamics according to the Poisson bracket relation

dg

dt= {g,h}

(8)=∞∑

m,m,n,n=0

ammbnn(mm − nn)gm+n−1,m+n−1.

The symplectic invariants associated with Hamiltonian flows ofthe q,p-moments were discovered and classified in [12]. Finitedimensional approximations of the whole q,p-moment hierar-chy were discussed in [13]. For discussions of the Lie-algebraicapproach to the control and steering of charged particle beams,see [11].

2.2. Dynamics of Vlasov p-moments

In contrast to the q,p-moments, the momentum moments,or “p-moments”, of the Vlasov function are defined as

(9)Am(q, t) =∫

pmf (q,p, t) dp, m = 0, 1, . . . .

That is, the p-moments are q-dependent integrals over p of theproduct of powers pm, m = 0, 1, . . ., times the Vlasov solutionf (q,p, t). We shall consider functionals of these p-momentsdefined by

g =∞∑

m=0

∫ ∫αm(q)pmf dq dp =

∞∑m=0

∫αm(q)Am(q)dq

(10)=:∞∑

m=0

〈Am,αm〉,

h =∞∑

n=0

∫ ∫βn(q)pnf dq dp =

∞∑n=0

∫βn(q)An(q) dq

(11)=:∞∑

n=0

〈An,βn〉,

where 〈·,·〉 is the L2 pairing on position space.The functions αm and βn with m,n = 0, 1, . . . are assumed

to be suitably smooth and integrable against the Vlasov p-mo-ments. To assure these properties, one may relate the p-mo-ments to the previous sums of Vlasov q,p-moments by choos-ing

(12)αm(q) =∞∑

m=0

ammqm and βn(q) =∞∑

n=0

bnnqn.

For these choices of αm(q) and βn(q), the sums of p-momentswill recover the full set of Vlasov (q,p)-moments. Thus, aslong as the q,p-moments of the distribution f (q,p) continue toexist under the Vlasov evolution, one may assume that the dualvariables αm(q) and βn(q) are smooth functions whose Tay-lor series expands the p-moments in the q,p-moments. Thesefunctions are dual to the p-moments Am(q) with m = 0, 1, . . .under the L2 pairing 〈·,·〉 in the spatial variable q . In what fol-lows we will assume homogeneous boundary conditions. Thismeans, for example, that we will ignore boundary terms arisingfrom integrations by parts.

The Poisson bracket among the p-moments is obtained fromthe LPV bracket and may be expressed as

{g,h}({A}

)

=∞∑

m,n=0

∫Am+n−1

(n

δh

δAn

∂

∂q

δg

δAm

− mδg

δAm

∂

∂q

δh

δAn

)dq

(13)=:∞∑

m,n=0

⟨Am+n−1,

�δg

δAm

,δh

δAn

�⟩

where we have introduced the compact notation {A} := {An}with n a non-negative integer. This is the Kupershmidt–ManinLie–Poisson (KMLP) bracket [14], which is defined for func-tions on the dual of the Lie algebra with bracket

(14)�αm,βn� = nβn∂qαm − mαm∂qβn.

This Lie algebra bracket inherits the Jacobi identity from itsdefinition in terms of the canonical Hamiltonian vector fields.Also, for n = m = 1 this Lie bracket reduces to minus theJacobi–Lie bracket for the vector fields α1 and β1. Thus, wehave recovered the following

Theorem (Gibbons [2]). The operation of taking p-momentsof Vlasov solutions is a Poisson map. It takes the LPV bracket

J. Gibbons et al. / Physics Letters A 372 (2008) 1024–1033 1027

describing the evolution of f (q,p) into the KMLP bracket, de-scribing the evolution of the p-moments An(x).

A result related to this, for the Benney hierarchy [1], wasalso presented by Lebedev and Manin [15,16].

The evolution of a particular p-moment Am(q, t) is obtainedfrom the KMLP bracket by

∂Am

∂t= ad∗

δh/δAnAm+n+1 = {Am,h}

= −∞∑

n=0

(n

∂

∂qAm+n−1 + mAm+n−1

∂

∂q

)δh

δAn

(15)=∞∑

n=0

{Am,An}δh

δAn

.

