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Reductions of Kinetic Equations to Finite Component Systems
Alexander Chesnokov1 , Maxim Pavlov2
L a v r e n t ' v In s t it u t e o f H y d r o d y n a m ic s , e S ib e r ia n D iv is io n o f R A S , N o v o s ib ir s k ; 2 S e c t or o f M a t h e m a t i c a l P h y s i c s , L e b e d e v P h y s i c a l I n s t i t u t e o f R A S , M o s c ow
1

27 April, 2012

A l e x a n d e r C h e s n o ko v ,

M axim Pavlo v (1 L avren t'en tndtico m po fsiH od ro p pno ach s, S ib erian D ivisio n o f R A S , prio,vo si1 2rsk ; 2 S ector o f m o m v I s e t u t e o t i y n a d y r a m ic e 27 A N l 20 bi 1 / 19


Vlasov (Collisionless Boltzmann) Kinetic Equation
t + p
x

p ux = 0,

where ( is a distribution function, x is a space coordinate, t is a time variable, p is a momentum) u=
Z

dp .



Following the approach developed in plasma physics, one can introduce an in...nite set of moments A= The Benney hydrodynamic chain is Ak + A t
A l e x a n d e r C h e s n o ko v ,

k

Z

p k dp .



k +1 x

+ kAk

1

A0 = 0, k = 0, 1, 2, ... x

M axim Pavlo v (1 L avren t'en tndtico m po fsiH od ro p pno ach s, S ib erian D ivisio n o f R A S , prio,vo si1 2rsk ; 2 S ector o f m o m v I s e t u t e o t i y n a d y r a m ic e 27 A N l 20 bi 2 / 19


Kinetic Equation Describing High Frequency Waves in Electron Plasma
t + p where Ex = 1
Z x

E p = 0,
Z

dp ,

Et =

p dp .

Z



Introducing moments of the distribution function A=
k

p k dp ,



the collisionless kinetic equation can be re-written as the nonlocal chain Ak + A t where dE = (1 followsChesnokov,(??).i A lexan d er from M ax A0 )dx + A1 dt
k +1 x

+ kAk

1

E = 0, k = 0, 1, 2, ...,

m Pavlo v (1 L avren t'en tndtico m po fsiH od ro p pno ach s, S ib erian D ivisio n o f R A S , prio,vo si1 2rsk ; 2 S ector o f m o m v I s e t u t e o t i y n a d y r a m ic e 27 A N l 20 bi 3 / 19


Waterbag Reduction

Ak =

1 k +1

n =1



N

n

(an )

k +1

,

n =1



N

n

= 0,

where i are arbitrary constants. Then Benney hydrodynamic chain reduces to the form ! N (ai )2 i n at + = 0, + na 2 n =1
x

where N is an arbitrary positive integer. Nonlocal chain reduces to the nonlocal N component system
i i at + a i ax + E = 0 ,

dE = (1

A0 )dx + A1 dt .

A l e x a n d e r C h e s n o ko v ,

M axim Pavlo v (1 L avren t'en tndtico m po fsiH od ro p pno ach s, S ib erian D ivisio n o f R A S , prio,vo si1 2rsk ; 2 S ector o f m o m v I s e t u t e o t i y n a d y r a m ic e 27 A N l 20 bi 4 / 19


Local Representation
The nonlocal chain Ak + A t where dE = (1 A0 )dx + A1 dt can be written in the evolution form E t = A1 , A1 + A2 + E ( 1 t x Ak + A t
k +1 x k +1 x

+ kAk

1

E = 0, k = 0, 1, 2, ...,

Ex ) = 0,

+ kAk

1

E = 0, k = 2, 3, ...,

where A0 = 1 Ex . In such a form this non-hydrodynamic chain has just one local conservation law of Energy

(A2 + E 2 )t + A3 = 0. x
A l e x a n d e r C h e s n o ko v ,

M axim Pavlo v (1 L avren t'en tndtico m po fsiH od ro p pno ach s, S ib erian D ivisio n o f R A S , prio,vo si1 2rsk ; 2 S ector o f m o m v I s e t u t e o t i y n a d y r a m ic e 27 A N l 20 bi 5 / 19


Local Representation
The nonlocal system
k k a t + a k ax + E = 0 ,

dE = (1

A0 )dx + A1 dt

transforms to N component evolution system (k = 2, 3, ..., N )
k k at + ak ax + E = 0,

1 Et = 2 where

n =2



N

n

1 (a ) + 2
n2

Ex +
1

n =2



N

n

a

n

1

!

