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Kinetic representation of the KP hierarchy
Volume 139, number 3,4
PHYSICS LETTERS A
31 July 1989
KINETIC REPRESENTATION OF THE KP HIERARCHY John GIBBONS Department of Mathematics, Imperial College, 180 Queen’s Gale, London SW7 2BZ, UK
and Boris A. KUPERSHMIDT The University of Tennessee Space Institute, Tullahoma, TN 37388, USA Received 17 April 1989; accepted for publication 26 May 1989 Communicatedby A.P. Fordy
The KP hierarchy results upon taking the moments of a one-particle distribution function satisfying a hierarchy of dispersive collisionless Boltzmann equations. The latter hierarchy is Hamiltonian and commuting, and the map which sends the kinetic hierarchy into the KP one is Hamiltonian.
1. Introduction
/
(\~ø~~’) := The K.P hierarchy is an infinite commuting set of systems of evolution equations, of the form (1) L,=[(Lm)÷,L], me7L~, where ~ ~
A~=A~(x,t);
(2)
n=O
~ is essentially just ô = ~
(l)øtr~i_~
~
+
/ \~
)
J_
:=
(5)
~
/
Thus, for each me7L~,the mth member of the KP hierarchy, KPrn, consists of an infinite-component system of evolution equations (1). For example, for the first nontrivial case m=2, the KP 2 system has the form:
Ø(r):=ar(Ø)
leZ,
(3)
~
+A~1
—
r
~ (n) (_l)rA~_,A~, r
r>O
and
ne7L÷.
(1 \ ~):=I(l_r)
~ t~O
...
(l—r+l)/r!,
:=0, are the binomial coefficients
If we take the zero-dispersion limit of the KP2 systern (6), i.e. change 8 into c~9and retain only terms of first order in , we get
rel~, r=0, ~
(6)
)
~ neZ÷, (7) which is (apart from the inessential scaling constant ~)the Benney system [1 ]; similarly, for every me 7L +, the zero-dispersion limit of the KP,~system (1) gives the mth Benney system [2]. Now, the Benney system (7) results from the ki-
0375-960 1/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
141
Volume 139, number 3,4
PHYSICS LETTERS A
netic equations [3]
31 July 1989
2. Dynamics
~ =if. —AO.XJP
(8)
for the distribution function f=f(x, p, t), when one sets
9
A~=Jfp’dp
and assumes that, at ± in p, f tends to zero faster than any power of p. Moreover [4], the same mo-
Since [L”~,L] =0 and L’~_-(L”) + + (L’~) the ,
mth KP system (1) can be rewritten in the form L1=[—(L’~)_,L]=[—(L”)~,L]~ m)±,LJ_. (13) =[(L This form is a particular case of a more general system L, 1=[Q±L]_, Qarbitrary,
ment map (9) produces all the higher flows in the Benney hierarchy from an appropriate hierarchy of kinetic equations, and these kinetic flows all cornmute between themselves. Also, the moment map (9) is a Hamiltonian (=canonical) map [4] between the Hamiltonian structure [4] ~ {H, F}zf[(-~y)
/~F ~ ~)]
(~F\ 73H\ k~y)—
(14)
which we will produce out of the corresponding kinetic system. Define Smbl
(~ ~
[R, S]()
:=
~ Ø1p’,
~0 ,~
-~
(15)
[a~(R)8~(S)—0~(R)8~(S)].
(16)
1
[~H 6F ~~L-~7’ ~fJ
(10)
Then the kinetic equation f~=[Smbl(Q±),.t1(I)
(17)
in the f-space, and the Hamiltonian structure [2] implies the evolution system (14) upon the identi{H,F}= ~
oH (11)
intheA-space. The purpose of this note is to generalise for the dispersive KP hierarchy (1) all the properties enjoyed by its zero-dispersion limit, and thereby cornplete the diagram
I
kinetic
the commutator
[R,s]~ ~~ ~r.
(18)
to make operations with infinite series rigorous. We do not explicitly do this in order to keep the grading parameter ~out of the Lax operator L (2). To show that (17) implies (14), let
)KP
1.