These moment equations may also be derived from varia-tional principles, as shown in [4], within the Hamilton–Poincaréframework [17].

Remark 1. The p-moments have an important geometric inter-pretation, that will be used in the next section. Indeed one canwrite the moments as

(16)An =∫p

⊗n(p dq)f (q,p)dq ∧ dp = An(q) ⊗n+1 dq

where ⊗n dq := dq ⊗ . . . ⊗ dq n times. Thus, moments An be-long to the vector space dual to the contravariant tensors of thetype βn = βn(q) ⊗n ∂q . These tensors are given a Lie algebrastructure by the Lie bracket

�αm,βn� =(nβn(q)α′

m(q) − mαm(q)β ′n(q)

) ⊗n+m−1 ∂q

(17)=: adαm βn

so that the ad∗ operator is defined by 〈ad∗βn

Ak ,αk−n+1〉 :=〈Ak , adβn αk−n+1〉 and is given explicitly as

(18)ad∗βn

Ak =(

(k + 1)Ak

∂βn

∂q+ nβn

∂Ak

∂q

)⊗k−n+2 dq.

3. Moments and cotangent lifts of diffeomorphisms

As explained in the introduction, a first order closure of themoment hierarchy leads to the equations of ideal fluid dynam-ics. Such equations represent coadjoint motion with respect tothe Lie group of smooth invertible maps (diffeomorphisms).This coadjoint evolution may be interpreted in terms of La-grangian variables, which are invariant under the action of dif-feomorphisms. In this section we investigate how the entire mo-ment hierarchy may be expressed in terms of the fluid quantitiesevolving under the diffeomorphisms and express the conserva-tion laws in this case.

3.1. Lagrangian variables and cotangent lifts

In order to look for Lagrangian variables, we consider thegeometric interpretation of the moments, regarded as fiber in-tegrals on the cotangent bundle T ∗Q of some configuration

manifold Q. A p-moment is defined as a fiber integral; thatis, an integral on the single fiber T ∗

q Q with base point q ∈ Q

kept fixed

(19)An(q) =∫

T ∗q Q

pnf (q,p)dp.

A similar approach is followed for gyrokinetics in [18]. Now,the problem is that in general the integrand does not stay ona single fiber under the action of canonical transformations,i.e. symplectomorphisms are not fiber-preserving in the gen-eral case. However, one may avoid this problem by restrictingto a subgroup of these canonical transformations whose actionis fiber preserving.

The transformations in this subgroup (indicated withT ∗ Diff(Q)) are called point transformations or cotangent liftsof diffeomorphisms and they arise from diffeomorphisms onpoints in configuration space [19], such that

(20)qt = qt (q0).

The fiber preserving nature of cotangent lifts is expressed bythe preservation of the canonical one-form:

(21)pt dqt = pt(q0,p0) dqt (q0) = p0 dq0.

This fact also reflects in the particular form assumed by thegenerating functions of cotangent lifts, which are linear in themomentum coordinate [19], i.e.

(22)H(q,p) = β(q)∂

∂q�p dq = pβ(q)

where the symbol � denotes contraction between the vectorfield β and the momentum one-form p. Restricting to cotan-gent lifts represents a limitation in comparison with consideringthe whole symplectic group. However, this is a natural way ofrecovering the Lagrangian approach, starting from the full mo-ment dynamics.

3.2. Characteristic equations for the moments

Once one restricts to cotangent lifts, Lagrangian momentvariables may be defined and conservation laws may be found,as in the context of fluid dynamics. The key idea is to use thepreservation of the canonical one-form for constructing invari-ant quantities. Indeed one may take n times the tensor productof the canonical one-form with itself and write:

(23)pnt (dqt )

n = pn0 (dq0)

n.

One then considers the preservation of the Vlasov density

(24)ft (qt ,pt ) dqt ∧ dpt = f0(q0,p0) dq0 ∧ dp0

and writes

pnt ft (qt ,pt )(dqt )

n ⊗ dqt ∧ dpt

(25)= pn0f0(q0,p0)(dq0)

n ⊗ dq0 ∧ dp0.

