Classification of integrable Vlasov-type equations

0.15 MB·English

by  A. V. Odesskii

Classification of integrable
Vlasov-type equations
A.V. Odesskii 1,2, M.V. Pavlov 3, V.V. Sokolov 2
1 L.D. Landau Institute for Theoretical Physics (Russia)
8 2 University of Manchester (UK)
0 3 P.N. Lebedev Physical Institute (Russia)
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Abstract Classification of integrable Vlasov-type equations is reduced to
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1 a functional equation for a generating function. A general solution of this

functional equation is found in terms of hypergeometric functions.
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MSC numbers: 17B80, 17B63, 32L81, 14H70
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Address: L.D. Landau Institute for Theoretical Physics of Russian Academy of Sciences,
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5 Kosygina 2, 119334, Moscow, Russia
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6 E-mail: [email protected], [email protected]
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1 Introduction
Consider the linear equation
Λ = Λ,ψ , (1.1)
t
{ }
where Λ,ψ = ψ Λ ψ Λ . Here the unknown function Λ(x,t,p) depends on the “spectral”
p x x p
{ } −
parameter p and a function ψ = ψ(U(x,t),p). Following [1], we call (1.1) the Vlasov-type
equation generated by ψ(U,p). Equation (1.1) is also known as the dispersionless Lax equation.
It is easy to check that a partial hodograph transformation Λ(x,t,p) p(x,t,Λ) reduces
→
(1.1) to the following conservative form
p = ψ(U,p) . (1.2)
t x
Here Λ plays a role of parameter.
For some functions ψ Vlasov-type equations are closely related to integrable hydrodynamic
chains [2, 3]. A hydrodynamic chain associated with Vlasov-type equation can be derived by
expanding Λ(x,t,p) at a singular point of the function ψ. In such a case, formula (1.2) yields
conservation laws for the hydrodynamic chain.
Example 1(The Benneychain)[4,5,6,1]. TheVlasovequation(orcollisionlessBoltzmann
equation) has the form
Λ +Λ U pΛ = 0, (1.3)
t p x x
−
where ψ = p2 +U. Substituting the expansion
2
A0 A1 A2 A3
Λ = p+ + + + +... (1.4)
p p2 p3 p4
into (1.3), one derives the famous Benney hydrodynamic chain
Ak = Ak+1 +kAk−1A0, k = 0,1,2,...,
t x x
where A0 = U. Let
H H H H
0 1 2 3
p = Λ ...
− Λ − Λ2 − Λ3 − Λ4 −
be the inverse series for (1.4). Functions H can be easily calculated: H = A0,H = A1,H =
i 0 1 2
A2 +(A0)2,.... The formula
p2
p = +U
t (cid:16) (cid:17)
2 x
generates infinitely many conservation laws for the Benney chain:
k−1
1
(H ) = H H H .
k t (cid:16) k+1 − 2 X i k−i−1(cid:17)x
i=0
2
Example 2 (The Kupershmidt chain) [7, 8]. The Kupershmidt hydrodynamic chain
Bk = Bk+1 +B0Bk +βkBkB0, k = 0,1,2,...
t x x x
is connected to the Vlasov-type equation (1.1), where
pβ+1
ψ = +U p,
β +1
by the expansion
B1 B2 B3
Λ = pβ +B0 + + + +...,
pβ p2β p3β
whereB0 = U.ConservationlawsforthischaincanbecalculatedinthesamewayasinExample
1.
The hydrodynamic chains described in Examples 1, 2 admit infinitely many hydrodynamic
reductions [9, 10]. The corresponding Vlasov-type equations admit the same reductions.
The following “integrable” functions ψ were found in [2]:
Case 1. ψ(U,p) = U +W(p), where
W′′ = c W′2 +c W′ +c
1 2 3
and
Case 2. ψ(U,p) = U W(p), where
1
W′′ = c W′2 +c W′ +c .
(cid:16) 1 2 3(cid:17)
W
Here c are arbitrary parameters. The Benney chain corresponds to Case 1 with W(p) = p2/2.
i
In this paper we describe all possible “integrable” functions ψ(U,p) using the method of
hydrodynamic reductions. The existence of hydrodynamic reductions have been proposed as a
definition of integrability for dispersionless multi-dimensional equations in [9]. We apply this
approach for Vlasov-type equations.
