J. Plasma Fusion Res. SERIES, Vol. 8 (2009)

Fusion Modeling in Vlasov-Like Models
Antonina N. FEDOROVA and Michael G. ZEITLINIPME RAS, Russian Academy of Sciences,
V.O. Bolshoj pr., 61, 199178, St. Petersburg, Russia
http://www.ipme.ru/zeitlin.html; http://mp.ipme.ru/zeitlin.html(Received: 1 September 2008 / Accepted: 9 December 2008)

The methods developed by the authors are applied to a few important reductions of BBGKY hierarchy,
namely, various examples of Vlasov-like systems important both for fusion modeling and for particular physicalproblems related to plasma/beam physics. As in companion paper [1] devoted to the general situation of kinetic
hierarchies, we concentrate mostly on the phenomena of localization and pattern formation to create a realistic
model for fusion state in plasma.
Keywords: fusion, non-equilibrium ensemles, kinetic equations, multiresolution, localization

1. Vlasov-Poisson System. Description

by powerful technique of local nonlinear harmonic analysis, based on representation theory of underlying symmetries of functional space such as aﬃne or more general.
The solution has the multiscale/multiresolution decomposition via nonlinear high-localized eigenmodes, which corresponds to the full multiresolution expansion in all underlying time/phase space scales. Starting from VlasovPoisson equations, we consider the approach based on
multiscale variational-wavelet formulation. We give theexplicit representation for all dynamical variables in thebase of compactly supported wavelets or nonlinear eigenmodes. Our solutions are parametrized by solutions of anumber of reduced algebraical problems, one from which
is nonlinear with the same degree of nonlinearity as initial problem and the others are the linear problems whichcorrespond to the particular method of calculations insideconcrete wavelet scheme. Because our approach started
from variational formulation we can control evolution of
instability on the pure algebraical level of reduced algebraical system of equations. This helps to control stability/unstability scenario of evolution in parameter spaceon pure algebraical level. In all these models numericalmodeling demonstrates the appearance of coherent highlocalized structures and as a result the (meta)stable patterns
formation or unstable chaotic behaviour. Analysis basedon the non-linear Vlasov equations leads to more clear understanding of collective eﬀects and nonlinear beam dynamics of high intensity beam propagation in periodicfocusing and uniform-focusing transport systems. We consider the following form of equations [4]:In this paper we consider the applications of our approach based on variational multiresolution technique [1,2]to the systems with collective type behaviour described bysome forms of Vlasov-Poisson/Maxwell equations, some
important reduction of general BBGKY hierarchy [3,4]. It
based on general ideology presented in companion paper[1], devoted to the general case of kinetic hierarchies. Suchapproach may be useful in all models in which it is possible
and reasonable to reduce all complicated problems related
to statistical distributions to the problems described by thesystems of (non)linear ordinary/partial diﬀerential/integral
equations with or without some (functional) constraints.
E.g., in periodic accelerators and transport systems at thehigh beam currents and charge densities the eﬀects of theintense self-ﬁelds, which are produced by the beam spacecharge and currents, determinine (possible) equilibrium
states, stability and transport properties according to underlying nonlinear dynamics. The dynamics of such spacecharge dominated high brightness beam systems can provide the understanding of the instability phenomena suchas emittance growth, mismatch, halo formation related tothe complicated behaviour of underlying hidden nonlinear
modes outside of perturbative tori-like KAM regions [4].
Our analysis based on the variational-wavelet approach allows to consider polynomial and rational type of nonlinearities. In some sense in this particular case this approach is
direct generalization of traditional nonlinear δF approach[4] in which weighted Klimontovich representation
δ fj = aj

Nj
�
i=1

w ji δ(x − x ji )δ(p − p ji )

(1)
∂
∂ �
∂ψ � ∂+ py − k x (s)x +
−
∂s
∂x
∂y
∂x ∂p x�∂ψ � ∂ �
fb (x, y, p x , py , s) = 0,ky (s)y +∂y ∂py
�
� ∂2
2πKb
∂2 �
ψ=−
dp x dpy fb ,+∂x2 ∂y2Nb�dxdydp xdpy fb = Nb .
�∂or self-similar decompostion like
δn j = b j