.J~0
(12)
~ Benney
in which the vertical arrows denote the limit of zerodispersion, and the horizontal arrows denote the Hamiltonian moment map (9). In the next section we construct the dispersive kinetic flows and show that they imply the corresponding KP flows. The subsequent section deals with the Hamiltonian aspects. 142
fication (9). Strictly speaking, Remark. Im should be defined as
Q+= Then ~ q,~’. [Q+’L] =
=
=
(19)
[~
q 1~,~+ ~
A~~’1]
[~
q~,~ ~~A~-’-’
—~
A1
(20a)
Volume 139, number 3,4
PHYSICS LETTERS A
[by(3) and since ~(~_i_l
-~,
iel÷],
J
=
r~0
[Q±,L]
—
=
dpp” ~ [(~)qjf(~_~ 1
(r)Pia~(~]
—
[~
~ A1( —1
qi(I)A)~J~~1 r
)~ (i±r)qV)~_I_
~[(1)qjA~j_q~(_ly(fl _~J],
(20b)
so that (14) becomes A~1= ~
Jf1p”dp
A,,,=
~ (—1 Y (i+r)
31 July 1989
[qj(J)A~
_AI(_lY(~r) q(~)]
i )A~+1~]~
which is the same as (21). 2)÷=~2+2A In particular, for m=2, (L 0, 2)~)=p2+2A Smbl((L 0,so that eq. (23) becomes &f=V+pf—
~ ~ r.
(25)
r>0
ne7L÷
(21)
.
On the other hand, [Smbl(Q+),~(I)
=
[~
qjpi,f]
[by (16)]
L,1= [(L”’)+,L](o), L= ([Q+,LJ(0))_ ,
(1)
= ~
-~
(22a)
[8,~(p)q~8)—q~p’8~(])]
and this equation implies the KP2 system (6). In the zero-dispersion limit, eqs. (1), (14), (16), (21) and (23) become, respectively:
[R,S](I)
(1’) (14’)
8~(R)8(S)—8(R)00(S) [R,S](o) ,
(16’)
and since A~1= ~
f,
8~(p’)=r!(I)pi_r
~
[jq1A~21_1+A~+1_1(n+j)qJ~], (21’) (23’)
Uq1f—qJ’~f~).
[Smbl(Q+),J]~1~ =
~[(I)qjp’)pj_,..._
(22b)
~
so that (17) becomes ~ [(jr) q1fj~’~— qjpJ8~(J)].
(23)
To show that (23) implies (21) we apply the operator Jp”dp to eq. (23), the limits of integration being always understood to be and X. Using the obvious formula
~ Hamiltonian formalism In this section we show that the kinetic equation (23) is a Hamiltonian system for every Q=L”’, and that the moment map (9) is a Hamiltonian map into the (first) HamiltonianstructureoftheKPhierarchy. Firstly, from formula (2.11) in ref. [5] we know that the mth KP system (1) can be put into the Hamiltonian form
—~
J
pi8~(fpk)dp=(—l~r!(I)AJ+k,.,
we get from (23)
(24)
A~1= —
~
[(~)ôrA~+k_~
/~\ A~+k_~f
1 OH J(_aY]~—~
\,rJ
(26)
with H= ~Res(Lm~), m+
Res(~ øi~t):=øi.
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:
Volume 139, number 3,4
PHYSICS LETTERS A
Secondly, from lemma 2.12 in ref. [61, we know that the multiplication law (3), in the form RoS
~0~(R)(—0)’(S),
(27)
makes functions of (x, p) into an associative ring. Therefore, these functions form a one-dimensional Lie algebra, with respect to the commutator
=
—
Jdpp”~ [a’uHk)(k)~~
~8,,(J~)0 k
= ~
[(—0y(R)0~(S)
=
p”f, dp
A,,, =
~,
[R, S](2) :=SoR_R~S —8~(R)(—ar(S)].
31 July 1989
—
(_
(Hk)
[(r)0~n±k_~~ 1
y
(28) Denoting byfthe coordinate on the dual space ~* to g, we can compute the natural Hamiltonian operator B=B(~)on g~by using formula VIII (3.4) in ref. [7]:
RB(S)—~f[R,S](2)
[by (28)] —
r —.R
r.