2 Hydrodynamic reductions
Suppose there exists a semi-Hamiltonian [11] hydrodynamic-type system
ri = vi(r)ri i = 1,2,...,N, (2.5)
t x
and functions U = u(r) and Λ = λ(r,p) such that these functions satisfy (1.1) for any solution
r(x,t) of system (2.5). Then (2.5) is called a hydrodynamic reduction for the Vlasov-type equa-
tion (1.1). The partial hodograph transformation λ(r,p) p(r,λ) leads to the corresponding
→
hydrodynamic reduction of (1.2).
3
Substituting Λ = λ(r,p) and U = u(r) in (1.1) and (1.2), we obtain the equations
λ = ψ λ ψ λ (2.6)
t p x x p
−
and
p = ψ(u,p) . (2.7)
t x
Calculating the derivatives by virtue of (2.5), we obtain from (2.6) that
λ ψ ∂ u+[vi(r) ψ ]∂ λ ri = 0,
X(cid:16) p u i − p i (cid:17) x
where we use the notation ∂ = ∂/∂ri. Since r is arbitrary solution of system (2.5), we get
i
λ ψ ∂ u = [ψ vi(r)]∂ λ, i = 1,2,...,N. (2.8)
p u i p i
−
Let us determine functions pi(r), i = 1,2,...,N as solutions of the equations
vi(r) = ψp p=pi. (2.9)
|
Then (2.8) implies that the equation λ = 0 has N solutions (pairwise distinct in the generic
p
case), i.e.
λp p=pi = 0, i = 1,2,...,N. (2.10)
|
Without loss of generality we can fix the Riemann invariants ri of the system (2.5) in such a
way that
ri = λ p=pi. (2.11)
|
Indeed, if we substitute p = pi into equation (2.6), then (2.6), (2.10) imply
(λ p=pi)t = (ψp p=pi)(λ p=pi)x.
| | |
This means (see (2.9)) thatλ p=pi satisfies (2.5) and therefore λ p=pi = Ri(ri) for some functions
| |
R . According to (2.10), the branch points of the Riemann surface determined by the equation
i
Λ = λ(r,p), are nothing but the Riemann invariants of system (2.5). This fact is well-known
for hydrodynamic-type systems that produced by the Whitham averaging procedure applied to
multi-phase solutions of both integrable continuous dispersive equations and integrable discrete
equations (see references in [3]).
Substituting functions p(r,λ), u(r) in (2.7), we obtain
ψ ∂ u
u i
∂ p = . (2.12)
i
ψp p=pi ψp
| −
If we fix λ = rk,k = i, then (2.11) implies p = pk and we obtain
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ψ ∂ u
∂ pk = u|p=pk i . (2.13)
i
ψp p=pi ψp p=pk
| − |
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Let us introduce the following notation:
ψu ψu p=pk
f = , f = | , i = k.
i ik
ψp p=pi ψp ψp p=pi ψp p=pk 6
| − | − |
The compatibility conditions ∂ (∂ p) = ∂ (∂ p), i = k are equivalent to the equations
k i i k
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2 fik∂pkfk fki∂pifi +∂u(fk fi)+fi∂pfk fk∂pfi
∂ u = − − − ∂ u∂ u. (2.14)
ik f f i k
i k
−
Equations(2.13), (2.14)formasystem ofequationsnamedin[2]the generalized Gibbons–Tsarev
system (cf. [10]).
Since u does not depend on p, (2.14) implies the following functional equation
fik∂pkfk fki∂pifi +∂u(fk fi)+fi∂pfk fk∂pfi
∂ − − − = 0 (2.15)
p(cid:16) (cid:17)
f f
i k
−
for the function ψ(u,p). In the next sections we study this functional equation and found its
general solution. The general solution is expressed in terms of a pair of arbitrary solutions of
the standard hypergeometric equation
u(u 1)y(u)′′+[(α+β +1)u γ]y(u)′+αβy(u) = 0. (2.16)
− −
Note that the general solution of the Chazy equation, which appears in the classification paper
[12], can also be parameterized by a pair of hypergeometric functions. Our solution ψ(u,p) is
a generalization of the solution h(ξ,u) found in [13] (see Example 4).
3 Particular solutions
In Sections 4, 5 we solve the functional equation (2.15) in terms of quadratures of hypergeo-
metric functions. In this section we consider some particular cases, where the result can be
written more explicitly.