1028 J. Gibbons et al. / Physics Letters A 372 (2008) 1024–1033

Integration over the canonical particle momenta yields the fol-lowing characteristic equations

(26)d

dt

[A(t)

n (qt )(dqt )n ⊗ dqt

]= 0 along qt =

∂H

∂p= β(q)

which recover the well known conservations for fluid densityand momentum (n = 0, 1) and can be equivalently written interms of the Lie–Poisson equations arising from the KMLPbracket, as shown in the next section. Indeed, if the vectorfield β is identified with the Lie algebra variable β = β1 =δh/δA1 and h(A1) is the moment Hamiltonian, the KMLP form(15) of the moment equations is

(27)∂An

∂t+ ad∗

β1An = 0.

In this case, the KM ad∗β1

operation coincides with the Liederivative £β1 ; so, one may write it equivalently as

(28)∂An

∂t+ £β1An = 0.

For n = 0, 1, one recovers the advection relations for the den-sity A0 and the momentum A1 in fluid dynamics. However,unlike fluid dynamics, all the moments are conserved quanti-ties. This equation is reminiscent of the so called b-equationintroduced in [20], for which the vector field β is nonlocal andmay be taken as β(q) = G ∗ An (for any n), where G is theGreen’s function of the Helmholtz operator and the star denotesconvolution. When the vector field β is sufficiently smooth, thisequation is known to possess singular solutions of the form

(29)An(q, t) =K∑

i=1

Pn,i (t)δ(q − Qi(t)

)

where the ith position Qi and weight Pn,i of the singular solu-tion for the nth moment satisfy the following equations

(30)Qi = β(q)|q=Qi, Pn,i = −nPn,i

∂β(q)

∂q

∣∣∣∣q=Qi

.

Interestingly, for n = 1 (with β(q) = G ∗ A1), these equa-tions recover the pulson solutions of the Camassa–Holm equa-tion [3], which play an important role in the following discus-sion. Moreover the particular case n = 1 represents the singleparticle solution of the Vlasov equation. However, when n �= 1the interpretation of these solutions as single-particle motionrequires the particular choice Pn,i = (Pi)

n. For this choice, thenth weight is identified with the nth power of the particle mo-mentum.

3.2.1. KMLP bracket and cotangent liftsWe have seen that restricting to cotangent lifts leads to a La-

grangian fluid-like formulation of the dynamics of the resultingp-moments. In this case, the moment equations are given by theKMLP bracket when the Hamiltonian depends only on the firstmoment (β1 = δh/δA1)

(31){g,h} =∑n

⟨An,

�δg

δAn

,δh

δA1

�⟩.

If one now restricts the bracket to functionals of only the firstmoments, one may check that the KMLP bracket yields the wellknown Lie–Poisson bracket on the group of diffeomorphisms

(32){g,h}[A1] = −⟨A1,

[δg

δA1

∂

∂q

δh

δA1− δh

δA1

∂

∂q

δg

δA1

]⟩.

This is a very natural step since diffeomorphisms and theircotangent lifts are isomorphic. In fact, this is the bracket usedfor ideal incompressible fluids as well as for the construction ofthe EPDiff equation, which will be discussed later as an appli-cation of moment dynamics.

4. Applications of the KMLP bracket and quadratic terms

4.1. The Benney equations and coasting beams in particleaccelerators

The KMLP bracket (13) was first derived in the context ofBenney long waves, whose Hamiltonian is

(33)H =1

2

∫ (A2(q) + gA2

0(q))dq.

The Hamiltonian form ∂tAn = {An,H} with the KMLP bracketleads to the moment equations

(34)∂An

∂t+

∂An+1

∂q+ gnAn−1

∂A0

∂q= 0

derived by Benney [1] as a description of long waves on a shal-low perfect fluid, with a free surface at y = h(q, t). In thisinterpretation, the An were vertical moments of the horizontalcomponent of the velocity p(q,y, t):

An =

h∫0

pn(q,y, t) dy.

The corresponding system of evolution equations for p(q,y, t)and h(q, t) is related by hodograph transformation, y =∫ p

−∞ f (q,p′, t) dp′, to the Vlasov equation

(35)∂f

∂t+ p

∂f

∂q− g

∂A0

∂q

∂f

∂p= 0.