1
[~
—~
~
—
~,
can be put into the Hamiltonian form (32) whenever Q=L,,,, it is enough to notice that, as we have
_—f~ [(—0 )‘(R)0~(S) 0~(R)(—0 Y(S)] -~
[(k)OrAh (~)A~±k_~.( _0)’] r r which is precisely the form (26). Thus, (32) implies (26). To show that eqs. (23) =
1 (8~f~~_0~,J8r)](S) ,
seen, both these systems produce the same KP,,, flow (1), and the moment map (9) is injective. Alternatively, the identification of eqs. (23) and (32) follows at once by substituting into eq. (23) the identities
so that 1 B=(0~,,—80~). r.
(29)
Hence, Hamiltonian systems in the f-space have the form (0’f—0~f8~)
~.
(30)
To show that the moments map (9) is Hamiltonian between the systems (30) and (26), take H in (30) as a pull back, via (9), of an arbitrary Hamiltonian in the A-space which enters into eq. (26). Then, since OH
phHk
Hk:=
OH
-~--,
(31)
(30) becomes =
[0~t)0(pk)
ref. [5] are written in the form ~q/~J=~kH~
(34)
As a corollary, we find that all the kinetic flows (32) which cover the KP hierarchy (1) commute between themselves, since their respective Hamiltonians are in involution: these Hamiltonians are pulled back by the Hamiltonian map (9) from the A-space where they are known to be in involution. Acknowledgement The second author was partially supported by the NSF.
_0~(fpk)0~(Hk)]. (32)
Using formulae (22) and (24), we find that (32) implies 144
q1= ~ (s±J)H(s (33) which follow when the formulae (2.7) and (2.8) in
Volume 139, number 3,4
PHYSICS LETTERS A
References [1] D.J. Benney, Stud. Appi. Math. 52 (1973) 45. [2] BA. Kupershmidt and Yu.I. Manin, Fund. Anal. Appl. 11 (1977) 188; 12 (1978) 120.
[3] [4] [5] [6] [7]
31 July 1989
V.E. Zakharov, Funct. Anal. AppI. 14 (1980) 15. J. Gibbons, Physica D 3 (1981) 503. BA. Kupershmidt, Commun. Math. Phys. 99 (1985) 51. Yu.I. Manin, J. Soy. Math. 1 I (1979)1. B.A. Kupershmidt, Discrete Lax equations and differentialdifference calculus (Astérisque, Paris, 1985).
145
—
PHYSICS LETTERS A
31 July 1989
KINETIC REPRESENTATION OF THE KP HIERARCHY John GIBBONS Department of Mathematics, Imperial College, 180 Queen’s Gale, London SW7 2BZ, UK
and Boris A. KUPERSHMIDT The University of Tennessee Space Institute, Tullahoma, TN 37388, USA Received 17 April 1989; accepted for publication 26 May 1989 Communicatedby A.P. Fordy
The KP hierarchy results upon taking the moments of a one-particle distribution function satisfying a hierarchy of dispersive collisionless Boltzmann equations. The latter hierarchy is Hamiltonian and commuting, and the map which sends the kinetic hierarchy into the KP one is Hamiltonian.
1. Introduction
/
(\~ø~~’) := The K.P hierarchy is an infinite commuting set of systems of evolution equations, of the form (1) L,=[(Lm)÷,L], me7L~, where ~ ~
A~=A~(x,t);
(2)
n=O
~ is essentially just ô = ~
(l)øtr~i_~
~
+
/ \~
)
J_
:=
(5)
~
/
Thus, for each me7L~,the mth member of the KP hierarchy, KPrn, consists of an infinite-component system of evolution equations (1). For example, for the first nontrivial case m=2, the KP 2 system has the form:
Ø(r):=ar(Ø)
leZ,
(3)
~
+A~1
—
r
~ (n) (_l)rA~_,A~, r
r>O
and
ne7L÷.
(1 \ ~):=I(l_r)
~ t~O
...
