A generalized hydrodynamical Gurevich-Zybin equation of Riemann type and its Lax type integrability
This paper is devoted to the study of a hydrodynamical equation of Riemann type, generalizing the remarkable Gurevich–Zybin system. This multi-component non-homogenous hydrodynamic equation is characterized by the only characteristic flow velocity. The compatible bi-Hamiltonian structures and Lax ty...
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| Cite this: | A generalized hydrodynamical Gurevich-Zybin equation of Riemann type and its Lax type integrability / M.V. Pavlov, A.K. Prykarpatsky // Condensed Matter Physics. — 2010. — Т. 13, № 4. — С. 43002:1-21. — Бібліогр.: 24 назв. — англ. |
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| citation_txt | A generalized hydrodynamical Gurevich-Zybin equation of Riemann type and its Lax type integrability / M.V. Pavlov, A.K. Prykarpatsky // Condensed Matter Physics. — 2010. — Т. 13, № 4. — С. 43002:1-21. — Бібліогр.: 24 назв. — англ. |
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| description | This paper is devoted to the study of a hydrodynamical equation of Riemann type, generalizing the remarkable Gurevich–Zybin system. This multi-component non-homogenous hydrodynamic equation is characterized by the only characteristic flow velocity. The compatible bi-Hamiltonian structures and Lax type representations of the 3-and 4-component generalized Riemann type hydrodynamical system are analyzed. For the first time the obtained results augment the theory of integrability of hydrodynamic type systems, originally developed only for distinct characteristic velocities in homogenous case.
Досліджено гідродинамічне рівняння типу Рімана, що узагальнює відому систему Гуревича - Зибіна. Це багатокомпонентне гідродинамічне рівняння характеризується єдиною характеристичною швидкістю потоку. Проаналізовані сумісні бі-гамільтонові структури та представлення Лакса для три- та чотирикомпонентної узагальненої гідродинамічної системи типу Рімана. Одержані результати вперше доповнюють теорію інтегровності систем гідродинамічного типу, раніше розвиненої тільки для відмінних швидкостей в однорідному випадку.
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Condensed Matter Physics 2010, Vol. 13, No 4, 43002: 1–21
http://www.icmp.lviv.ua/journal
A generalized hydrodynamical Gurevich-Zybin equation
of Riemann type and its Lax type integrability
M.V. Pavlov1∗, A.K. Prykarpatsky2,3
1 Department of Mathematical Physics, Lebedev Physics Institute of RAS, Moscow, Russian Federation
2 The AGH University of Science and Technology, Krakow, 30–059, Poland
3 The Ivan Franko State Pedagogical University, Drohobych of Lviv region, Ukraine
Received July 2, 2010, in final form September 1, 2010
This paper is devoted to the study of a hydrodynamical equation of Riemann type, generalizing the remarkable
Gurevich–Zybin system. This multi-component non-homogenous hydrodynamic equation is characterized by
the only characteristic flow velocity. The compatible bi-Hamiltonian structures and Lax type representations of
the 3-and 4-component generalized Riemann type hydrodynamical system are analyzed. For the first time the
obtained results augment the theory of integrability of hydrodynamic type systems, originally developed only
for distinct characteristic velocities in homogenous case.
Key words: Riemann type hydrodynamical equations, Lax type integrability, conservation laws
PACS: 02.30.Ik, 02.30.Jr
1. Introduction
Evolution differential equations of special type are capable of describing [1–3] many important
problems of wave propagation in nonlinear media with distributed parameters, for instance, in-
visible non-dissipative dark matter, playing a decisive role [4, 5] in the formation of large scale
structure in the Universe like galaxies, clusters of galaxies, super-clusters. In particular, if the
nonlinear medium is endowed with some regularity no-blow up properties, the propagation of the
corresponding waves can be modeled by means of the so-called Gurevich-Zybin dynamical system
Dtu = z, Dtz = 0, (1)
where, by definition, Dt := ∂/∂t + u∂/∂x, u := dx/dt, (x, t) ∈ R2, is the corresponding charac-
teristic flow velocity and z ∈ C∞(R2;R), is the related self-dual inhomogeneity magnitude. It was
amazing to see that the inhomogeneous Riemann type hydrodynamic system (1) can be integrated,
up to the first singularity, using the hodograph method (see [4, 6]).
Below in the second section we first construct a general solution to (1) using the method of
reciprocal transformations. In the third and fourth sections we will analyze the related infinite
hierarchies of conservation laws, the bi-Hamiltonian structures and Lax type integrability of a
Riemann type hydrodynamical system DN
t u = 0 at N = 3 and N = 4, naturally generalizing the
system (1). In the Conclusion, the obtained results subject to the well-known C- and S-integrability
schemes of nonlinear dynamical systems are discussed.
2. A general solution to the Gurevich-Zybin hydrodynamical system
of Riemann type
Let us introduce the auxiliary field variable ρ := zx for some smooth mapping ρ ∈ C∞(R2;R).
Then the second equation of (1) reduces to the so-called continuity equation
ρt + ∂x(ρu) = 0, (2)
∗
Email: maxim@math.sinica.edu.tw
c© M.V. Pavlov, A.K. Prykarpatsky 43002-1
http://www.icmp.lviv.ua/journal
M.V. Pavlov, A.K. Prykarpatsky
which can be utilized in the construction of a simple reciprocal transformation
dz = ρdx− ρudt, dy = dt. (3)
From (3) one easily obtains that ∂x = ρ∂z and ∂t = ∂y − ρu∂z. Thus, system (1) reduces to the
form
(
ρ−1
)
y
= uz, uy = z. (4)
The second equation can be easily integrated to
u = yz + f(z), (5)
where f ∈ C∞(R;R) is an arbitrary smooth function. Then, the first equation in (4) reduces to
the form
(
ρ−1
)
y
= y + f ′(z),
which can be easily integrated as
ρ−1 = y2/2 + f ′(z)y + ϕ′(z), (6)
where ϕ ∈ C∞(R;R) is another arbitrary smooth function. Taking into account (5) and (6), an
independent spatial variable x ∈ R can be found by integrating the inverse to (3) reciprocal
transformation
dx = ρ−1dz + udy, dt = dy. (7)
Thus, the general solution to the dynamical system (1) is given implicitly in the following para-
metric form:
u = zt+ f(z), x =
zt2
2!
+ f(z)t+ ϕ(z).
3. An N -component generalized one-dimensional Riemann type hydrody-
namical equation
The inhomogeneous hydrodynamic type system (1) can be written in a compact form (thanks
to Darryl Holm for this observation)
D2
tu = 0, (8)
where the flow operator Dt = ∂t + u∂x, (x, t) ∈ R2, is well known in fluid dynamics [10]. Indeed,
the aforementioned equation can be written as two interrelated equations of the first order
Dtu = z, Dtz = 0, (9)
which is nothing else but exactly (1). Thus, an obvious generalization of (1) for an N -component
case is written as
DN
t u = 0, (10)
where N ∈ Z+ is arbitrary.
Thus, we can formulate our result as the following proposition.
Proposition 3.1 The generalized dynamical system (10) is also integrable by means of a suitable
reciprocal transformation (see (3)), possessing the related infinite hierarchies of conservation laws.
Proof. Indeed, let us write (10) as N−component quasilinear system of the first order
Dtu1 = u2, Dtu2 = u3, . . . , DtuN−1 = uN , DtuN = 0, (11)
where mappings u1 = u, u2, u3, . . . , uN−1, uN ∈ C∞(R2;R) are the corresponding intermediate
smooth field variables.
43002-2
A hydrodynamical Riemann type equation integrability
Let us introduce an auxiliary field variable ρ = zx, where z := uN . The inhomogeneous hydro-
dynamic Riemann type system (11), upon its rewriting as
∂tuN + u1∂xuN = 0, . . . , ∂tuk + u1∂xuk = uk+1 , ∂tu1 + u1∂xu1 = u1+1 , (12)
reduces (by means of the reciprocal transformation (3), based on the continuity equation (2)), to
the following form
(
ρ−1
)
y
= ∂zu1 , ∂yu1 = u2 , ∂yu2 = u3 , . . . , ∂yuN−2 = uN−1 , ∂yuN−1 = z,
which is, evidently, equivalent to a pair of simple equations
(
ρ−1
)
y
= ∂zu, ∂N−1
y u = z. (13)
The last equation can be easily integrated to
u =
zyN−1
(N − 1)!