Nj
�
i=1

w ji s(x − x ji ),(2)where s(x − x ji ) is a shape function of distributing particles on the grids in conﬁguration space, are replaced

+ px

(3)

author’s e-mail: zeitlin@math.ipme.ru, anton@math.ipme.ru

126©2009 by The Japan Society of Plasma
Science and Nuclear Fusion Research

A.N. Fedorova and M.G. Zeitlin, Fusion Modeling in Vlasov-Like Models

where x are the generalized space coordinates or phasespace coordinates, and s is ”time” coordinate. After someanzatzes [2] the main reduced problem may be formulated
as the system of ordinary diﬀerential equations

The corresponding Hamiltonian for transverse singleparticle motion is given by
1 2
1(p x + p2y ) + [k x (s)x222+ky (s)y2 ] + H1 (x, y, p x , py , s) + ψ(x, y, s),H(x, y, p x , py , s) =d fi
= Pi ( f, s); f = ( f1 , ..., fn ),ds
i = 1, . . . , n; max deg Pi = p; max deg Qi = q(4)

Qi ( f )where H1 is nonlinear (polynomial/rational) part of thefull Hamiltonian and corresponding characteristic equations are:2d x∂
+ k x (s)x + ψ(x, y, s)
2
∂x
ds
d2 y
∂
+ ky (s)y + ψ(x, y, s)
∂y
ds2

i

(5)

2. Multiscale Representation

We obtain our multiscale/multiresolution representations for solutions of these equations via variationalwavelet approach. We decompose the solutions as
fb (s, x, y, p x , py ) =∞�L(Qi j , λ, αI ) = M(Pi j , λ, β J ),ψ(s, x, y) =⊕δi f (s, x, y, p x , py ),⊕δ j ψ(s, x, y),j= jc

x(s) =∞
�⊕δk x(s),y(s) =k=kc∞
�⊕δ� y(s),

(6)

�=�c

� = max{p, q}

where set
(ic , jc , kc , �c )corresponds to the coarsest level of resolution c in the fullmultiresolution decomposition [5](8)fi (s) = fi (0) +

∞
�

(10)

W j.

(11)

j=c

In some sense it is some generalization of the old δFapproach [4]. Let L be an arbitrary (non) linear diﬀerential/integral operator with matrix dimension d, which acts
on some set of functions��Ψ ≡ Ψ(s, x) = Ψ1 (s, x), . . . , Ψd (s, x) ,s, x ∈ Ω ⊂ Rn+1

(17)

x = (x, y, p x , py )(18)

we will haveΨ(s, x) =�ai j Ui ⊗ V j (s, x),(19)(i, j)∈Z 2(12)

from L2 (Ω):
LΨ ≡ L(R(s, x), s, x)Ψ(s, x) = 0,

λki fk (s),where coeﬃcients λki are the roots of the corresponding
reduced algebraical (polynomial) problem RSAE. Consequently, we have a parametrization of solution of initial problem by the solution of reduced algebraical problem. The obtained solutions are given in this form, where
fk (t) are basis functions obtained via multiresolution expansions and represented by some compactly supported
wavelets. As a result the solution of equations has the following multiscale/multiresolution decomposition via nonlinear high-localized eigenmodes, which corresponds tothe full multiresolution expansion in all underlying scales
starting from coarsest one. For

we have forL2 (R) = Vc

N
�
k=1

Introducing detail space W j as the orthonormal complement of V j with respect to�(9)
V j+1 : V j+1 = V j
W j,f, ψ, x, y ⊂ L2 (R)(16)

and the degree of this algebraical system coincides with degree of initial diﬀerential system. So, we have the solutionof the initial nonlinear (rational) problem in the form

(7)

Vc ⊂ Vc+1 ⊂ Vc+2 ⊂ . . .