2

,

a=

1

1
1

1

n =2



N

n

a

n

Ex

!

.

A l e x a n d e r C h e s n o ko v ,

M axim Pavlo v (1 L avren t'en tndtico m po fsiH od ro p pno ach s, S ib erian D ivisio n o f R A S , prio,vo si1 2rsk ; 2 S ector o f m o m v I s e t u t e o t i y n a d y r a m ic e 27 A N l 20 bi 6 / 19


Hamiltonian Structure
The waterbag reduction can be written in the Hamiltonian form Et =

n =2



N

H 1 H H i , i = 2, 3, ..., N , , at = x i + n a a E i !3 3 1 5 dx ,
2 Ex

which is determined by the local Hamiltonian 2 Z 1 42 1N 1 H= E + n (an )3 + 2 Ex + 2 3 n =2 31 the momentum 2 Z 1N P= 4 2 n =2
n

n =2



N

n

a

n

(an )2 +

1 2

1

n =2



N

n

a

n

!

2

+

E

x

1 n =2



N

n

an +

2

1

3

and N

2 parametric Casimir Q=
Z

5 dx

n =2



N

~ n an ,

n =2



N

~ n = 0.

A l e x a n d e r C h e s n o ko v ,

M axim Pavlo v (1 L avren t'en tndtico m po fsiH od ro p pno ach s, S ib erian D ivisio n o f R A S , prio,vo si1 2rsk ; 2 S ector o f m o m v I s e t u t e o t i y n a d y r a m ic e 27 A N l 20 bi 7 / 19


Periodic Solutions
Each ...eld variable ak depend on a sole phase = x Then the nonlocal system
k k a t + a k ax + E = 0 ,

t .

dE = (1
N

A0 )dx + A1 dt

reduces to an ODE system ( and k are arbitrary constants) u0 = where a1 = u + , ak =
2

u2

+

2u 3

1

2 3u

2

k =2



k

(u 2 + 2 ) k

3 /2

,

[u 2 + 2 ] k

1 /2

, k = 2, 3, ..., N ,

This is N parametric periodic solution, whose characteristic velocity is determined by 1N 2 = k k , 2 k =2 which follows from comparison both above expressions for E 0 .
A l e x a n d e r C h e s n o ko v ,

M axim Pavlo v (1 L avren t'en tndtico m po fsiH od ro p pno ach s, S ib erian D ivisio n o f R A S , prio,vo si1 2rsk ; 2 S ector o f m o m v I s e t u t e o t i y n a d y r a m ic e 27 A N l 20 bi 8 / 19


Periodic Solutions
Since E = uu 0 , its explicit expression is v u u 2u 3 2 E = t u 2 + 1 3 3

k =2



N

k

(u 2 + 2 ) k

3 /2

.

Then a corresponding solution of the distribution function (x , t , p ) can be found due to the inverse formula 1 (x , t , p ) = 2 where Ak = 1 k +1
Z

e

iq p



k =0





( iq )k k A dq , k!

n =2



N

n

[(

[u 2 + 2 ] n

1 /2 k +1

)

(u + )

k +1

].

A l e x a n d e r C h e s n o ko v ,

M axim Pavlo v (1 L avren t'en tndtico m po fsiH od ro p pno ach s, S ib erian D ivisio n o f R A S , prio,vo si1 2rsk ; 2 S ector o f m o m v I s e t u t e o t i y n a d y r a m ic e 27 A N l 20 bi 9 / 19


Kodama Reduction
N component Kodama reductions 1 0 0 k at + a0 ax + E = 0, Ex = + aN , at + 2 possess the local Hamiltonian structure H(N ) H(N ) H( k , Et = , at = x N 0 E a a where the Hamiltonian is given by " Z 1 a0 N 1 k N H(N ) = E 2 + ( a 0 ) 2 ( Ex ) aa 2 3 k =1
0 at = N) k k

m =0



am a

km

!

= 0, k = 1, ..., N
x

, k = 1, 2, ..., N

1,

k

1 3

N1n

n =1 k =0



ak a

nkNn

a

#

and the Momentum is P=
A l e x a n d e r C h e s n o ko v ,

Z

"

1 2

N1 k =1



aa

kNk

+ a Ex dx .

0

#

M axim Pavlo v (1 L avren t'en tndtico m po fsiH od ro p pno ach s, S ib erian D ivisio n o f 2 7AA prN,o2o si2 irsk ; 2 S ector o f m o m v I s e t u t e o t i y n a d y r a m ic e R S , il v 0 1 b 10 / 19


Periodic Solutions
Each ...eld variable ai depend on a sole phase = x reduction a +a
0 t 00 ax

t . Then Kodama !