+
N−2
∑
n=0
fn+1(z)
yn
n!
,
where fn ∈ C∞(R2;R), n = 1, N − 1, are arbitrary smooth functions. Then, the first equation
reduces to the form
(
ρ−1
)
y
=
yN−1
(N − 1)!
+
N−2
∑
n=0
f ′
n+1(z)
yn
n!
one can easily integrate as follows:
ρ−1 =
yN
N !
+
N−2
∑
n=0
f ′
n+1(z)
yn+1
(n+ 1)!
+ f ′
0(z),
where f0 ∈ C∞(R2;R) is also an arbitrary smooth function. Thus, an independent spatial variable
x ∈ R can be found from the inverse reciprocal transformation (7) as
x =
zyN
N !
+
N−2
∑
n=0
fn+1(z)
yn+1
(n+ 1)!
+ f0(z).
As a result, a general solution to (10) is given implicitly by the parametric form
x =
ztN
N !
+
N−2
∑
n=0
fn+1(z)
tn+1
(n+ 1)!
+ f0(z),
u =
ztN−1
(N − 1)!
+
N−2
∑
n=0
fn+1(z)
tn
n!
, (14)
finishing the proof.
The next remark demonstrates a very deep symmetry degeneracy of the generalized Riemann
type hydrodynamical equation (10).
Remark: The inhomogeneous hydrodynamic type system (12) with a common characteristic
velocity dx/dt = u1 := u can be generalized for the case of an arbitrary characteristic veloc-
ity dx/dt = a(u1, u2, . . . , uN) := a(û), still preserving the reciprocal transformation integrability
described above. Indeed, such an inhomogeneous hydrodynamic type system
∂tuN + a(û)∂xuN = 0, . . . , ∂tuk + a(û)∂xuk = uk+1 ,
under the reciprocal transformation
dz = ρdx− ρa(û)dt, dy = dt,
43002-3
M.V. Pavlov, A.K. Prykarpatsky
reduces to the form
(
ρ−1
)
y
= ∂za(û), uk =
zyN−k
(N − k)!
+
N−2
∑
n=k−1
fn+1(z)
yn+1−k
(n+ 1 − k)!
,
where k = 1, N. Since all functions uk ∈ C∞(R2;R), k = 1, N, are found explicitly in terms of the
new independent variables z ∈ R and y ∈ R, the first equation can be easily integrated for any
functional dependence a ∈ C∞(RN ;R). Then, the functional dependence x := x(z, y) can be also
found in quadratures.
4. The generalized Riemann type hydrodynamical equation at N = 2:
conservation laws, bi-Hamiltonian structure and Lax type representation
Consider the generalized Riemann type hydrodynamical equation (10) at N = 2:
D2
tu = 0, (15)
where Dt = ∂/∂t+ u∂/∂x, which is equivalent to the following dynamical system:
ut = v − uux
vt = −uvx
}
:= K[u, v], (16)
where K : M → T (M) is a related vector field on the 2π-periodic smooth nonsingular functional
phase space M := {(u, v)⊺ ∈ C∞(R/2πZ;R2) : u2x − 2vx 6= 0, x ∈ R}. As we are interested first in
the conservation laws for the system (16), the following proposition holds.
Proposition 4.1 Let H(λ) :=
∫ 2π
0 h(x;λ)dx ∈ D(M) be an almost everywhere smooth functional
on the manifold M, depending parametrically on λ ∈ C, and whose density satisfies the differential
condition
ht = λ(uh)x (17)
for all t ∈ R and λ ∈ C on the solution set of dynamical system (16). Then the following iterative
differential relationship
(f/h)t = λ(uf/h)x (18)
holds, if a smooth function f ∈ C∞(R;R) (parametrically depending on λ ∈ C) satisfies for all
t ∈ R the linear equation
ft = 2λuxf + λufx . (19)
Proof. We have from (17)-(20) that
(f/h)t = ft/h− fht/h
2 = ft/h− λfux/h− λfuhx/h
2
= ft/h+ λfu(1/h)x − λuxf/h
= λ(uf)x/h+ λuf(1/h)x = λ(uf/h)x , (20)
proving the proposition.
The obvious generalization of the previous proposition is read as follows.
Proposition 4.2 If a smooth function h ∈ C∞(R;R) satisfies the relationship
ht = kuxh+ uhx , (21)
where k ∈ R, then
H =
2π
∫
0
h1/kdx (22)
is a conservation law for the Riemann type hydrodynamical system (16).
43002-4
A hydrodynamical Riemann type equation integrability
Remark 4.3 Let ĥ ∈ C∞(R;R) satisfy the differential relationship ĥt = (ĥu)x, then f = ĥ2 is a
solution to equation (19).
Remark 4.4 If functions hj ∈ C∞(R;R), j ∈ Z+, satisfy the relationships hj,t = λ(hju)x, j ∈ Z+,
λ ∈ C, then the functionals
H(i,j) =
∑
n∈ Z+
k(i,j)n
2π
∫
0
h2
n
i h
(1−2n)
j (23)
with k
(i,j)
n ∈ R, n ∈ Z+, i, j ∈ Z+, being arbitrary constants, are conserved quantities to equa-
tion (16). This formula, in particular, makes it possible to construct an infinite hierarchy of non-
polynomial conserved quantities for the Riemann type hydrodynamical system (16).
Example 4.5 The following non-polynomial functionals
H
( 1
3 )
4 =
2π
∫
0
√
u2x − 2vxdx, H
( 1
3 )
7 =
2π
∫
0
(uxvxx − uxxvx)
1/3
dx,
H
( 1
2 )
7 =
2π
∫
0
√
v(u2x − 2vx)dx,
H
( 1
3 )
8 =
2π
∫
0
(
k1u(uxxvx − uxvxx) + k1vxxv + k2(u2xv − 2v2x)
)1/3
dx,
H
( 1
6 )
9 =
2π
∫
0
(uxxvxxx − uxxxvxx)
1
6 dx,
H
( 1
4 )
9 =
2π
∫
0
(ux(uxxvx − uxvxx) + vxxvx))
1
4 ,
H
( 1
6 )
10 =
2π
∫
0
(
2uxx(uxvxx − uxxvx) − v2xx
)
1
6 (24)
are conservation laws for the Riemann type dynamical system (16).
Quite different conservation laws have been obtained in [21–23] using the recursion operator
technique. The corresponding recursion operator proves to generate no new conservation law, if
one applies it to the non-polynomial conservations laws (24).
We also notice that dynamical system (16), as it was shown before in [21, 22], can be transformed
via the substitution
v =
1
2
∂−1(u2x + η2) (25)
into the generalized two-component Hunter-Saxton equation:
ux,t = −1
2
u2x − uuxx +
1
2
η2,
ηt = −(uη)x. (26)
This equation allows a simple reduction to the Hunter-Saxton dynamical system [13, 16, 21, 22, 24]
at η = 0 :
uxt = −1
2
u2x − uuxx . (27)
43002-5
M.V. Pavlov, A.K. Prykarpatsky
The non-polynomial conservation laws (24), upon rewriting with respect to the substitution (25),
give rise to the related non-polynomial conservation laws for a generalized two-component Hunter-
Saxton dynamical system (27). Moreover, if we further apply the reduction η = 0, we obtain,
respectively, new non-polynomial conservation laws for the Hunter-Saxton dynamical system (27),
supplementing those found before in [22, 24].
Example 4.6 The following functionals
H
( 1
3 )
7 =
2π
∫
0
(
uxxu
2
x
)
1
3 dx, H
( 1
6 )
9 =
2π
∫
0
uxxxu+ 2uxxux√
uxx
dx,
H
( 1
3 )
8 =
2π
∫
0
[
uxxux(∂−1u2x) − uxxu
2
xu
]
1
3 dx (28)
are the conservation laws for the Hunter-Saxton dynamical system (27).
All of these and many other non-polynomial conservation laws can be easily obtained using
Proposition (4.2). For example, the following functionals
H(n,m) =
2π
∫
0
(
unxxu
m
x
)
2
m+4n dx, H
(2)
1 =
2π
∫
0
u2x(∂−1u2x)2dx,
H
(1/2)
2 =
2π
∫
0
√
uxxdx, H
(1/3)
3 =
2π
∫
0
√
uxx(∂−1u2x)dx,
H
(2/9)
4 =
2π
∫
0
[
(∂−1u2x)(uuxu
2
xx − u2xx(∂−1u2x))
]
2
9 dx (29)
are also conservation laws for the Hunter-Saxton dynamical system (27), where m 6= −4n and
n,m ∈ Z.