The most important fact about the Benney hierarchy is that it iscompletely integrable.

4.1.1. Applications to coasting accelerator beamsInterestingly, the equation that regulates coasting proton

beams in particle accelerators takes exactly the same form asthe Vlasov equation (35) resulting from the hodograph trans-formation of the Benney equation. (See for example [21] wherea linear bunching term is also included.) The integrability ofthe Vlasov–Benney equation implies coherent structures. Thesestructures are indeed found experimentally at CERN [22],BNL [23], LANL [24] and FermiLab [25]. (In the last case co-herent structures are shown to appear even when a bunchingforce is present.) These structures have attracted the attentionof the accelerator community and considerable analytical workhas been carried out over the last decade (see for example [26]).

J. Gibbons et al. / Physics Letters A 372 (2008) 1024–1033 1029

To our knowledge, the existence of coherent structures in coast-ing proton beams has never been related to the integrability ofthe governing Vlasov equation via its connection to the Ben-ney hierarchy. However, this connection would explain verynaturally why robust coherent structures are seen in these ex-periments as fully nonlinear excitations. We plan to pursue thisdirection in future research.

4.2. The Vlasov–Poisson system, the wake-field model andsingular solutions

Besides integrability of the Vlasov–Benney equation, thereare other important applications of the Vlasov equation thathave in common the presence of a quadratic term in A0 withinthe Hamiltonian:

(36)

H =1

2

∫A2(q) dq +

1

2

∫ ∫A0(q)G(q,q ′)A0(q

′) dq dq ′.

For example, when G = (∂2q )−1, this Hamiltonian leads to the

Vlasov–Poisson system, which is of fundamental importancein many areas of plasma physics. More generally, this Hamil-tonian is widely used for beam dynamics in particle accelera-tors: in this case G is related to the electromagnetic interactionof a beam with the vacuum chamber. The wake field is origi-nated by the image charges induced on the walls by the passageof a moving particle: while the beam passes, the charges in thewalls are attracted towards the inner surfaces and generate afield that acts back on the beam. This affects the dynamics of thebeam, thereby causing several problems such as beam energyspread and instabilities. In the literature, the wake function W

is introduced so that [21]

(37)G(q,q ′) =

q∫−∞

W(x,q ′) dx.

Wake functions usually depend only on the properties of theaccelerator chamber.

An interesting wake-field model has been presented in [26]where G is chosen to be the Green’s function of the Helmholtzoperator (1 − α2∂2

q ): this generates a Vlasov–Helmholtz (VH)equation [27] that is particularly interesting for future work.Connections of this equation with the well known integrableKdV equation have been proposed. However we believe thatthis is not a natural step since integrability appears alreadywith no further approximations in the Vlasov–Benney systemthat governs the collective motion of the beam. In particularwe would like to understand the VH equation as a special de-formation of the integrable Vlasov–Benney case that allowsthe existence of singular solutions. Indeed, the presence of theGreen’s function G above is a key ingredient for the existenceof the single-particle solution, which is not allowed in the VBcase. In particular, the single-particle solutions for the Vlasov–Helmholtz equation may be of great interest, since these sin-gular solutions arise from a deformation of an integrable sys-tem. The Vlasov–Helmholtz solutions may differ considerablyfrom the well-known particle behavior of the Vlasov–Poisson

case. However, in the limit as the deformation parameter (thelength-scale α) in the Helmholtz Green’s function passes tozero (α → 0) in the wake-field equations, one recovers the in-tegrable Vlasov–Benney case.

4.3. The EPDiff equation and singular solutions

Another interesting moment equation is given by the Eu-ler–Poincaré equation on the group of diffeomorphisms(EPDiff) [3]. In this case, the Hamiltonian is purely quadraticin the first moments:

(38)H =1

2

∫ ∫A1(q)G(q,q ′)A1(q

′) dq dq ′

and the EPDiff equation [28]

∂A1

∂t+

∂A1

∂q

∫G(q,q ′)A1(q

′, t) dq ′

(39)+ 2A1∂

∂q

∫G(q,q ′)A1(q

′, t) dq ′ = 0

comes from the closure of the KMLP bracket given by cotan-gent lifts. (Without this restriction we would obtain again theequations (26) with β = G ∗ A1.) Thus this EPDiff equationis a geodesic equation on the group of diffeomorphisms. TheCamassa–Holm equation is a particular case in which G is theGreen’s function of the Helmholtz operator 1 − α2∂2

q . Both theCH and the EPDiff equations are completely integrable andhave a large number of applications in fluid dynamics (shallowwater theory, averaged fluid models, etc.) and imaging tech-niques [29] (medical imaging, contour dynamics, etc.).