Computing the numerator of the left hand side of (2.15) and expanding it at pi = p, pk = p,
we obtain that the vanishing of the coefficients up to 8-th degree in the corresponding Taylor
series is equivalent to the following system of PDEs for the function ψ(u,p):
3ψ3 ψ3 4ψ ψ ψ ψ3 +ψ2 ψ ψ3 3ψ ψ2 ψ2ψ +2ψ2 ψ ψ2ψ +
ppp u − pp ppp pppp u pp ppppp u − pp ppp u pu pp pppp u pu
(3.17)
6ψ2 ψ ψ2ψ 5ψ3 ψ2ψ 6ψ4 ψ ψ +6ψ4 ψ ψ = 0,
pp ppp u ppu − pp u pppu − pp pu uu pp u puu
3ψ2 ψ3ψ ψ ψ ψ3ψ 3ψ ψ ψ3ψ +ψ2 ψ3ψ 3ψ2 ψ ψ ψ ψ
ppp u ppu − pp pppp u ppu − pp ppp u pppu pp u ppppu − pp ppp u pu uu−
6ψ3 ψ2 ψ +3ψ3 ψ ψ ψ +3ψ2 ψ ψ2ψ +6ψ3 ψ ψ ψ 3ψ3 ψ2ψ = 0,
pp pu uu pp u ppu uu pp ppp u puu pp u pu puu − pp u ppuu
(3.18)
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and
3ψ2 ψ2ψ ψ +ψ ψ ψ2ψ ψ 6ψ ψ ψ ψ2 ψ 6ψ2 ψ3 ψ +
− ppp u pu uu pp pppp u pu uu − pp ppp u pu uu − pp pu uu
3ψ ψ ψ2ψ ψ +6ψ2 ψ ψ ψ ψ ψ2 ψ2ψ ψ +3ψ2 ψ3ψ
pp ppp u ppu uu pp u pu ppu uu − pp u pppu uu ppp u puu−
(3.19)
ψ ψ ψ3ψ +6ψ ψ ψ2ψ ψ +6ψ2 ψ ψ2 ψ 3ψ2 ψ2ψ ψ
pp pppp u puu pp ppp u pu puu pp u pu puu − pp u ppu puu−
3ψ ψ ψ3ψ 3ψ2 ψ2ψ ψ +ψ2 ψ3ψ = 0.
pp ppp u ppuu − pp u pu ppuu pp u pppuu
This system and the functional equation (2.15) are invariant under any transformation of the
form u f(u). They are also invariant with respect to the following symmetry group:
→
ψ c ψ +c p+c , p k ψ +k p+k . (3.20)
2 1 0 2 1 0
→ →
These symmetries are associated with linear transformations of independent variables in (1.1),
(1.2). Notice that any function ψ = W(u)+pV(u) satisfies equation (2.15) and system (3.17)-
(3.19).
The integrable cases ψ(u,p) = u + W(p) and ψ(u,p) = uW(p) described in Introduction
can be found directly from (3.17). We denote these cases as Case 1 and Case 2, respectively.
We mention two more particular integrable cases:
Case 3. ψ(u,p) = W(u p), where
−
W′′ = c W′3 +c W′2 +c W′;
1 2 3
Case 4. ψ(u,p) = pu+W(p), where
1
W′′ = c W′2 +c W′ +c .
(cid:16) 1 2 3(cid:17)
p
Here c are arbitrary constants. The Kuperhsmidt chain belongs to Case 4. It is easy to see
i
that Case 3 is connected to Case 1 by the transformation ψ p. In all these four particular
↔
cases system (3.17)-(3.19) is equivalent to some ordinary differential equation of the fifth order
for the function W.
System (3.17)-(3.19) admits the substitution
ψ = F(p,ψ), (3.21)
p
corresponding to the factorization with respect to the symmetry group u f(u). As the result,
→
one gets an overdetermined system of three PDEs for the function F(x,y). We do not present
this system here because of its complexity.
Case 1 corresponds to the case 1¯: F(x,y) = V(x), where V = W′. Case 3 corresponds
to 3¯: F(x,y) = V(y). Case 2 corresponds to 2¯: F(x,y) = yV(x), where V = W′/W. Case 4
corresponds to 4¯: F(x,y) = y/x+V(x), where V = W′ W/x.
−
6
Another particular solution is determined by the function
2x+α y2 +γy +δ
F(x,y) = . (3.22)
x2 +αx+β · 2y +γ
The corresponding equation (3.21) leads to equation (2.7) of the form
p = ∂ u(p2+t ) t ,
t xp 1 − 2
where t are arbitrary constants. In this example the function F has the form
i
V(y)
F(x,y) = . (3.23)
W(x)
An investigation of this ansatz leads to the following result. The case, where V or W is a
linear function, was considered above. Namely, the cases W = 1, V = 1 and V = y coincide
with Cases 3, 1 and 2, respectively. The case W = x transforms to Case 2 by the substitution
ψ p. If V′′ = 0 and W′′ = 0, then the following three classes of solutions (3.23) exist:
↔ 6 6
Case 5. VV′′ = 2V′2 +c V′ +c , WW′′2W′2 +c W′ +c ,
1 2 1 2
Case 6. VV′′ = V′2 +c V′ +c , WW′′ = W′2 +c W′ +c ,
1 2 1 2
Case 7. VV′′ = V′2 +c V′ +c , WW′′ = W′2 +c W′ +c .