Now we proceed to the analysis of the Hamiltonian properties of the dynamical system (16),
for which we will search for solutions [7–9] of Nöther equation
LKϑ = ϑt − ϑK ′,∗ −K ′ϑ = 0. (30)
where LK denotes the corresponding Lie derivative on M subject to the vector field K : M →
T (M), K ′ : T (M) →T (M) is its Frechet derivative, K ′∗ : T ∗(M) →T ∗(M) is its conjugation
with respect to the standard bilinear form (·, ·) on T ∗(M)×T (M), and ϑ : T ∗(M) → T (M) is a
suitable implectic operator on M, with respect to which the following Hamiltonian representation
K = −ϑ grad Hϑ (31)
for some smooth functional Hϑ ∈ D(M) holds. To show this, it is enough to find, for instance by
means of the small parameter method [7, 8], a non-symmetric (ψ′ 6= ψ′,∗) solution ψ ∈ T ∗(M) to
the following Lie-Lax equation:
ψt +K ′,∗ψ = grad L (32)
for some suitably chosen smooth functional L ∈ D(M). As a result of easy calculations one obtains
that
ψ = (v, 0)⊺, L =
1
2
2π
∫
0
v2dx. (33)
43002-6
A hydrodynamical Riemann type equation integrability
Making use of (32) jointly with the classical Legendrian relationship
Hϑ := (ψ,K) − L (34)
for the suitable Hamiltonian function, one easily obtains the corresponding symplectic structure
ϑ−1 := ψ′ − ψ′,∗ =
(
0 1
−1 0
)
(35)
and the non-singular Hamilton function
Hϑ :=
1
2
2π
∫
0
(v2 + vxu
2)dx. (36)
Since the operator (35) is nonsingular, we obtain the corresponding implectic operator
ϑ =
(
0 −1
1 0
)
, (37)
necessarily satisfying the Nöther equation (43).
Here it is worth to observe that the Lie-Lax equation (32) possesses another solution
ψ =
(
ux
2
,− u2x
2vx
)
, L =
1
4
2π
∫
0
uvxdx, (38)
giving rise, owing to expressions (35) and (34), to the new co-implectic (singular “symplectic”)
structure
η−1 := ψ′ − ψ′,∗ =
(
∂ −∂uxv−1
x
−uxv−1
x ∂ u2xv
−2
x ∂ + ∂u2xv
−2
x
)
(39)
on the manifold M, subject to which the Hamiltonian functional equals
Hη :=
1
2
2π
∫
0
(uxv − vxu)dx, (40)
supplying the second Hamiltonian representation
K = −η grad Hη (41)
of the Riemann type hydrodynamical system (16). The co-implectic structure (39) is singular, since
η̂−1(ux, vx)⊺ = 0, nonetheless one can calculate its inverse expression
η :=
(
−∂−1 ux∂
−1
∂−1ux vx∂
−1 + ∂−1vx
)
. (42)
Moreover, the corresponding implectic structure η : T ∗(M) → T ∗(M) satisfies the Nöther equation
LKη = ηt − ηK ′,∗ −K ′η = 0, (43)
whose solutions can also be obtained by means of the small parameter method [7, 8]. We also
remark that, owing to the general symplectic theory results [7–9] for nonlinear dynamical systems
on smooth functional manifolds, operator (39) defines on the manifold M a closed functional
differential two-form. Thereby it is a priori co-implectic (in general, singular symplectic), satisfying
on M the standard Jacobi brackets condition.
43002-7
M.V. Pavlov, A.K. Prykarpatsky
As a result, the second implectic operator (42), being compatible [7, 9] with the implectic
operator (37), gives rise to a new infinite hierarchy of polynomial conservation laws
γn :=
1
∫
0
dλ
〈
(ϑ−1η)n grad Hϑ[uλ}, u
〉
(44)
for all n ∈ Z+. Having defined the recursion operator Λ := ϑ−1η, one also finds from (44), (30)
and (43) that the following Lax type relationship
LKΛ = Λt − [Λ,K
′,∗] = 0 (45)
holds. If we construct the asymptotical expansion ϕ(x;λ) ≃ ∑
j∈Z+
λ1−2j grad γj−1[u, v] as λ→ ∞,
it is easy to obtain from (44) that the gradient like relationship
λ2ϑϕ(x;λ = ηϕ(x;λ) (46)
holds. The latter relationship, making use of the implectic operators (37) and (42), can be repre-
sented in the two factorized forms:
ϕ(x;λ) :=
(
ϕ1(x;λ)
ϕ2(x;λ)
)
=
(
−4λ3f2
1 + 2λvxf
2
2
−4λ2f1f2 − 2λuxf
2
2
)
=
(
−2λ(f1f2)x
−(f2
2 )x
)
, (47)
where a vector f ∈ C∞(R2;C2) lies in an associated to manifold M vector bundle L(M;E2), whose
fibers are isomorphic to the complex Euclidean vector space E2. Take now into account [7, 8] that
the Lie-Lax equation
LKϕ(x;λ) = dϕ(x;λ)/dt+K ′,∗ϕ(x;λ) = 0 (48)
can be transformed equivalently for all x, t ∈ R and λ ∈ C into the following evolution system:
Dtϕ =
(
0 vx
−1 −ux
)
ϕ, Dt = ∂/∂t+ u∂/∂x. (49)
The equation (49), owing to the relationship (46) and the obvious identity
Dtfx + uxfx = (Dtf)x , (50)
can be further split into the adjoint to (49) system
Dtf = q(λ)f, q(λ) :=
(
0 0
−λ 0
)
, (51)
where a vector f ∈ C∞(R2;C2) satisfies the following linear equation
fx = ℓ[u, v;λ]f, ℓ[u, v;λ] :=
(
−λux −vx
2λ2 λux
)
, (52)
compatible with (51). Moreover, as a result of (51) and (50), the general solution to (52) allows
the following functional representation:
f1(x, t) = g̃1(u− tv, x− tu+ vt2/2),
f2(x, t) = −tλg̃1(u− tv, x− tu+ vt2/2) + g̃2(u − tv, x− tu+ vt2/2), (53)
where g̃j ∈ C∞(R2;C), j = 1, 2, are arbitrary smooth complex valued functions. Now combining
together the obtained relationships (51) and (52), we can formulate the following proposition.
Proposition 4.7 The Riemann type hydrodynamical system (16) is equivalent to a completely
integrable bi-Hamiltonian flow on the functional manifold M, allowing the Lax type representation
43002-8
A hydrodynamical Riemann type equation integrability
fx = ℓ[u, v;λ]f, ft = p(ℓ)f, p(ℓ) := −uℓ[u, v;λ] + q(λ),
ℓ[u, v;λ] :=
(
−λux −vx
2λ2 λux
)
, q(λ) :=
(
0 0
−λ 0
)
,
p(ℓ) =
(
λuxu vxu
−λ− 2λ2u −λuxu
)
, (54)
where f ∈ C∞(R2;C2) and λ ∈ C is an arbitrary spectral parameter.
Remark 4.8 It is worth to mention here that equation (51) is equivalent on the solution set of
the Riemann type hydrodynamical system (16) to the single equation
D2
t f2 = 0 ⇐⇒ Dtf1 = 0, Dtf2 = −λf1 , (55)
where vector f ∈ C∞(R2;C2) satisfies for all λ ∈ C the compatibility condition (52) and whose
general solution is represented in the functional form (53).
Concerning the set of conservation laws {H(1/2)
2 , H
(1/2)
3 }, constructed above in (29), they can
be extended to an infinite hierarchy {γ(1/2)j ∈ D(M) : j ∈ Z+}, where
γ
(1/2)
j :=
2π
∫
0
σ2j−1[u, v]dx, (56)
and the affine generating function σ(x;λ) := d/dx ln f2(x;λ) ≃ ∑
j=Z+∪{−1} σj [u, v]λ−j as λ→ ∞
satisfies the following functional equation:
(σ − λux)x + σ2 + λ2(2vx − u2x) = 0. (57)
In addition, the gradient functional ϕ(x;λ) := grad γ(x;λ) ∈ T ∗(M), where γ(λ) :=
∫ 2π
0
σ(x;λ)dx,
satisfies for all λ ∈ C the gradient relationship (46).