(15)

where operators L and M are algebraization of RHS and
LHS of initial problem and are unknowns of reduced system of algebraical equations (RSAE). After solution of
RSAE (15) we determine the coeﬃcients of wavelet expansion and therefore obtain the solution of our initial problem. It should be noted that if we consider only truncatedexpansion with N terms then we have the system of N × n
algebraical equations with degree

i=ic

∞
�

i

or a set of such systems corresponding to each independent coordinate in phase space. They have the ﬁxed initial(or boundary) conditions fi (0), where Pi , Qi are not morethan polynomial functions of dynamical variables f j andhave arbitrary dependence on time. As result we have thefollowing reduced algebraical system of equations on theset of unknown coeﬃcients λki of localized eigenmode expansion:

= 0,
= 0.

(14)

V j (s) = VNj,slow (s) +�Vlj (ωl s),ωl ∼ 2l

U (x) = Ui,slow
(x) +
M

�

Uim (km x),

km ∼ 2 m ,

i

(13)

127

l≥N

m≥M

A.N. Fedorova and M.G. Zeitlin, Fusion Modeling in Vlasov-Like Models

slow
These formulae give us expansion into the slow part Ψ N,M
and fast oscillating parts for arbitrary N, M. So, we may
move from coarse scales of resolution to the ﬁnest one
for obtaining more detailed information about our dynamical process. The ﬁrst terms in the RHS correspond on
the global level of function space decomposition to resolution space and the second ones to detail space. It should be
noted that such representations give the best possible localization properties in the corresponding (phase)space/time
coordinates. In contrast with other approaches this formulae do not use perturbation technique or linearization
procedures. So, by using wavelet bases with their good(phase) space/time localization properties we can describe
high-localized (coherent) structures in spatially-extended
stochastic systems with collective behaviour. Modelingdemonstrates the appearance of (meta)stable patterns formation from high-localized coherent structures (fundamental eigenmodes, Fig. 1) or chaotic behaviour. On Fig.
2 we present contribution to the full expansion from coarsest level of decomposition. Fig. 3 shows the representations for full solutions, constructed from the ﬁrst six scales(dilations) and demonstrates (meta) stable localized pattern formation in comparison with chaotic-like behaviour(Fig. 4) outside of KAM region. We can control the type of
behaviour on the level of reduced algebraical system (15)which plays the role of Generalized Dispersion Relations(GDR).

resolution to the ﬁnest one to obtain more detailed information about our dynamical process. In this way we givecontribution to our full solution from each scale of resolution or each time/space scale or from each nonlinear eigenmode. Starting from some electrostatic approximation of
Vlasov-Maxwell system and rms/rate dynamical models
we consider the approach based on variational-wavelet formulation. We give explicit representation for all dynamicalvariables in the bases of compactly supported wavelets or
nonlinear eigenmodes. Our solutions are parametrized by
the solutions of a number of reduced standard algebraicalproblems like GDR mentioned above.

4. Rate Equations

In thin-beam approximation with negligibly small
spread in axial momentum for beam particles we have inLarmor frame the following electrostatic approximation
for Vlasov–Maxwell equations [4]:∂F∂F �∂ψ � ∂F
∂F
+ x�
+ y�
− k(s)x +∂s∂x∂y∂x ∂x���∂ψ ∂F− k(s)y += 0,(20)∂y ∂y�
� ∂2
∂x

+
2

2πK
∂2 �
ψ=−
2
N
∂y

�

dx� dy� F,(21)where ψ(x, y, s) is normalized electrostatic potential andF(x, y, x� , y� , s) is distribution function in transverse phasespace (x, y, x� , y� , s) with normalization
�
�
(22)
N=
dxdyn,
n(x, y, s) =dx� dy� F,