Besides its complete integrability, the EPDiff equation hasthe important feature of allowing singular delta-function solu-tions. The connection between the CH and EPDiff equationsand the moment dynamics lies in the fact that singular solutionsappear in both contexts. The existence of this kind of solutionfor EPDiff leads to investigate its origin in the context of Vlasovmoments. More particularly we wonder whether there is a nat-ural extension of the EPDiff equation to all the moments. Thiswould again be a geodesic (hierarchy of) equation, which wouldperhaps explain how the singular solutions for EPDiff arise inthis larger context.

Remark 2. It should be pointed out that the KMLP and VLPformulations are not wholly equivalent; in particular the mapfrom the distribution function f (q,p) to the moments {An} isexplicit, but it is not a trivial problem to reconstruct the distribu-tion from its moments. Simple fluid-like closures of the systemarise very naturally in the KMLP framework, as with the exam-ple in Section 3.

Moreover the KMLP bracket has been recently applied inthe context of double bracket dissipation. In [30], a dissipativebracket for moment dynamics has been formulated, which hasbeen shown to recover the well known Darcy’s Law for parti-cles in porous media. Also, in [31] the KMLP bracket has beenextended to include the dynamics of particles with anisotropicinteractions. Both these formulations exhibit the spontaneousemergence of singular solutions.

1030 J. Gibbons et al. / Physics Letters A 372 (2008) 1024–1033

5. Geodesic motion and singular solutions

5.1. Quadratic Hamiltonians

The previous examples show how quadratic terms in theHamiltonian produce interesting behavior in various contexts.This suggests that a deeper analysis of the role of quadraticterms may be worthwhile particularly in connections betweenVlasov p-moment dynamics and the EPDiff equation, with itssingular solutions. Purely quadratic Hamiltonians are consid-ered in [4], leading to the problem of geodesic motion on thespace of p-moments.

In this problem the Hamiltonian is the norm on the p-mo-ments given by the following metric and inner product,

h =1

2‖A‖2

(40)=1

2

∞∑n,s=0

∫ ∫An(q)Gns(q,q ′)As(q

′) dq dq ′.

The metric Gns(q,q ′) in (40) is chosen to be positive definite,so it defines a norm for {A} ∈ g∗. The corresponding geodesicequation with respect to this norm is found as in the previoussection to be,

∂Am

∂t= {Am,h}

(41)

= −∞∑

n=0

(nβn

∂

∂qAm+n−1 + (m + n)Am+n−1

∂

∂qβn

)

with dual variables βn ∈ g defined by

βn =δh

δAn

=∞∑

s=0

∫Gns(q,q ′)As(q

′) dq ′

(42)=∞∑

s=0

Gns ∗ As.

Thus, evolution under (41) may be rewritten as formal coadjointmotion on g∗,

(43)∂Am

∂t= {Am,h} =: −

∞∑n=0

ad∗βn

Am+n−1.

This system comprises an infinite system of nonlinear, non-local, coupled evolutionary equations for the p-moments. Inthis system, evolution of the mth moment is governed by thepotentially infinite sum of contributions of the velocities βn as-sociated with nth moment sweeping the (m + n − 1)th momentby a type of coadjoint action. Moreover, by Eq. (42), each ofthe βn potentially depends nonlocally on all of the moments.

Remark 3. When the metric Gnm is diagonal (Gnm =Knmδm

n =: Gn), one can find the inverse Legendre transformand the problem admits a Lagrangian formulation in terms ofEuler–Poincaré equations, following [4]. This is not possible inthe general case.