4.1. The Lax type representation
Here we proceed to the analysis of conservation laws and bi-Hamiltonian structure of the
generalized Riemann type equation (10) at N = 3 :
ut = v − uux
vt = z − uvx
zt = −uzx
:= K[u, v, z], (58)
where K : M → T (M) is a suitable vector field on the periodic functional manifold M :=
C∞(R/2πZ;R3) and t ∈ R is an evolution parameter. The system (58) also proves to possess
infinite hierarchies of polynomial conservation laws, being suspicious for complete and Lax type
integrability.
Namely, the following polynomial functionals, found by means of the algorithm described in
43002-9
M.V. Pavlov, A.K. Prykarpatsky
section 2, are conserved with respect to the flow (58):
H(1)
n :=
2π
∫
0
dxzn
(
vux − vxu− n+ 2
n+ 1
z
)
,
H(4) :=
2π
∫
0
dx
[
−7vxv
2u+ z(6zu+ 2vxu
2 − 3v2 − 4vuux)
]
,
H(5) :=
2π
∫
0
dx(z2ux − 2zvvx), H(6) :=
2π
∫
0
dx(zzv
3 + 3z2vxu+ z3),
H(7) :=
2π
∫
0
dx(zxv
3 + 3z2vux − 3z3),
H(8) :=
2π
∫
0
dxz(6z2u+ 3zvxu
2 − 3zv2 − 4zvux − 2vxv
2u+ 2v3ux),
H(9) :=
2π
∫
0
dx[1001vxv
4u+ (1092z2u2 + 364zvxu
3
−1092zv2u− 728zvuxu
2 − 364vxv
2u2 + 273v4 + 728v3uxu]),
H(2)
n :=
2π
∫
0
dxzxvz
n, H(3)
n :=
2π
∫
0
dxzx(v2 − 2zu)n, (59)
where n ∈ Z+. In particular, as n = 1, 2, . . . , from (59) one obtains that
H
(2)
0 :=
2π
∫
0
dxzxv, H
(2)
1 :=
2π
∫
0
dxzxzv, . . . ,
H
(3)
1 :=
2π
∫
0
dxzx(v2 − 2uz),
H
(3)
2 :=
2π
∫
0
dxzx(v4 + 4z2u2 − 4zv2u), . . . , (60)
and so on.
Making use of the iterative property, similar to that, formulated above in Proposition 4.1, one
can construct the following hierarchy of non-polynomial dispersive and dispersionless conservation
laws:
H
(1/4)
1 =
2π
∫
0
dx(−2uxxuxzx + uxxv
2
x + 2u2xzxx − uxvxxvx + 3vxxzx − 3vxzxx)1/4,
H
(1/3)
2 =
2π
∫
0
dx(−vxxzx + vxzxx)1/3,
H
(1/3)
3 =
2π
∫
0
dx(vxxux − vxuxx − zxx)1/3,
43002-10
A hydrodynamical Riemann type equation integrability
H
(1/2)
1 =
2π
∫
0
dx[−2vuxzx + v2x + z(−uxvx + 3zx)]1/2,
H
(1/2)
2 =
2π
∫
0
dx(8u3xzx − 3u2xv
2
x − 18uxvxzx + 6v3x + 9zx)1/2,
H
(1/5)
1 =
2π
∫
0
dx(−2uxxxuxzx + uxxxv
2
x + 6u2xxzx − 6uxxuxzxx
− 3uxxvxxvx + 2u2xzxxx − uxvxxxvx + 3uxv
2
xx + 3vxxxzx − 3vxzxxx)1/5,
H(1/3) =
2π
∫
0
dx[k1u(−vxxzx + vxzxx) + k1v(uxxzx − uxzxx)
+ z(k2uxxvx − k2uxvxx + k1zxx + k2zxx) + k3(−3uxvxzx + v3x + 3z2x)]1/3, (61)
where kj ∈ R, j = 1, 3, are arbitrary real numbers. Below we will attempt to generalize the crucial
relationship (51) from section 2 on the case of the Riemann type hydrodynamical system (58).
Namely, we will assume, based on the Remark (4.3), that there exists its following linearization:
D3
t f3(λ) = 0, (62)
modeling the generalized Riemann type hydrodynamical equation (10) at N = 3, and where
f3(λ) ∈ C∞(R2;C) for all values of the parameter λ ∈ C. The scalar equation (62) can be easily
rewritten as the system of three linear equations
Dtf1 = 0, Dtf2 = µ1f1, Dtf3 = µ2f2 (63)
where we have defined a vector f := (f1, f2, f2)
⊺ ∈ C∞(R2;C3) and naturally introduced constant
numbers µj := µj(λ) ∈ C, j = 1, 2. It is easy to observe now that, owing to the former result (14),
the system of equations (63) allows the following solution representation:
f1(x, t) = g̃1(u − tv + zt2/2, v − zt, x− tu+ vt2/2 − zt3/6),
f2(x, t) = tµ1g̃1(u− tv + zt2/2, v − zt, x− tu+ vt2/2 − zt3/6)
+ g̃2(u− tv + zt2/2, v − zt, x− tu+ vt2/2 − zt3/6),
f3(x, t) = µ1µ2
t2
2
g̃1(u− tv + zt2/2, v − zt, x− tu+ vt2/2 − zt3/6)
+ tµ2g̃2(u− tv + zt2/2, v − zt, x− tu+ vt2/2 − zt3/6)
+ g̃3(u− tv + zt2/2, v − zt, x− tu+ vt2/2 − zt3/6), (64)
where g̃j ∈ C∞(R3;C), j = 1, 3, are arbitrary smooth complex valued functions. The system (63)
transforms into the equivalent vector equation
Dtf = q(µ)f, q(λ) :=
0 0 0
µ1(λ) 0 0
0 µ2(λ) 0
, (65)
which should be compatible both with a suitably chosen equation for derivative
fx = ℓ[u, v, z;λ]f (66)
with some matrix ℓ[u, v, z;λ] ∈ SL(3;C), defined on the functional manifold M, and with the
Lie-Lax equation (48), rewritten as the following system of equations
Dtϕ =
0 vx zx
−1 −ux 0
0 −1 −ux
ϕ, Dt = ∂/∂t+ u∂/∂x, (67)
43002-11
M.V. Pavlov, A.K. Prykarpatsky
where the vector ϕ := ϕ(x;λ) ∈ T ∗(M) is considered as the one factorized by means of a solution
f ∈ C∞(R2;C3) to (66), satisfying the identity (50). Namely, it is assumed that the following
quadratic trace-relationship
ϕ(x;λ) = tr(Φ f ⊗ f⊺) (68)
holds for some vector valued matrix functional Φ := Φ[λ;u, v, z] ∈ E
3 ⊗ End E
3, defined on the
manifold M, where “⊗” means the standard tensor product of vectors from the Euclidean space
E3. Making now use of the expressions (50), (68) and (65), one can find by means of somewhat cum-
bersome and tedious calculations that µ1(λ) = λ, µ2(λ) = λ, λ ∈ C, and the matrix representation
of the derivative (66)
ℓ[u, v, z;λ] =
λ2ux −λvx zx
3λ3 −2λ2ux λvx
6λ4r[u, v, z] −3λ3 λ2ux
, (69)
compatible with equation (67), where a smooth mapping r : M → R satisfies the differential
relationship
Dtr + uxr = 1. (70)
The latter possesses a wide set R of different solutions amongst which there are the following:
r ∈ R :=
{
[
(xv − u2/2)/z
]
x
, (vx − u2x/6)z−1
x ,
u3x/3 − uxvx + 3zx/2
2uxzx − v2x
,
(vxv
3/6 − uxv
2z/2 + uzx(uz − v2)/6 + vz2)z−3
}
. (71)
Note here that only the third element from the set (71) allows the reduction z = 0 to the case
N = 2. Thus, the resulting Lax type representation for the Riemann type dynamical system (58)
ensues in the form:
fx = ℓ[u, v, z;λ]f, ft = p(ℓ)f, p(ℓ) := −uℓ[u, v, z;λ] + q(λ),
ℓ[u, v, z;λ] =
λ2ux −λvx zx
3λ3 −2λ2ux λvx
6λ4r[u, v, z] −3λ3 λ2ux
, q(λ) :=
0 0 0
λ 0 0
0 λ 0
,
p(ℓ) =
−λ2uux λuvx −uzx
−3uλ3 + λ 2λ2uux −λuvx
−6λ4u r[u, v, z] λ+ 3uλ3 −λ2uux
, (72)
where f ∈ C∞(R2;C3) and λ ∈ C is a spectral parameter.