3. Rate/RMS Models. Description

In this part we consider the applications of our technique based on the methods of local nonlinear harmonic
analysis to nonlinear rms/rate equations for averaged quantities related to some particular case of nonlinear Vlasov–Maxwell equations [4]. We consider electrostatic approximation for a thin beam. This approximation is a particular important case of the general reduction from statistical collective description based on Vlasov–Maxwell equations to a ﬁnite number of ordinary diﬀerential equations
for the second moments related quantities (beam radiusand emittance). In our case these reduced rms/rate equations also contain some distribution averaged quantities besides the second moments, e.g. self-ﬁeld energy of thebeam particles. Such model is very eﬃcient for analysis
of many problems related to periodic focusing accelerators, e.g. heavy ion fusion and tritium production. So,
we are interested in the understanding of collective properties, nonlinear dynamics and transport processes of intense
non-neutral beams propagating through a periodic focusing ﬁeld. Our approach allows to consider rational typeof nonlinearities in rms/rate dynamical equations containing statistically averaged quantities also. The solution has
the multiscale/multiresolution decomposition via nonlinear high-localized eigenmodes (Fig. 1), which corresponds
to the full multiresolution expansion in all underlying internal hidden scales. We may move from coarse scales ofwhere K is self-ﬁeld perveance which measures self-ﬁeld
intensity. Introducing self-ﬁeld energy�1dxdy|∂2ψ/∂x2 + ∂2 ψ/∂y2 |(23)E(s) =
4πK
we have obvious equations for root-mean-square beam radius R(s)R(s) =< x2 + y2 >1/2

(24)

and unnormalized beam emittance
ε2 (s) = 4(< x�2 + y�2 >< x2 + y2 >− < xx� − yy� >),(25)

which appear after averaging second-moments quantities
regarding distribution function F [4]:d2 R(s) �ε2 (s)
K(1 + Δ) �R(s) =+
k(s)R(s)−22ds
2R (s)
4R3 (s)��2dε (s)dR K(1 + Δ) dE(s)+ 8R2 (s)−= 0, (26)ds
ds 2R(s)dswhere the term
K(1 + Δ)/2

128

(27)

A.N. Fedorova and M.G. Zeitlin, Fusion Modeling in Vlasov-Like Models

may be ﬁxed in some interesting cases, but generally wehave it only as averageK(1 + Δ)/2 = − < x∂ψ/∂x + y∂ψ/∂y >where
R ≡ R(s, ∂/∂s, Ψ).

(28)

Let us consider now the N mode approximation forsolution as the following expansion in some high-localized
wavelet-like basis:w.r.t. distribution F. Anyway, the rate equations represent reasonable reductions for the second-moments related
quantities from the full nonlinear Vlasov-Poisson system.
For trivial postulated distributions Davidson e.a. [4] foundadditional reductions. For KV distribution (step–function
density) the second rate equation is trivial,ε(s) = constΨN (s) =Accordingly to our approach which allows us to ﬁnd
exact solutions as for Vlasov- like systems as for rms-likesystems we need not to ﬁx particular case of distributionfunction F(x, y, x� , y� , s). Our consideration is based on thefollowing multiscale N-mode anzatz:ψN (x, y, s) =N�b j1 , j2 , j3

j1 , j2 , j3 =1

3�B jk (x, y, s).

H(Qi j , aiN , αI ) = M(Pi j , aiN , β J ),(30)

I = (i1 , ..., iq+2 ),These formulae provide multiresolution representation forvariational solutions of our system. Each high-localized
mode/harmonics A j (s) corresponds to level j of resolution
from the whole underlying inﬁnite scale of spaces:

where p is the degree of polynomial operator� �
φi1 ...φ˙is ...φiq+2 ,αI = {αi1 ...αiq+2 } =where q is the degree of polynomial operator Q,i� = (1, ..., q + 2),

φ˙is = dφis /ds.