5.2. A geodesic Vlasov equation: EPSymp

Importantly, geodesic motion for the p-moments is equiva-lent to geodesic motion for the Euler–Poincaré equations on thesymplectomorphisms (EPSymp) given by the following Hamil-tonian

(44)H [f ] =

1

2

∫ ∫f (q,p)G(q,p,q ′,p′)f (q ′,p′) dq dp dq ′ dp′.

The equivalence with EPSymp emerges when the function G iswritten as

(45)G(q,q ′,p,p′) =∑n,m

pnGnm(q,q ′)p′m

and the corresponding Vlasov equation reads as

(46)∂f

∂t+ [f ,G ∗ f ] = 0,

where [·,·] denotes the canonical Poisson bracket.Thus, whenever the metric G for EPSymp has a Taylor se-

ries, its solutions may be expressed in terms of the geodesicmotion for the p-moments. More particularly the geodesicVlasov equation presented here is nonlocal in both position andmomentum and is equivalent to the vorticity equation in two-dimensions, for a particular choice of the metric. However thisequation extends to more dimensions [4] and to any kind of geo-desic motion, no matters how the metric is expressed explicitly.

5.3. Singular geodesic solutions

We have now clarified the geometric meaning of the momentequations and we can therefore characterize singular solutions,since the geodesic Vlasov equation (EPSymp) essentially de-scribes advection in phase space. Indeed, the geodesic Vlasovequation possesses the single particle solution

(47)f (q,p, t) =∑j

δ(q − Qj(t)

)δ(p − Pj (t)

)

which is a well-known singular solution that is admitted when-ever the phase-space density is advected along a smoothHamiltonian vector field. This happens, for example, in theVlasov–Poisson system and in the general wake-field model.On the other hand, these singular solutions do not appear in theVlasov–Benney equation.

Eq. (41) admits singular solutions of the form

(48)An(q, t) =N∑

j=1

P nj (t)δ

(q − Qj(t)

).

In order to show this is a solution in one dimension, onechecks that these singular solutions satisfy a system of partialdifferential equations in Hamiltonian form, whose Hamiltoniancouples all the moments

(49)HN =1

2

∞∑n,s=0

N∑j ,k=1

P sj (t)P n

k (t)Gns

(Qj(t),Qk(t)

).

J. Gibbons et al. / Physics Letters A 372 (2008) 1024–1033 1031

Explicitly, one takes the pairing of the coadjoint equation

(50)Am = −∑n,s

ad∗Gns∗As

Am+n−1

with a sequence of smooth functions {ϕm(q)} and finally ob-tains the equations for Qj and Pj in canonical form,

(51)dQj

dt=

∂HN

∂Pj

,dPj

dt= −∂HN

∂Qj

.

These singular solutions of EPSymp are also solutions of theEuler–Poincaré equations on the diffeomorphisms (EPDiff). Inthe latter case, the single-particle solutions reduce to the pulsonsolutions for EPDiff [3]. Thus, the singular pulson solutions ofthe EPDiff equation arise naturally from the single-particle dy-namics on phase-space. A similar result also holds in higherdimensions [4].

5.3.1. Further remarks on singular solutionsAnother kind of singular solution for the moments may be

obtained by considering the cold-plasma solution of the Vlasovequation

(52)f (q,p, t) =∑j

ρj (q, t)δ(p − Pj (q, t)

).

Indeed exchanging the variables q ↔ p in the single particlePDF leads to the following expression

(53)f (q,p, t) =∑j

ψj (p, t)δ(q − λj (p, t)

)

which is always a solution of the Vlasov equation because ofthe symmetry in q and p. This leads to the following singularsolutions for the moments:

(54)An(q, t) =∑j

∫dp pnψj (p, t)δ

(q − λj (p, t)

).

At this point, if one considers a Hamiltonian depending onlyon A1 (i.e. one considers the action of cotangent lifts of dif-feomorphism), then it is possible to drop the p-dependence inthe λ’s and thereby recover to the singular solutions previouslyfound for Eq. (26).

5.4. Examples of simplifying truncations and specializations

The problem presented by the coadjoint motion equa-tion (43) for geodesic evolution of p-moments under EPDiffmay be simplified, by truncating the Poisson bracket to a finiteset. Such truncations are not in general consistent with the fulldynamics; in the rarer cases where they are consistent, they willbe referred to as ‘reductions’ as in [32]. These moment dynam-ics may be truncated to a Hamiltonian system, at any stage bysimply modifying the Lie algebra in the KMLP bracket to van-ish for weights m + n − 1 greater than a chosen cut-off value.