The next problem, which is of great interest, consists in proving that the generalized hydro-
dynamical system (58) is a completely integrable bi-Hamiltonian flow on the periodic functional
manifold M, as it was proved above for the system (16).
That dynamical system (58) is bi-Hamiltonian which easily follows as a simple corollary from
the fact that it possesses the Lax type representation (72) and from the general Lie-algebraic in-
tegrability theory [7–9]. Taking into account that dynamical system (58) possesses many (at least
4) Lax type representations, one derives that it possesses many (at least 4) different pairs of com-
patible co-symplectic structures, each of which generates its own infinite hierarchy of conservation
laws commuting to each other. Moreover, the involution of conservation laws belonging to different
hierarchies fails owing to their non-compatibility. As the procedure of finding these structures is
adjoint with quite cumbersome analytical calculations, hereinbelow we present only one pair of
related co-symplectic structures, making use of the standard properties of determining Lie-Lax
equation (32).
43002-12
A hydrodynamical Riemann type equation integrability
To tackle with the related task of retrieving the Hamiltonian structure of the dynamical system
(58), it is enough, as in section 2, to construct [7, 8] exact non-symmetric solutions to the Lie-Lax
equation
ψt +K ′,∗ψ = grad L, ψ′ 6= ψ′,∗, (73)
for some functional L ∈ D(M), where ψ ∈ T ∗(M) is, in general, a quasi-local vector, such that
the system (58) allows for the following Hamiltonian representation:
K[u, v, z] = −η grad Hη[u, v, z], Hη = (ψη,K) − L, η−1 = ψ
′
η − ψ′,∗
η . (74)
As a test solution to (73) one can take the one
ψη = (ux/2, 0,−z−1
x u2x/2 + z−1
x vx)⊺, L =
1
2
2π
∫
0
(2z + vux)dx,
which gives rise to the following co-implectic operator:
η−1 := ψ′
η − ψ′,∗
η =
∂ 0 −∂uxz−1
x
0 0 ∂z−1
x
−uxz−1
x ∂ z−1
x ∂ 1
2 (u2xz
−2
x ∂ + ∂u2xz
−2
x ) − (vxz−2
x ∂ + ∂vxz−2
x )
. (75)
This expression is not strictly invertible, as its kernel possesses the translation vector field d/dx :
M → T (M) with components (ux, vx, zx)⊺ ∈ T (M), that is η−1(ux, vx, zx)⊺ = 0.
Nonetheless, upon formally inverting the operator expression (75), using quite simple, but a bit
cumbersome, direct calculations, we obtain that the Hamiltonian function is equal to:
Hη :=
2π
∫
0
dx(uxv − z) (76)
and the implectic η-operator looks as follows:
η :=
∂−1 ux∂
−1 0
∂−1ux vx∂
−1 + ∂−1vx ∂−1zx
0 zx∂
−1 0
. (77)
The same way, representing the Hamiltonian function (76) in the scalar form
Hη = (ψϑ, (ux, vx, zx)⊺), ψϑ = 1
2 (−v, u+ · · · ,− 1√
z
∂−1
√
z)⊺,
dψϑ/dt+K ′,∗ψϑ = grad Lϑ (78)
for some functional Lϑ ∈ D(M), one can construct a second implectic (co-symplectic) operator
ϑ : T ∗(M) → T (M), looking up to O(µ2) terms, as follows:
ϑ =
µ
( (u(1))2
z(1) ∂ + ∂ (u(1))2
z(1)
) 1 + 2µ
3
(
u(1)v(1)
z(1) ∂
+2∂ u(1)v(1)
z(1)
)
2µ
3
(
∂ (v(1))2
z(1) + ∂u(1)
)
−1 + 2µ
3
(
∂ u(1)v(1)
z(1)
+2u(1)v(1)
z(1) ∂
)
2µ
3
( (v(1))2
z(1) ∂ + ∂ (v(1))2
z(1)
)
+ 2µ
3
(
u(1)∂ + ∂u(1)
) 2µ∂v(1)
2µ
3 ( (v
(1))2
z(1) ∂ + u(1)∂) 2µv(1)∂ µ(∂z(1) + z(1)∂)
+O(µ2),
(79)
where we put, by definition, ϑ−1 := (ψ′
ϑ − ψ′,∗
ϑ ), u := µu(1), v := µv(1), z := µz(1) as µ → 0,
and whose exact form needs some additional simple but cumbersome calulations, which will be
presented in a work under preparation.
43002-13
M.V. Pavlov, A.K. Prykarpatsky
The operator (79) satisfies the Hamiltonian vector field condition:
(ux, vx, zx)⊺ = −ϑ grad Hη , (80)
following easily from (78). Now, making use of the expressions (74) and (78), one can derive that
ϑ−1η grad Hη = −ϑ−1K := ϕϑ , ϕ′
ϑ = ϕ′,∗
ϑ . (81)
Owing to the second equality of (81) and the classical homology relationship ϕϑ = grad Hϑ for
some function Hϑ ∈ D(M), one can calculate the expression
Hϑ =
1
∫
0
(ϕϑ[su, sv, sz], (u, v, z)⊺)ds, (82)
satisfying the Hamiltonian condition
K = −ϑ grad Hϑ . (83)
Remark 4.9 We mention here that the exact expression of the Hamiltonian function (82) can be
easily calculated modulo the exact form of the element ϕϑ ∈ T ∗(M) and the co-implectic operator
ϑ−1 : T (M) → T ∗(M), constructed by formulae (81) and (79), respectively.
The results obtained above can be formulated as the following proposition.
Proposition 4.10 The Riemann type hydrodynamical system (10) at N = 3 is equivalent to a
completely integrable bi-Hamiltonian flow on the functional manifold M, allowing the Lax type
representation (72) and the compatible pair of co-symplectic structures (77) and (79).
4.2. The hierarchies of conservation laws and their origin analysis
The infinite hierarchy of conservation laws like (61) and related recurrent relationships can be
regularly reconstructed, if we compute the asymptotical solutions to the following Lie-Lax equation:
LK̃ ϕ̃ = ϕ̃τ + K̃ ′,∗ϕ̃ = 0,
ϕ̃ ≃ ã(x;λ) exp{λ2τ + ∂−1σ̃(x;λ)}, (84)
where, by definition, ã(x;λ) ≃ ∑
j∈Z+
ãj[u, v, z]λ−j , σ̃(x;λ) ≃ ∑
j∈Z+∪{−2} σ̃j [u, v, z]λ−j as λ →
∞, and
d
dτ
(u, v, z)⊺ := −3η grad H
(1/3)
3 [u, v, z] =
−(u2xh
−2)x + v−1
x (v2xh
−2)x
−vxu−1
x (u2xh
−2)x + z−1
x (z2xh
−2)x
−zxu−1
x (z2xh
−2)x
:= K̃[u, v, z],
H
(1/3)
3 :=
2π
∫
0
h[u, v, z]dx, h[u, v, z] = (vxxux − uxxvx − zxx)1/3, (85)
is a Hamiltonian vector field on the functional manifold M with respect to a suitable evolution
parameter τ ∈ R. Since the vector fields (85) and (58) are commuting to each other on the whole
manifold M, the functionals
H̃
(1/3)
j :=
2π
∫
0
σ̃j−2[u, v, z]dx, (86)
j ∈ Z+,will be functionally independent conservation laws for both these dynamical systems.
Moreover, as one can check by means of quite cumbersome calculations, the conservation laws
43002-14
A hydrodynamical Riemann type equation integrability
H̃
(1/3)
j , j ∈ Z+, coincide up to constant coefficients with the conservation laws H
(1/3)
j , j ∈ Z+,
given by suitable elements of (61). But here a question arises – how they are related with the Lax
pair (72), strongly depending on the r-solutions (71) to the differential-functional equation (70)?