1 + E = 0, Ex = + a , a + 2
N k t

m =0



k

aa

mkm

= 0, k = 1, ..., N
x

becomes to the form of two ordinary di¤erential equations of the ...rst order (u = a0 ) uu 0 + E = 0, E 0 = + aN , while all other functions can be found iteratively a1 = 1 2 ,a = u 1 u 1 2 + (a1 ) 2
2

, ..., aN =

1 u

N +

1 2

N1 m =1



am a

Nm

where k are integration constants.
A l e x a n d e r C h e s n o ko v ,

M axim Pavlo v (1 L avren t'en tndtico m po fsiH od ro p pno ach s, S ib erian D ivisio n o f 2 7AA prN,o2o si2 irsk ; 2 S ector o f m o m v I s e t u t e o t i y n a d y r a m ic e R S , il v 0 1 b 11 / 19


Periodic Solutions
Then periodic solution has the form ( is an arbitrary constant) u0 = +
2

2 +2 2 u u

Z

uaN (u )du

and can be integrated in hyperelliptic functions. If N = 1, then u0 = if N = 2, then u0 = if N = 3, then u0 = and so on.
A l e x a n d e r C h e s n o ko v , 2 2 2

21 + 2; u u 2 22 1 + 2 + 3; u u u

23 2 3 + 2 + 13 2 + 15 ; u u u 3u

M axim Pavlo v (1 L avren t'en tndtico m po fsiH od ro p pno ach s, S ib erian D ivisio n o f 2 7AA prN,o2o si2 irsk ; 2 S ector o f m o m v I s e t u t e o t i y n a d y r a m ic e R S , il v 0 1 b 12 / 19


Waterbag Reduction. Heaviside Step Function
Let us consider the ansatz
N1

=

i =1


i

fi ( p
i +1

ai (t , x ) )

(p

a

i +1

(t , x ) ) .

We assume that ai < a fi = Then we have

, fi = const > 0. Let us suppose that

n =1 N





n

(i = 1, ..., N

1),

fN =

n =1



N

n = 0.

=

i =1



i (p

ai (t , x ) ).

Substitution to the kinetic equation t + p
A l e x a n d e r C h e s n o ko v ,

x

E p = 0,

Ex = 1

Z

dp ,

Et =



Z

p dp



M axim Pavlo v (1 L avren t'en tndtico m po fsiH od ro p pno ach s, S ib erian D ivisio n o f 2 7AA prN,o2o si2 irsk ; 2 S ector o f m o m v I s e t u t e o t i y n a d y r a m ic e R S , il v 0 1 b 13 / 19


Waterbag Reduction. Heaviside Step Function
yields N component reduction
i i at + ai ax + E = 0, N1

(i = 1, ..., N )
an ) = 1

Ex = 1 1 Et = 2

n =1 N1 n =1



fn (a

n +1



fn ( a

n +1 2

)

(an )

2

n =1 N



N

n an , n (an )2 ,

=

1 2

n =1



which coincide with already found waterbag reduction via the moment decomposition Ak = 1 k +1

n =1



N

n (an )

k +1

,

n =1



N

n = 0.

A l e x a n d e r C h e s n o ko v ,

M axim Pavlo v (1 L avren t'en tndtico m po fsiH od ro p pno ach s, S ib erian D ivisio n o f 2 7AA prN,o2o si2 irsk ; 2 S ector o f m o m v I s e t u t e o t i y n a d y r a m ic e R S , il v 0 1 b 14 / 19


Cold Plasma. Multi-ow Hydrodynamics
=

i =1



N

bi ( t , x ) ( p

ai (t , x ) )

Substitution to the kinetic equation t + p implies E p = 0, Ex = 1
Z

x

dp ,

Et =



Z

p dp



i =1



N

i i (bt + pbx )(p

ai )

i i b i (at + pax + E )0 (p

ai ) = 0.

Taking into account the following equalities (p ) (p
A l e x a n d e r C h e s n o ko v ,

ai ) = (ai ),

(p ) 0 (p

ai ) =

0 (ai ),

M axim Pavlo v (1 L avren t'en tndtico m po fsiH od ro p pno ach s, S ib erian D ivisio n o f 2 7AA prN,o2o si2 irsk ; 2 S ector o f m o m v I s e t u t e o t i y n a d y r a m ic e R S , il v 0 1 b 15 / 19


Cold Plasma. Multi-ow Hydrodynamics

we obtain
i i at + ai ax + E = 0, i bt + (ai b i )x = 0,

(i = 1, ..., N )

Ex = 1

n =1



N

bn ,

Et =

n =1



N

an b n .