1 2 1 2
− −
For all these cases the generic solution depends on 6 arbitrary parameters. Function (3.22)
belongs to Case 5 with c = 3, c = 1.
1 2
−
4 General solution
Letus expandtheleft handside of(2.15) intotheTaylor series atpk = p. Denoteby S(pi,p)the
first nontrivial coefficient of this expansion. For fixed p consider S = 0, S = 0 as a system of
p
linear algebraic equations with respect to derivatives ψu,ψupi. Its determinant does not vanish
if ψ ψ = 0. Solving this system, we obtain
u pp
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Q(ψ )
p
ψ = , (4.24)
u
ψ
pp
where Q is a polynomial with respect to ψ of degree not greater than 4 with coefficients
p
depending on u only. Taking into account (4.24), it is easy to extract from the equation Spi = 0
that
ψ R(ψ )
ppp p
= , (4.25)
ψ2 Q(ψ )
pp p
where R is a polynomial of degree not greater than three. The compatibility condition of
equations (4.24) and (4.25) has the form
2 2 2 2
Q Q RQQ +RQ (R +R Q)Q +RQ R Q +(2RR R )Q = 0, (4.26)
xxx − xx x − x x u − xx x − u
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where x = ψ .
p
Assume that the polynomial
Q(x) = a(x b )(x b )(x b )(x b )
1 2 3 4
− − − −
has distinct roots and rewrite (4.25) as
ψ k k
ppp 1 4
= +...+ . (4.27)
ψ2 ψ b ψ b
pp p − 1 p − 4
One can verify that the system consisting of equations (3.17)-(3.19) and (4.26) is equivalent to
the following :
1. The functions k (u) are arbitrary constants such that k +...+k = 3.
i 1 4
2. The functions a(u), b (u) satisfy the following system of ODEs:
i
b′ = a(1 k ) (b b ), i = 1,...,4. (4.28)
i − i Y i − j
j6=i
The function a(u) can be chosen arbitrarily due to the admissible transformations u s(u).
→
Consider the double ratio
(b b )(b b )
1 2 3 4
ρ = − − .
(b b )(b b )
1 3 2 4
− −
Differentiating ρ by virtue of (4.28), it is easy to check that ρ′ = 0. Let us change u in such a
6
way that ρ = u. This means that we choose
1 1
a = + .
(b b )(b b ) (b b )(b b )
2 3 1 4 1 2 3 4
− − − −
One can verify that the formulae
z +uy y z +y z
2 2 2 2 2 2
b = , b = , b = , b = ,
1 2 3 4
z +uy y z +y z
1 1 1 1 1 1
where (y ,z ), (y ,z ) are two arbitrary solutions of the linear system
1 1 2 2
k +k +k 2 k +k +k 2 1 k 1 k k +k u
y′ = 1 2 3 − y + 1 2 3 − z, z′ = − 1 y + − 1 − 2 2 z, (4.29)
u 1 u(u 1) u 1 u(u 1)
− − − −
define a general solution of (4.28). Notice that if k +k +k = 2, then
1 2 3
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u(u 1)
z = uy + − y′
− k +k +k 2
1 2 3
−
and system (4.29) is equivalent to the hypergeometric equation (2.16), where k = 1+α γ,
1
−
k = 1 α, k = γ β.
2 3
− −
8
System (4.24), (4.25) can be reduced to quadratures by the following way. Let us determine
a function φ(u,p) as the solution of the over-determined system:
φ(φ 1)y′ φk1(φ u)k2(φ 1)k3
1
φ = − , φ = − − . (4.30)
u p
−β(y φ+z ) y φ+z
1 1 1 1
It is easy to check that this system is consistent. Then the solution of the following system in
involution
y y′ y y′ y φ+z
ψ = 2 1 − 1 2 φ1−k1(φ u)1−k2(φ 1)1−k3, ψ = 2 2, (4.31)
u p
β(y φ+z ) − − y φ+z
1 1 1 1
is a general solution of (2.15). This fact can be verified by a direct calculation. It turns out
that the expression under differentiating in the left hand side of (2.15) is equal to
(1 u β)α2 +(1 u α)β2 +u(4αβ α β)
− − − − − − ,
u(u 1)(α β)2
− −
where α = φ(u,pi),β = φ(u,pk).