(We cordially thank a Referee of the article for posing this question.) To reply to this question, it is
enough to construct the corresponding hierarchy of conservation laws making use of the standard
Riccati type procedure, applied to the first equation of (72). Namely, having put, by definition,
∂f3/∂x := σ(x;λ)f3 , f2 := b(x;λ)f3 , f1 := a(x;λ)f3 , (87)
where the following asymptotical expansions
σ(x;λ) ≃
∞
∑
j>−2
σj [u, v, z; r]λ−j, a(x;λ) ≃
∞
∑
j>2
aj [u, v, z; r]λ−j ,
b(x;λ) ≃
∞
∑
j>1
bj [u, v, z; r]λ−j , (88)
hold as |λ| → ∞ and whose coefficients satisfy the sequences of reccurent differential-functional
equations
∂aj/∂x+
∑
k
aj−kσk = uxaj+2 − vxbj+1 + zxδj,0 ,
∂bj/∂x+
∑
k
bj−kσk = 3aj+3 − 2uxbj+2 + vxδj,−1 ,
σj = raj+4 − 3bj+3 + uxδj,−2 , (89)
for all integers j+4 ∈ Z+, we easily obtain that the initial local functionals σ−2[u, v, z; r], a2[u, v, z; r]
and b1[u, v, z; r] solve the system of equations
σ−2 + 3b1 − ra2 = ux , (90)
b1(3ux + ra2 − 3b1) − 3a2 = vx ,
a2(ra2 − 3b1) + vxb1 = zx ,
easily reducing to a one cubic equation on the local functional σ−2[u, v, z; r]. Since the latter makes
it possible, owing to (89), to recurrently calculate all other functionals σj [u, v, z; r], j > 1, we can
obtain this way an infinite hierarchy of functionals
γ
(1/3)
j :=
2π
∫
0
σj−2[u, v, z; r]dx (91)
for all j ∈ Z+, being, owing to the first equation of (87) and the second one of (72), conservation
laws for a dynamical system (58). Moreover, these conservation laws at r := (vx − u2x/6)z−1
x
coincide, up to constant coefficients, with those (86) constructed above. Similar calculations can
be also performed for other r-solutions of (71), but owing to their cumbersomeness, we do not
present them in detail.
Remark 4.11 Based on the Lax type representation (87) one can state on the manifold M by
means of direct analytical calculations the well known gradient-like identity (46)
λ2ϑϕ(x;λ) = ηϕ(x;λ) (92)
for the gradient functional ϕ(x;λ) := gradλ[u, v, z; r] ∈ T ∗(M), where the implectic operators
η, ϑ : T ∗(M) → T (M) coincide at some r ∈ R with those, given by expressions (77) and (79).
43002-15
M.V. Pavlov, A.K. Prykarpatsky
The Lax type integrability of the Riemann type hydrodynamical equation (10) at N = 2 and
N = 3, stated above, allows one to speculate that this property holds for arbitrary N ∈ Z+.
Concerning the evident difference between analytical properties of the cases N = 2 and N = 3,
we can easily observe that it is related with structures of the corresponding Lax type operators (52)
and (72): in the first case the corresponding r-equation (70) is trivial (that is empty), but in the
second case it is already nontrivial, allowing many different solutions. This situation generalizes,
as we will see below, to the case N > 4, thereby explaining the appearing diversity of the related
Lax type representations.
To support this hypothesis we will prove below that also at N = 4 it is equivalent to a Lax
type integrable bi-Hamiltonian dynamical system on the suitable smooth 2π-periodic functional
manifold M := C∞(R/2πZ;R4), possesses infinite hierarchies of polynomial dispersionless and
dispersive non-polynomial conservation laws.
5. The case N=4: conservation laws, bi-Hamiltonian structure and Lax type
representation
The Riemann type hydrodynamical equation (10) at N = 4 is equivalent to the nonlinear
dynamical system
ut = v − uux
vt = w − uvx
wt = z − uwx
zt = −uzx
:= K[u, v, w, z], (93)
where K : M → T (M) is a suitable vector field on the smooth 2π-periodic functional manifold
M := C∞(R/2πZ;R4). To state its Hamiltonian structure, we need to find a functional solution
to the Lie-Lax equation (32):
ψt +K
′,∗
ψ = grad L (94)
for some smooth functional L ∈ D(M), where
K
′
=
−∂u 1 0 0
−vx −u∂ 1 0
−wx 0 −u∂ 1
−zx 0 0 −u∂
, K
′,∗
=
u∂ −vx −wx −zx
1 ∂u 0 0
0 1 ∂u 0
0 0 1 ∂u
(95)
are, respectively, the Frechet derivative of the mapping K : M → T (M) and its conjugate. The
small parameter method [7], applied to equation (94), gives rise to the following exact solution:
ψ =
(
−wx, vx/2, 0,−
v2x
2zx
+
uxwx
zx
)⊺
, L =
2π
∫
0
(zux − vwx/2)dx. (96)
As a result, right away from (94) we obtain that dynamical system (93) is a Hamiltonian system
on the functional manifold M, that is
K = −ϑ grad H, (97)
where the Hamiltonian functional is equal to
H := (ψ,K) − L =
2π
∫
0
(uzx − vwx)dx (98)
and the co-implectic operator is equal to
ϑ−1 := ψ
′ − ψ′,∗ =
0 0 −∂ ∂ wx
zx
0 ∂ 0 −∂ vx
zx
−∂ 0 0 ∂ ux
zx
wx
zx
∂ − vx
zx
∂ ux
zx
∂
1
2 [z−2
x (v2x − 2uxwx)∂+
+∂(v2x − 2uxwx)z−2
x ]
. (99)
43002-16
A hydrodynamical Riemann type equation integrability
The latter is degenerate: the relationship ϑ−1(ux, vx, wx, zx)⊺ = 0 exactly on the whole manifold
M, but the inverse to (99) exists and can be calculated analytically.
In order to state the Lax type integrability of Hamiltonian system (93), we will apply the
standard gradient-holonomic scheme of [7, 8] to it and find its following linearization:
D4
t f4(λ) = 0, (100)
where f4(λ) ∈ C∞(R2;C) for all λ ∈ C. Having rewritten (100) in the form of a linear system
Dtf = q(λ)f, q(λ) :=
0 0 0 0
λ 0 0 0
0 λ 0 0
0 0 λ 0
, (101)
where λ ∈ C is a spectral parameter and the vector f ∈ C∞(R2;C4) allows, owing to the relation-
ship (100), the following functional representation:
f1(x, t) = g̃1
(
u− tv +
wt2
2
− xt3
3!
, v − wt+
zt2
2
, w − zt, x− tu+
vt2
2
− wt3
3!
+
zt4
4!
)
,
f2(x, t) = tµ1g̃1
(
u− tv +
wt2
2
− xt3
3!
, v − wt+
zt2
2
, w − zt, x− tu+
vt2
2
− wt3
3!
+
zt4
4!
)
+ g̃2
(
u− tv +
wt2
2
− xt3
3!
, v − wt+
zt2
2
, w − zt, x− tu+
vt2
2
− wt3
3!
+
zt4
4!
)
,
f3(x, t) = µ1µ2
t2
2
g̃1
(
u− tv +
wt2
2
− xt3
3!
, v − wt+
zt2
2
, w − zt, x− tu+
vt2
2
− wt3
3!
+
zt4
4!
)
+ tµ2g̃2
(
u− tv +
wt2
2
− xt3
3!
, v − wt+
zt2
2
, w − zt, x− tu+
vt2
2
− wt3
3!
+
zt4
4!
)
+ g̃3
(
u− tv +
wt2
2
− xt3
3!
, v − wt+
zt2
2
, w − zt, x− tu+
vt2
2
− wt3
3!
+
zt4
4!
)
,
f4(x, t) = µ1µ2µ3
t3
3!
g̃1
(
u−tv +
wt2
2
−xt
3
3!
, v − wt+
zt2
2
, w − zt, x− tu+
vt2
2
−wt3
3!
+
zt4
4!
)
+ µ2µ3
t2
2
g̃2
(
u− tv +
wt2
2
− xt3
3!
, v − wt+
zt2
2
, w − zt, x− tu+
vt2
2
−wt3
3!
+
zt4
4!
)
+ tµ3g̃3
(
u− tv +
wt2
2
− xt3
3!