(l—r+l)/r!,
:=0, are the binomial coefficients
If we take the zero-dispersion limit of the KP2 systern (6), i.e. change 8 into c~9and retain only terms of first order in , we get
rel~, r=0, ~
(6)
)
~ neZ÷, (7) which is (apart from the inessential scaling constant ~)the Benney system [1 ]; similarly, for every me 7L +, the zero-dispersion limit of the KP,~system (1) gives the mth Benney system [2]. Now, the Benney system (7) results from the ki-
0375-960 1/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
141
Volume 139, number 3,4
PHYSICS LETTERS A
netic equations [3]
31 July 1989
2. Dynamics
~ =if. —AO.XJP
(8)
for the distribution function f=f(x, p, t), when one sets
9
A~=Jfp’dp
and assumes that, at ± in p, f tends to zero faster than any power of p. Moreover [4], the same mo-
Since [L”~,L] =0 and L’~_-(L”) + + (L’~) the ,
mth KP system (1) can be rewritten in the form L1=[—(L’~)_,L]=[—(L”)~,L]~ m)±,LJ_. (13) =[(L This form is a particular case of a more general system L, 1=[Q±L]_, Qarbitrary,
ment map (9) produces all the higher flows in the Benney hierarchy from an appropriate hierarchy of kinetic equations, and these kinetic flows all cornmute between themselves. Also, the moment map (9) is a Hamiltonian (=canonical) map [4] between the Hamiltonian structure [4] ~ {H, F}zf[(-~y)
/~F ~ ~)]
(~F\ 73H\ k~y)—
(14)
which we will produce out of the corresponding kinetic system. Define Smbl
(~ ~
[R, S]()
:=
~ Ø1p’,
~0 ,~
-~
(15)
[a~(R)8~(S)—0~(R)8~(S)].
(16)
1
[~H 6F ~~L-~7’ ~fJ
(10)
Then the kinetic equation f~=[Smbl(Q±),.t1(I)
(17)
in the f-space, and the Hamiltonian structure [2] implies the evolution system (14) upon the identi{H,F}= ~
oH (11)
intheA-space. The purpose of this note is to generalise for the dispersive KP hierarchy (1) all the properties enjoyed by its zero-dispersion limit, and thereby cornplete the diagram
I
kinetic
the commutator
[R,s]~ ~~ ~r.
(18)
to make operations with infinite series rigorous. We do not explicitly do this in order to keep the grading parameter ~out of the Lax operator L (2). To show that (17) implies (14), let
)KP
1.
.J~0
(12)
~ Benney
in which the vertical arrows denote the limit of zerodispersion, and the horizontal arrows denote the Hamiltonian moment map (9). In the next section we construct the dispersive kinetic flows and show that they imply the corresponding KP flows. The subsequent section deals with the Hamiltonian aspects. 142
fication (9). Strictly speaking, Remark. Im should be defined as
Q+= Then ~ q,~’. [Q+’L] =
=
=
(19)
[~
q 1~,~+ ~
A~~’1]
[~
q~,~ ~~A~-’-’
—~
A1
(20a)
Volume 139, number 3,4
PHYSICS LETTERS A
[by(3) and since ~(~_i_l
-~,
iel÷],
J
=
r~0
[Q±,L]
—
=
dpp” ~ [(~)qjf(~_~ 1
(r)Pia~(~]
—
[~
~ A1( —1
qi(I)A)~J~~1 r
)~ (i±r)qV)~_I_
~[(1)qjA~j_q~(_ly(fl _~J],
(20b)
so that (14) becomes A~1= ~
Jf1p”dp
A,,,=
~ (—1 Y (i+r)
31 July 1989
[qj(J)A~
_AI(_lY(~r) q(~)]
i )A~+1~]~
which is the same as (21). 2)÷=~2+2A In particular, for m=2, (L 0, 2)~)=p2+2A Smbl((L 0,so that eq. (23) becomes &f=V+pf—
~ ~ r.
(25)
r>0
ne7L÷
(21)
.
On the other hand, [Smbl(Q+),~(I)
=
[~
qjpi,f]
[by (16)]
L,1= [(L”’)+,L](o), L= ([Q+,LJ(0))_ ,
(1)
= ~
-~
(22a)
[8,~(p)q~8)—q~p’8~(])]
and this equation implies the KP2 system (6). In the zero-dispersion limit, eqs. (1), (14), (16), (21) and (23) become, respectively:
[R,S](I)
(1’) (14’)
8~(R)8(S)—8(R)00(S) [R,S](o) ,
(16’)
and since A~1= ~
f,
8~(p’)=r!(I)pi_r
~
[jq1A~21_1+A~+1_1(n+j)qJ~], (21’) (23’)
Uq1f—qJ’~f~).
[Smbl(Q+),J]~1~ =
~[(I)qjp’)pj_,..._
(22b)
~
so that (17) becomes ~ [(jr) q1fj~’~— qjpJ8~(J)].