5.4.1. Simplifying truncationsFor example, if we truncate the sums to m,n = 0, 1, 2 only,

then Eq. (43) produces the coupled system of partial differentialequations,

∂A0

∂t= −∂q(A0β1) − 2A1∂qβ2 − 2β2∂qA1,

∂A1

∂t= −A0∂qβ0 − 2A1∂qβ1 − β1∂qA1

− 3A2∂qβ2 − 2β2∂qA2,

(55)∂A2

∂t= −2A1∂qβ0 − 3A2∂qβ1 − β1∂qA2.

The fluid closure of system (55) neglects A2 and may be writtenas

∂A0

∂t= −∂q(A0β1),

(56)∂A1

∂t= −A0∂qβ0 − 2A1∂qβ1 − β1∂qA1.

When A1 = (1−α2∂2q )β1 and β0 = A0, this system becomes

the two-component Camassa–Holm system studied in [33–35].For this case, the fluid closure system (56) is equivalent to thecompatibility for dλ/dt = 0 of a system of two linear equa-tions,

(57)∂2xψ +

(−1

4+ A1λ + A2

0λ2)

ψ = 0,

(58)∂tψ = −(

1

2λ+ β0

)∂xψ +

1

2ψ∂xβ1.

The first of these (57) is an eigenvalue problem known asthe Schrödinger equation with energy dependent potential. Be-cause the eigenvalue λ is time independent, the evolution ofthe nonlinear fluid closure system (56) is said to be isospec-tral. The second equation (58) is the evolution equation for theeigenfunction ψ .

The fluid closure system for geodesic flow of the first twoVlasov moments also has a semidirect product structure [36]which allows for singular solutions for both A0 and A1 in thecase that βs = G ∗ As , s = 0, 1. The behavior of these singu-lar solutions will be investigated in future work. In particularwe would like to understand whether these singularities mayemerge spontaneously as for the EPDiff equation. Remarkably,a similar system of equations also arises in the study of imag-ing using a process of template matching with active templates,known as metamorphosis [37].

Setting A0 and A2 both initially to zero in (55) reduces thesethree equations to the single equation

(59)∂A1

∂t= −β1∂qA1 − 2A1∂qβ1.

Finally, if we assume that G in the convolution β1 = G ∗ A1 isthe Green’s function for the operator relation

(60)A1 =(1 − α2∂2

q

)β1

for a constant length-scale α, then the evolution equation forA1 reduces to the integrable Camassa–Holm (CH) equation [3]

1032 J. Gibbons et al. / Physics Letters A 372 (2008) 1024–1033

in the absence of linear dispersion. This is the one-dimensionalEPDiff equation, which has singular (peakon) solutions.

Thus, even very drastic restrictions of the p-moment systemstill lead to interesting special cases, some of which are inte-grable and possess emergent coherent structures among theirsolutions. This feature bodes well for future investigations ofthe EPSymp p-moment equations.

For example, one might consider extending phase space toconsider orientation dependence.

5.4.2. Geodesic moment equations for anisotropic interactionsTake the purely quadratic Hamiltonian on s∗ ⊕ so(3)∗ (with

s := TeSymp) defined by

H [f ] =∫ ∫ ∫

f (q,p,g)(G ∗ f )(q,p,g)dq dp d3g

with notation g = gaea , pairing 〈ea , eb〉 = δa

b and Lie bracket[gb,gc] = εa

bcga ,

(G ∗ f )(q,p,g)

=∫ ∫ ∫

G(q,q ′,p,p′,g,g′)f (q ′,p′,g′) dq ′ dp′ d3g′.