, v − wt+
zt2
2
, w − zt, x− tu+
vt2
2
− wt3
3!
+
zt4
4!
)
+ g̃4
(
u− tv +
wt2
2
− xt3
3!
, v − wt+
zt2
2
, w − zt, x− tu+
vt2
2
− wt3
3!
+
zt4
4!
)
, (102)
where g̃j ∈ C∞(R4;C), j = 1, 4, are arbitrary smooth complex valued functions.
Based on the expressions (101) and (102), one can construct the related linear representation
of the expression ∂f/∂x ∈ C∞(R4;C4) in the following matrix form:
fx = ℓ[u, v, w, z;λ]f, (103)
where
ℓ[u, v, w, z;λ] :=
−λ3ux λ2vx −λwx zx
−4λ2 3λ3ux −2λ2vx λwx
−10λ5r1 6λ4 −3λ3ux λ2vx
−20λ6r2 10λ5r1 −4λ4 λ3ux
(104)
f ∈ C∞(R2;C4) and which is compatible with (101). Thus, we can formulate the following theorem
about the Lax integrability of the generalized Riemann type hydrodynamical system (93).
43002-17
M.V. Pavlov, A.K. Prykarpatsky
Theorem 5.1 The dynamical system (10) at N = 4, equivalent to the generalized Riemann type
hydrodynamical system (93), possesses the Lax type representation
fx = ℓ[u, v, z, w;λ]f, ft = p(ℓ)f, p(ℓ) := −uℓ[u, v, w, z;λ] + q(λ), (105)
where
ℓ[u, v, w, z;λ] :=
−λ3ux λ2vx −λwx zx
−4λ4 3λ3ux −2λ2vx λwx
−10λ5r1 6λ4 −3λ3ux λ2vx
−20λ6r2 10λ5r1 −4λ4 λ3ux
, q(λ) :=
0 0 0 0
λ 0 0 0
0 λ 0 0
0 0 λ 0
,
p(ℓ) =
λuux −λ2uvx λuwx −uzx
λ+ 4λ4u −3λ3uux 2λ2uvx −λuwx
10λ5ur1 λ− 6λ4u 3λ3uux −λ2uvx
20λ6ur2 −10λ5ur1 λ+ 4λ4u −λ3uux
, (106)
thus being a Lax type integrable dynamical system on the functional manifold M.
Owing to the existence of the Lax type representation (105) and the related gradient such as
relationship [7, 8], we can easily derive that the Hamiltonian system (93) is at the same time a
bi-Hamiltonian flow on the functional manifold M. In addition, making use of the above results
and the approach of work [11], we can construct the infinite hierarchies of conservation laws for
(93), both dispersionless polynomial and dispersive non-polynomial ones:
a) polynomial conservation laws:
H9 =
2π
∫
0
dx(vwx − uzx), H(13) =
2π
∫
0
dx(zxw − zwx),
H14 =
2π
∫
0
dx
(
k1
(
zx(v2 − 2uw) − z2
)
+ k2
(
− zxv
2 + 2wx(vw − uz) − z2
)
+ k3
(
2zxv
2 + 4wx(4z − vw) + 2z2
))
,
H16 =
2π
∫
0
dx(3uz − vw)zx, H18 =
2π
∫
0
dxzx(w2 − 2vz),
H17 =
2π
∫
0
dx[12vxuzw + zx(9u2z + 16uvw− 2v3) + 6wwx(v2 − 2uw) + 6z(2vz − w2)],
H19 =
2π
∫
0
dx
[
k1
(
10vxuz
2 + zx(12uvz − uw2 − 2v2w) + 5wwx(vw − 2uz)
)
+ k2
(
zx(6uvz − 3uw2 − v2w) + 5wxv(w2 − vz)
)]
; (107)
b) non-polynomial conservation laws:
H11 =
2π
∫
0
dx
(
uxxzx − uxzxx + vxwxx − vxxwxx
)
1
3
, H
(1/2)
12 =
2π
∫
0
dx
√
w2
x − 2vxzx ,
H
(1/3)
12 =
2π
∫
0
dx
(
9u2xzx − 6uxvxwx + 2v3x − 12vxzx + 6w2
x
)
1
3
,
43002-18
A hydrodynamical Riemann type equation integrability
H
(1/3)
13 =
2π
∫
0
dx
(
u(2vxzx − w2
x) + v(vxwx − 3uxzx) + w(uxwx − v2x + 2zx) + z(uxvx − 2wx)
)
1
3
,
H
(1/3)
15 =
2π
∫
0
dx
(
zxwxx − zxxwx
)
1
3
,
H
(1/2)
15 =
2π
∫
0
dx
(
k1
(
v(2vxzx − w2
x) + z(4zx − uxwx) + w(vxwx − 3uxzx)
)
+ k2z
(
2zx + v2x − uxwx
)
)
1
2
,
H
(1/5)
16 =
2π
∫
0
dx
(
uxxx(2vxzx−w2
x)+vxxx(vxww−3uxzx)
+ zxxx(uxvx − 2wx) + wxxx(uxwx − v2x + 2zx)
+ 3uxx(vxxzx − 3vxzxx + wxxwx) + 3vxx(2uxzxx
− vxxwx + vxwxx) − 3w2
xxux
)
1
5
,
5H
(1/4)
16 =
2π
∫
0
dx
(
4u2xw
2
x − 4uxv
2
xwx − 8uxzxwx + v4x − 4v2xzx + 4z2x
)
1
4
,
H
(1/3)
16 =
2π
∫
0
dx
(
k1
(
u(zxwxx − zxxwx) + v(vxzxx − vxxzx) + zzxx + w(uxxzx − uxzxx)
)
+ k2
(
z(uxxwx − uxwxx + 2zxx) + w(uxxzx − uxzxx − vxxwx + vxwxx)
)
+ k3zx(v2x − 2uxwx + 2zx)
)
1
3
, (108)
where kj , j = 1, 3, are arbitrary constants. We also observe that the Hamiltonian functional (98)
coincides exactly up to the sign with the polynomial conservation law H(9) ∈ D(M).
As concerns the general case N ∈ Z+, successively applying the above devised method, one
can also obtain for the Riemann type hydrodynamical system (10) the corresponding Lax type
representation, construct infinite hierarchies of dispersive and dispersionless conservation laws,
their symplectic structures and the related Lax type representations, which is a topic of the next
work under preparation.
6. Conclusion
As follows from the results obtained in this work, the generalized Riemann type hydrodynamical
system (10) at N = 2, 3 and N = 4 possesses many infinite hierarchies of conservation laws, both
dispersive non-polynomial and dispersionless polynomial ones. This fact can be easily explained
by the fact that the corresponding dynamical systems (16), (58) and (93) possess many, plausibly,
infinite sets of algebraically independent compatible implectic structures, which generate via the
corresponding gradient like relationships [7, 8] the related infinite hierarchies of conservation laws,
and as a by-product, infinite hierarchies of the associated Lax type representations. The existence
of many Lax type representations for the generalized Riemann type equation 10 for N ∈ Z+ was
recently justified by means of differential-algebraic tools in [12].
It is also worth to mention that the generalized Riemann type equation (10) is an example of
integrable dynamical systems belonging [14, 15] at the same time to two different classes: C- and
43002-19
M.V. Pavlov, A.K. Prykarpatsky
S- integrable. Really, these systems are linearizable and have exact general solutions though in an
unwieldy form. Thus, the Riemann type systems belong to the C-integrable class. Similar properties
had been analyzed earlier for [17–20] for the case of the Monge-Ampere equations. Moreover, these
systems have also infinite sets of compatible Hamiltonian structures, Lax type representations and
respectively commuting flows. So, they belong to the S- integrable class too. Such a situation
within the theory of Lax type integrable nonlinear dynamical systems is encountered, virtually, for
the first time and is interesting from different points of view, both theoretically and practically.
Keeping in mind these and some other important aspects of the Riemann type hydrodynamical
systems (10), we consider that they deserve additional thorough investigation in the future.