(43)

We may extend our approach to the case when we haveadditional constraints on the set of our dynamical variables
Ψ = {R, ε}(44)

and additional averaged terms also. In this case by using
the method of Lagrangian multipliers we may again apply
the same approach but for the extended set of variables.
As a result we obtain the expanded system of algebraicalequations analogous to our system. Then, after reduction
we can again extract from its solution the coeﬃcients ofthe expansion. It should be noted that if we consider only
truncated expansion with N terms then we have the system
of N × d algebraical equations with the degreefrom L2 (Ω):

(35)

(42)

i1 ,...,iq+2

where the closed subspace V j ( j ∈ Z) corresponds to level jof resolution, or to scale j. The construction of tensor algebra based on the multiscale bases [5] is considered in [2].
We will consider rate equations as the following operator
equation. Let L, P, Q be an arbitrary nonlinear (rational in
dynamical variables) ﬁrst-order matrix diﬀerential operators with matrix dimension (d = 4 in our case) corresponding to the system of equations, which act on some set offunctions��Ψ ≡ Ψ(s) = Ψ1 (s), . . . , Ψd (s) , s ∈ Ω ⊂ R (33)LΨ ≡ L(R, s)Ψ(s) = 0(40)

1≤ jk ≤p+1

(32)

(34)

J = ( j1 , ..., j p+1 )are multiindices, by means of which are labelled α I and βI ,the other coeﬃcients of RSAE:� �
φ jk ,(41)
β J = {β j1 ... j p+1 } =(31)

Q(R, s)Ψ(s) = P(R, s)Ψ(s)(39)

where operators H and M are algebraization of RHS and
LHS of initial problem. Qi j (Pi j ) are the coeﬃcients of
LHS (RHS) of the initial system of diﬀerential equations
and as consequence are coeﬃcients of RSAE (39).

k=1. . . V−2 ⊂ V−1 ⊂ V0 ⊂ V1 ⊂ V2 ⊂ . . . ,(37)

We have exactly dN algebraical equations for dN unknowns ar . So, variational approach reduced the initialproblem to the problem of solution of functional equations
at the ﬁrst stage and some algebraical problems at the second stage. As a result we have the following reduced algebraical system of equations (RSAE) on the set of unknown
coeﬃcients aiN of the expansion:

5. Multiscale Representation

k=1

arN φr (s).

We will determine the coeﬃcients of expansion from thefollowing variational condition:
�N(38)
Lk ≡ (LΨN )φk (s)ds = 0

and we have only one nontrivial rate equation for rms beam
radius. The ﬁxed-shape density proﬁle ansatz for axisymmetric distributions also leads to similar situation: emittance conservation and the same envelope equation with
two shifted constants only.

i1 ,...,i5 =1

N
�
r=1

(29)

F N (x, y, x� , y� , s) =N5��
ai1 ,...,i5
Aik (x, y, x� , y� , s),

(36)

or

� = max{p, q}

129

(45)

A.N. Fedorova and M.G. Zeitlin, Fusion Modeling in Vlasov-Like Models

and the degree of this algebraical system coincides withthe degree of the initial system. So, after all we have thesolution of the initial nonlinear (rational) problem in theform
RN (s) = R(0) +N�

the full space of hierarchy of partitions deﬁned on a domain of the deﬁnition in the whole phase space. Nevertheless, our Generalized Dispersion Relations, like (15) or(39), give some chance to build the controllable localizedstate (Fig. 7) starting from initial chaotic- like partition(Fig. 4) via process of controllable self-organization. Figures 5 and 6 demonstrate two subsequent steps towards
creation important fusion or conﬁnement state, Fig. 7,
which can be characterized by the presence of a few important physical modes only in contrast with the opposite,
chaotic-like state, described by inﬁnite number of contributed modes. Of course, such conﬁnement states, characterized by zero measure and minimum entropy (we namethem waveletons), can be only metastable. But these longliving ﬂuctuations can and must be very important fromthe practical point of view, because the averaged time ofexistence of such states may be even more than needed forpractical realization, e.g., in controllable fusion processes.
Further details will be considered elsewhere.