This reduction is determined by the so called Zakharov moment decomposition Ak =
n =1



N

(an )k b n .

A l e x a n d e r C h e s n o ko v ,

M axim Pavlo v (1 L avren t'en tndtico m po fsiH od ro p pno ach s, S ib erian D ivisio n o f 2 7AA prN,o2o si2 irsk ; 2 S ector o f m o m v I s e t u t e o t i y n a d y r a m ic e R S , il v 0 1 b 16 / 19


Generalizations
Let us consider more general ansatz =

i =1



N

c i (t , x ) (p

ai (t , x ) ) + e i (t , x ) 0 (p

ai (t , x ) ) .

Substitution to the kinetic equation t + p implies E p = 0, Ex = 1
Z

x

dp ,

Et =



Z

p dp



i =1



N

i (cti + pcx )(p

ai ) + e

i t

i c i at + p ( e

i x

i c i ax )

Ec i 0 (p

ai )

i i e i (at + pax + E )00 (p

ai ) = 0.

A l e x a n d e r C h e s n o ko v ,

M axim Pavlo v (1 L avren t'en tndtico m po fsiH od ro p pno ach s, S ib erian D ivisio n o f 2 7AA prN,o2o si2 irsk ; 2 S ector o f m o m v I s e t u t e o t i y n a d y r a m ic e R S , il v 0 1 b 17 / 19


Then we obtain cti + (ai c i )
x i ex = 0 , i i (eti + ai ex + 2e i ax ) = 0, i i ai (eti + ai ex + 2e i ax ) = 0.

i i c i ( at + a i ax + E )

(ai c

i

i i e i ) ( a t + a i ax + E )

Introducing new functions b i such that e i = the system
i i at + ai ax + E = 0, i bt + ( a i b i ) x = 0,

(bi )2 /2 we come back to (b i ) 2
2

cti + ai c i +
2

= 0,
x

Ex = 1

n =1



N

cn,

Et =

n =1



N

an c n +

(b n ) 2

is associated with the moment decomposition Ak =
A l e x a n d e r C h e s n o ko v ,

M axim Pavlo v ( L avren t'en tndtico m po fsiH od ro p pno ach s, S ib erian D ivisio n o f 2 7AA prN,o2o si2 irsk ; 2 S ector o f m o m v I s e t u t e o t i y n a d y r a m ic e R S , il v 0 1 b 18 / 19

n1 1 =



N

k (an )k c n + (an ) 2

k1

(b n )2 .


Generalizations

Next 4N component nonlocal system
i i at + a i ax + E

= 0, = 0,

i bt + (ai b i )x = 0, cti + ai c i +

(b i ) 2

2 x

= 0,

gti + ai g i + b i c

i x

dE = (1

A0 )dx + A1 dt

is associated with the moment decomposition Ak =

n =1



N

(an )k g

n

k

n =1



N

(an )

k1nn

bc +

k (k 6

1)

n =1



N

(an )

k2

(b n )3 .

A l e x a n d e r C h e s n o ko v ,

M axim Pavlo v (1 L avren t'en tndtico m po fsiH od ro p pno ach s, S ib erian D ivisio n o f 2 7AA prN,o2o si2 irsk ; 2 S ector o f m o m v I s e t u t e o t i y n a d y r a m ic e R S , il v 0 1 b 18 / 19


References
S.A. Akhmanov, Yu.E. Diakov, A.S. Chirkin, Introduction in statistical radiophysics and optics, Moscow. Nauka 1981. R.C. Davidson, Methods in nonlinear plasma theory. Vol. 37 in Pure and Applied Physics. Academic Press. New York. 1972. T.P. Co¤ey, Breaking of large amplitude plasma oscillations, Physics of Fluids, 1971, vol. 14, pp. 1402-1406. A.K. Khe, A.A. Chesnokov, Propagation of nonlinear perturbations in a quasineutral collisionless plasma. Journal of Applied Mechanics and Technical Physics 52 No. 5 (2011) 677-688. V.P. Silin, Introduction to the kinetic theory of gases. Moscow, FIAN (in Russian) (1998). ...rst edition. Nauka (1971).
A l e x a n d e r C h e s n o ko v ,

M axim Pavlo v (1 L avren t'en tndtico m po fsiH od ro p pno ach s, S ib erian D ivisio n o f 2 7AA prN,o2o si2 irsk ; 2 S ector o f m o m v I s e t u t e o t i y n a d y r a m ic e R S , il v 0 1 b 19 / 19