The geodesic Vlasov equation is given in [6] as

∂f

∂t= −{f ,G ∗ f }1,

where {·,·}1 is the sum of the canonical bracket on s∗ and theLie–Poisson bracket on so(3)∗,

{f ,k}1 = {f ,k} +⟨g,

[δf

δg,δk

δg

]⟩= {f ,k} + g · δf

δg× δk

δg,

in vector notation for elements of so(3)∗. Now, assume that thekernel G can be expanded as

G(q,q ′,p,p′,g,g′)= K0(q,q ′) + pK1(q,q ′)p′ + gaK

ab(q,q ′)g′b

so that the quadratic Hamiltonian becomes

H =∫

ρ(q)(K0 ∗ ρ)(q) dq +∫

M(q)(K1 ∗ M)(q)dq

+∫ ⟨

G(q), (K • G)(q)⟩dq

where we have defined

K • G(q) :=∫

Kab(q,q ′)Gb(q′) dq ′ea ∈ so(3).

The moment equations for mass density ρ(q, t) =∫

f dp d3g,momentum density M(q, t) =

∫pf dp d3g and orientation

density G(q, t) =∫

gf dp d3g are presented in [6]. For thequadratic Hamiltonian above these become

(61)∂ρ

∂t= − ∂

∂q(ρu),

(62)∂G

∂t= − ∂

∂q(Gu) + ad∗

K•G G,

(63)∂M

= −£uM − ρ∂

(K0 ∗ ρ) −⟨G,

∂(K • G)

⟩

∂t ∂q ∂q

where u = K1 ∗ M and ad∗K•G G = −(K • G) × G. When G ∈

Den(R) ⊗ so(3)∗, one recognizes the Hamiltonian part of theLandau–Lifschitz equation on the right hand side in the secondequation. For K1 = (1−∂2

q )−1 and K0 = δ(q −q ′), this extendsthe Camassa–Holm system to several components.

Further specializations and truncations of these equationswill be explored elsewhere.

6. Open questions for future work

Before closing, we mention two other open questions aboutthe solution behavior of the p-moments of EPSymp.

6.1. Emergence of singular solutions

Several open questions remain for future work. The first ofthese is whether the singular solutions found here will emergespontaneously in EPSymp dynamics from a smooth initialVlasov PDF. This spontaneous emergence of the singular so-lutions does occur for EPDiff. In fact, integrability of EPDiff inone dimension by the inverse scattering transform shows thatonly the singular solutions (peakons) are allowed to emergefrom any confined initial distribution in that case [3] (this alsohappens in higher dimensions as it is shown by numerical sim-ulations). In contrast, the point vortex solutions of Euler’s fluidequations (which are isomorphic to the cold plasma singularsolutions of the Vlasov–Poisson equation) while comprising aninvariant manifold of singular solutions, do not spontaneouslyemerge from smooth initial conditions. Nonetheless, somethingquite analogous to the singular solutions is seen experimentallyfor cold plasma in a Malmberg–Penning trap [38]. Therefore,one may ask which outcome will prevail for the singular so-lutions of EPSymp. Will they emerge from a confined smoothinitial distribution, or will they only exist as an invariant mani-fold for special initial conditions? One might argue that in twodimensions, the EPSymp equation encompasses the equationof vorticity and thus spontaneous emergence of point vorticesshould not occur. However it is possible that the choice of themetric plays an important role in this matter. Of course, the in-teractions of these singular solutions for various metrics and theproperties of their collective dynamics is a question for futurework.

6.2. Possible connections with the Bloch–Iserles system

The EPSymp equation is surprisingly similar in constructionto another important integrable geodesic equation on the linearHamiltonian vector fields (Hamiltonian matrices), which has re-cently been proposed [39]. This is a finite dimensional equationwhose dynamical variables are symmetric matrices. Recently ithas been shown that this system may be written as the geodesicequation on the group of the linear canonical transformationsSp(R;2n) [40]. This association to canonical transformationsraises the question whether the Bloch–Iserles system is also re-lated to the geodesic Vlasov equations discussed here.

J. Gibbons et al. / Physics Letters A 372 (2008) 1024–1033 1033

Acknowledgements

We are grateful to U. Amaldi, P. Channell, V. Putkaradzeand R. Zennaro for helpful discussions during the course of thiswork. The work of D.D.H. was partially supported by the RoyalSociety of London Wolfson Award and by the US DOE, Officeof Science, Applied Mathematics Research program. The au-thors are also grateful for their participation in the ESF programMISGAM and the EU program ENIGMA.

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