Remark. The standard Wronskian formula for second order linear ODE implies that the
expression y y′ y y′ from (4.31) equals Cuk1+k2−2(u 1)k2+k3−2 for some constant C.
2 1 1 2
− −
5 Degenerations
In Section 4 we have considered the general case. This means that the polynomial Q has
degree 4 and all its roots b are distinct for the generic value u. In this section we consider
i
degenerations. It is easy to see that the degree of the polynomial Q can be fixed by 4 with the
help of transformations (3.20). It turns out that in all cases the result can be parameterized
by a pair of solutions of some degenerations of the hypergeometric equation.
Degeneration 1. Suppose Q = a(x b )2(x b )(x b ); then
1 2 3
− − −
R k f k k
1 1 2 3
= + + + ,
Q x b (x b )2 x b x b
1 1 2 3
− − − −
where k ,k ,k are constants such that k +k +k = 3, and
1 2 3 1 2 3
b′ = a(b b )(b b )f , b′ = a(b b )2(b b )(1 k ),
1 1 2 1 3 1 2 2 1 2 3 2
− − − − − −
b′ = a(b b )2(b b )(1 k ), f′ = af (b b )(b b )(2 k )+(b +b 2b )f . (5.32)
3 3 1 3 2 3 1 1(cid:16) 1 2 1 3 1 2 3 1 1(cid:17)
− − − − − − −
A general solution of system (5.32) can be parameterized in the following way:
y z y +uz
1 1 1 1
b = , b = , b = ,
2 3 1
y z y +uz
2 2 2 2
9
where (y ,z ), (y ,z ) are two arbitrary solutions of the linear system
1 1 2 2
1 3k +k 4 k +k 2 1 3k +k 4
y′ = + 2 1 − y + z, z′ = 1 2 − y + 2 1 − z. (5.33)
(cid:16)− 2 2(k 1)u (cid:17) (1 k )u2 (cid:16)− 2 − 2(k 1)u (cid:17)
2 2 2
− − −
The functions a,f are determined by system (5.32). Note that system (5.33) is equivalent to
1
the Bessel equation
1 (k k )(k +k 2)
y′′+y′ + + 2 − 1 1 2 − y = 0.
(cid:16)4 4(k 1)2u2 (cid:17)
2
−
Degeneration 2. Suppose Q = a(x b )2(x b )2; then
1 2
− −
R k f k f
1 1 2 2
= + + + ,
Q x b (x b )2 x b (x b )2
1 1 2 2
− − − −
where k ,k are constants such that k +k = 3, and
1 2 1 2
b′ = a(b b )2f , b′ = a(b b )2f ,
1 1 2 1 2 1 2 2
− − − −
f′ = a(b b )f (b b )(k 1)+2f , f′ = a(b b )f (b b )(k 1)+2f .
1 2 1 1(cid:16) 2 1 2 1(cid:17) 2 1 2 2(cid:16) 1 2 1 2(cid:17)
− − − − − −
The general solution is given by
y z
1 1
b = , b = ,
1 2
y z
2 2
where (y ,z ), (y ,z ) are two arbitrary solutions of the linear system
1 1 2 2
3k 5 1 3(1 k )
y′ = 1 − y +uz, z′ = y + − 1 z. (5.34)
2u −3 2u
Notice that the function y(u) satisfies the following second order equation:
u 3(3k 5)(3k 7)
y′′ + 1 − 1 − y = 0.
(cid:16)3 − u2 (cid:17)
Degeneration 3. Suppose Q = a(x b )3(x b ); then
1 2
− −
R k f f k
1 1 2 2
= + + + ,
Q x b (x b )2 (x b )3 x b
1 1 1 2
− − − −
where k ,k are constants, such that k +k = 3, and
1 2 1 2
b′ = a(b b )f , b′ = a(b b )3(1 k ),
1 2 1 2 2 2 1 2
− − −
f′ = a (b b )(f2 k f ) 2f f , f′ = 2af (b b )f f .
1 (cid:16) 2 1 1 2 2 1 2(cid:17) 2 2(cid:16) 2 1 1 2(cid:17)
− − − − −
The general solution can be written in the following form
y y′ +uy
1 1 1
b = , b = ,
1 2
y y′ +uy
2 2 2
10