(23)
To show that (23) implies (21) we apply the operator Jp”dp to eq. (23), the limits of integration being always understood to be and X. Using the obvious formula
~ Hamiltonian formalism In this section we show that the kinetic equation (23) is a Hamiltonian system for every Q=L”’, and that the moment map (9) is a Hamiltonian map into the (first) HamiltonianstructureoftheKPhierarchy. Firstly, from formula (2.11) in ref. [5] we know that the mth KP system (1) can be put into the Hamiltonian form
—~
J
pi8~(fpk)dp=(—l~r!(I)AJ+k,.,
we get from (23)
(24)
A~1= —
~
[(~)ôrA~+k_~
/~\ A~+k_~f
1 OH J(_aY]~—~
\,rJ
(26)
with H= ~Res(Lm~), m+
Res(~ øi~t):=øi.
143
:
Volume 139, number 3,4
PHYSICS LETTERS A
Secondly, from lemma 2.12 in ref. [61, we know that the multiplication law (3), in the form RoS
~0~(R)(—0)’(S),
(27)
makes functions of (x, p) into an associative ring. Therefore, these functions form a one-dimensional Lie algebra, with respect to the commutator
=
—
Jdpp”~ [a’uHk)(k)~~
~8,,(J~)0 k
= ~
[(—0y(R)0~(S)
=
p”f, dp
A,,, =
~,
[R, S](2) :=SoR_R~S —8~(R)(—ar(S)].
31 July 1989
—
(_
(Hk)
[(r)0~n±k_~~ 1
y
(28) Denoting byfthe coordinate on the dual space ~* to g, we can compute the natural Hamiltonian operator B=B(~)on g~by using formula VIII (3.4) in ref. [7]:
RB(S)—~f[R,S](2)
[by (28)] —
r —.R
r.
1
[~
—~
~
—
~,
can be put into the Hamiltonian form (32) whenever Q=L,,,, it is enough to notice that, as we have
_—f~ [(—0 )‘(R)0~(S) 0~(R)(—0 Y(S)] -~
[(k)OrAh (~)A~±k_~.( _0)’] r r which is precisely the form (26). Thus, (32) implies (26). To show that eqs. (23) =
1 (8~f~~_0~,J8r)](S) ,
seen, both these systems produce the same KP,,, flow (1), and the moment map (9) is injective. Alternatively, the identification of eqs. (23) and (32) follows at once by substituting into eq. (23) the identities
so that 1 B=(0~,,—80~). r.
(29)
Hence, Hamiltonian systems in the f-space have the form (0’f—0~f8~)
~.
(30)
To show that the moments map (9) is Hamiltonian between the systems (30) and (26), take H in (30) as a pull back, via (9), of an arbitrary Hamiltonian in the A-space which enters into eq. (26). Then, since OH
phHk
Hk:=
OH
-~--,
(31)
(30) becomes =
[0~t)0(pk)
ref. [5] are written in the form ~q/~J=~kH~
(34)
As a corollary, we find that all the kinetic flows (32) which cover the KP hierarchy (1) commute between themselves, since their respective Hamiltonians are in involution: these Hamiltonians are pulled back by the Hamiltonian map (9) from the A-space where they are known to be in involution. Acknowledgement The second author was partially supported by the NSF.
_0~(fpk)0~(Hk)]. (32)
Using formulae (22) and (24), we find that (32) implies 144
q1= ~ (s±J)H(s (33) which follow when the formulae (2.7) and (2.8) in
Volume 139, number 3,4
PHYSICS LETTERS A
References [1] D.J. Benney, Stud. Appi. Math. 52 (1973) 45. [2] BA. Kupershmidt and Yu.I. Manin, Fund. Anal. Appl. 11 (1977) 188; 12 (1978) 120.
[3] [4] [5] [6] [7]
31 July 1989
V.E. Zakharov, Funct. Anal. AppI. 14 (1980) 15. J. Gibbons, Physica D 3 (1981) 503. BA. Kupershmidt, Commun. Math. Phys. 99 (1985) 51. Yu.I. Manin, J. Soy. Math. 1 I (1979)1. B.A. Kupershmidt, Discrete Lax equations and differentialdifference calculus (Astérisque, Paris, 1985).
145
—