akN φk (s),k=1

εN (s) = ε(0) +N�bkN φk (s),

(46)

k=1

where coeﬃcients akN , bkN are the roots of the corresponding reduced algebraical (polynomial) problem RSAE. Consequently, we have a parametrization of the solution of theinitial problem by solution of reduced algebraical problem. The problem of computations of coeﬃcients α I , β Jof reduced algebraical system may be explicitly solved inwavelet framework via various approaches [5]. The obtained solutions are given in the form, where φk (s) areproper wavelet bases functions (e.g., periodic or boundary). It should be noted that such representations give thebest possible localization properties in the corresponding(phase)space/time coordinates. In contrast with diﬀerent
approaches these formulae do not use perturbation technique or linearization procedures and represent dynamics
via generalized nonlinear localized eigenmodes expansion.
Our N mode construction gives the following general multiscale representation:�Ri (ωi s), ωi ∼ 2i ,R(s) = RNslow (s) +
i≥N

ε(s)

=

εNslow (s) +�ε j (ω j s),ω j ∼ 2 j,

(47)

j≥N

where Ri (s), ε j (s) are represented by some family of(nonlinear) eigenmodes and gives the full multiresolution/multiscale representation in the high-localized
wavelet bases. As a result we can construct various unstable (Fig. 4) or (meta) stable (non-gaussian!) patterns(Fig. 3, Fig. 7) from high-localized (coherent) fundamental modes (Fig. 1) in complicated stochastic systems withcomplex collective behaviour. Deﬁnitely, partition(s) as
generic dynamical variable(s) cannot be postulated (e.g.,
as some sort of gaussians or something else ﬁxed apriori)
but need to be computed as solutions of proper stochastic
dynamical evolution model. Only after that, it is possible
to calculate other dynamical quantities and physically interesting averages.

Fig. 1 Localized modes.
[1] A.N. Fedorova and M.G. Zeitlin, Thorny Path to Fusion in
Plasma: Fusion State as a Waveleton, This Volume.
[2] A.N. Fedorova and M.G. Zeitlin, a full list of papers/preprints (including links to arXiv) can be found on web
sites above.
[3] A. H. Boozer, Rev. Mod. Phys., 76, 1071 (2004).
[4] R.C. Davidson and H. Qin, Physics of Intense Charged Particle Beams in High Energy Accelerators (World Scientiﬁc,
Singapore, 2001); A.W. Chao, Physics of Collective Beam
Instabilities in High Energy Accelerators (Wiley, New York,
1993); R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics, (Wiley, New York, 1975).
[5] Y. Meyer, Wavelets and Operators (Cambridge Univ. Press,
1990); D. Donoho, WaveLab (Stanford, 2000).

6. Towards Energy Conﬁnement

Analysis and modeling considered in this and companion paper [1] describes, in principle, a scenario for thegeneration of controllable localized (meta) stable fusionlike state (Fig.3 and Fig.7). Deﬁnitely, chaotic-like unstable partitions/states (Fig. 4) dominate during nonequilibrium evolution. It means that (possible) localized(meta) stable partitions have measure equal to zero a.e. on

130

A.N. Fedorova and M.G. Zeitlin, Fusion Modeling in Vlasov-Like Models

−3

x 10
10

0.6
0.4

5
0.2
0

0
−0.2
−0.4
30

−5
30
20
10
0

0

10

5

15

25

20

20

30

10
0

0

10

5

15

25

20

30

Fig. 5 Self-organization, step 1: towards conﬁnement.
Fig. 2 Localized two-dimensional partition.

8

10

6
5

4
0

2
−5

0
−10
60

−2
30
20
10
0

0

10

5

15

25

20

40

30

20
0

0

20

10

30

50

40

60

Fig. 6 Self-organization, step 2: towards conﬁnement.
Fig. 3 Metastable pattern.

0.5

10

0.4

8
6

0.3

4

0.2

2

0.1

0
−2
40

0
80

40

30

80

60
60

40

10
0

20
0

20

10

40

20

30

20

0

0

Fig. 7 Localized pattern(waveleton): energy conﬁnement state.
Fig. 4 Chaotic pattern.

131