Acknowledgements
Authors express their sincere appreciation to Prof. Z. Popowicz and Dr. J. Golenia for many
discussions of the work and instrumental help in studying the infinite hierarchies of the conserva-
tion laws used in this paper. We are also sincerely appreciated to Profs. M. B laszak, Z. Peradzyński,
J. S lawianowski, N. Bogolubov (jr.) and D.L. Blackmore for useful discussions of the results ob-
tained. M.P. is indebted to Professor Y. Nutku for his hospitality at TUBITAK Marmara Research
Centre and in the Feza Gursey Institute (Istanbul), for his helpful explanations of the relation-
ship between Monge-Ampere equation and its Hamiltonian structures. He is grateful to Professor
A. Gurevich for his explanations of the physical nature of these equations and Prof. K. Zybin for
his remark that their system describing a dynamics of dark matter in the Universe can be derived
from the Vlasov kinetic equation (which can be derived from the Liouville equation for the distri-
bution function). He also would like to thank Prof. E. Ferapontov and Prof. G. El for their interest
and valuable discussions.
Especially M.V.P. is grateful to the SISSA in Trieste (Italy) where some part of this work has
been done. This research was in part supported by the RFBR grant 08–01–00464–a and by the
grant of Presidium of RAS “Fundamental Problems of Nonlinear Dynamics”.
References
1. Jeans J.H., Astronomy and Cosmology. Cambridge Univ. Press, London and New York, 1969.
2. Zeldovich Ya.B., Novikov I.D., Structure and Evolution of the Universe. Moscow, Nauka, 1975.
3. Sagdeev R.Z. – In: Reviews of Plasma Physics, Leontovich M.A., ed. Consultants Bureau, New York,
1968, 4, 20.
4. Gurevich A.V., Zybin K.P., Soviet Phys. JETP, 1988, 67, No. 1, 1–12.
5. Gurevich A.V., Zybin K.P., Soviet Phys. Usp., 1995, 38, No. 7, 687–722.
6. Whitham G.B., Linear and Nonlinear Waves. Willey-Interscience, New York, 1974.
7. Prykarpatsky A., Mykytyuk I., Algebraic Integrability of Nonlinear Dynamical Systems on Manifolds:
Classical and Quantum Aspects. Kluwer Academic Publishers, Netherlands, 1998.
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Investigating Nonlinear Dynamical Systems on Functional Manifolds. The Second edition. Lviv Uni-
versity Publ., 2006 (in Ukrainian).
9. Blaszak M., Multi-Hamiltonian Theory of Dynamical Systems. Springer, Berlin, 1998.
10. Marsden J., Chorin R., Mathematical Theory of Hydrodynamics. ???
11. Golenia J., Popowicz Z., Pavlov M., Prykarapatsky A., SIGMA, 2010, 6, 1–13.
12. Prykarpatsky A.K., Artemovych O.D., Popowicz Z., Pavlov M., J. Phys. A: Math. Theor., 2010, 43,
295205.
13. Das A., Brunelli J.C., J. Math. Phys., 2004, 45, No. 7, 2633–2645.
14. Calogero F., Stud. Appl. Math., 1984, 70, No. 3, 189–199.
15. Pavlov M.V., Theor. Math. Phys., 2001, 128, No. 1, 927–932.
16. Hunter J., Saxton R., SIAM J. Appl. Math., 1991, 51, No. 6, 1498–1521.
17. Juras M., Anderson I., Duke Math. J., 1997, 89, No. 2, 351–375.
18. Donato A., Oliveri F., Transport Theor. Stat., 1996, 25, No. 3–5, 303–322.
19. Nutku Y., Sarioglu O. Phys. Lett. A, 1993, 173, 270–274.
20. Pavlov M.V., J. Nonlinear Math. Phys, Supplement, 2002, 1, No. 9, 173–191.
43002-20
A hydrodynamical Riemann type equation integrability
21. Pavlov M.V., J. Phys. A: Math. Gen., 2005, 38, 3823–3840.
22. Bogolubov N.(jr.), Prykarpatsky A., Gucwa I., Golenia J. Preprint ICTP–IC/2007/109, Trieste, Italy
(available at: http://publications.ictp.it).
23. Golenia J., Bogolubov N. (jr.), Popowicz Z., Pavlov M., Prykarpatsky A. Preprint ICTP, IC/2009/095,
2009.
24. Prykarpatsky A., Prytula M.M., Nonlinearity, 2006, 19, 2115–2122.
Узагальнене гiдродинамiчне рiвняння Гуревича-Зибiна типу
Рiмана i його iнтегровнiсть типу Лакса
М.В. Павлов1, А.К. Прикарпатський2,3
1 Факультет математичної фiзики, Iнститут фiзики Лєбєдєва РАН, Москва, Росiя
2 Гiрничо-Металургiйна академiя iм. Ст. Сташiца, 30–059 Кракiв, Польща
3 Державний педагогiчний унiверситет iм. I. Франка, Дрогобич, Україна
Стаття присвячена дослiдженню гiдродинамiчного рiвняння типу Рiмана, що узагальнює вiдому
систему Гуревича-Зибiна. Це багатокомпонентне гiдродинамiчне рiвняння характеризується єдиною
характеристичною швидкiстю потоку. Проаналiзованi сумiснi бi-гамiльтоновi структури та представлення
Лакса для 3-та 4-компонентної узагальненої гiдродинамiчної системи типу Рiмана. Отриманi
результати вперше доповнюють теорiю iнтегровностi систем гiдродинамiчного типу, ранiше
розвинутої тiльки для вiдмiнних швидкостей в однорiдному випадку.
Ключовi слова: гiдродинамiчнi рiвняння типу Рiмана, iнтегровнiсть типу Лакса, закони збереження
43002-21
Introduction
A general solution to the Gurevich-Zybin hydrodynamical system of Riemann type
An N-component generalized one-dimensional Riemann type hydrodynamical equation
The generalized Riemann type hydrodynamical equation at N=2: conservation laws, bi-Hamiltonian structure and Lax type representation
The Lax type representation
The hierarchies of conservation laws and their origin analysis
The case N=4: conservation laws, bi-Hamiltonian structure and Lax type representation
Conclusion
|
| id | nasplib_isofts_kiev_ua-123456789-32119 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-07T18:54:34Z |
| publishDate | 2010 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Pavlov, M.V. Prykarpatsky, A.K. 2012-04-09T20:38:18Z 2012-04-09T20:38:18Z 2010 A generalized hydrodynamical Gurevich-Zybin equation of Riemann type and its Lax type integrability / M.V. Pavlov, A.K. Prykarpatsky // Condensed Matter Physics. — 2010. — Т. 13, № 4. — С. 43002:1-21. — Бібліогр.: 24 назв. — англ. 1607-324X PACS: 02.30.Ik, 02.30.Jr https://nasplib.isofts.kiev.ua/handle/123456789/32119 This paper is devoted to the study of a hydrodynamical equation of Riemann type, generalizing the remarkable Gurevich–Zybin system. This multi-component non-homogenous hydrodynamic equation is characterized by the only characteristic flow velocity. The compatible bi-Hamiltonian structures and Lax type representations of the 3-and 4-component generalized Riemann type hydrodynamical system are analyzed. For the first time the obtained results augment the theory of integrability of hydrodynamic type systems, originally developed only for distinct characteristic velocities in homogenous case. Досліджено гідродинамічне рівняння типу Рімана, що узагальнює відому систему Гуревича - Зибіна. Це багатокомпонентне гідродинамічне рівняння характеризується єдиною характеристичною швидкістю потоку. Проаналізовані сумісні бі-гамільтонові структури та представлення Лакса для три- та чотирикомпонентної узагальненої гідродинамічної системи типу Рімана. Одержані результати вперше доповнюють теорію інтегровності систем гідродинамічного типу, раніше розвиненої тільки для відмінних швидкостей в однорідному випадку. Authors express their sincere appreciation to Prof. Z. Popowicz and Dr. J. Golenia for many discussions of the work and instrumental help in studying the infinite hierarchies of the conservation laws used in this paper.We are also sincerely appreciated to Profs.M. B laszak, Z. Peradzy´nski, J. S lawianowski, N. Bogolubov (jr.) and D.L. Blackmore for useful discussions of the results obtained. M.P. is indebted to Professor Y. Nutku for his hospitality at TUBITAK Marmara Research Centre and in the Feza Gursey Institute (Istanbul), for his helpful explanations of the relationship between Monge-Ampere equation and its Hamiltonian structures. He is grateful to Professor A. Gurevich for his explanations of the physical nature of these equations and Prof. K. Zybin for his remark that their system describing a dynamics of dark matter in the Universe can be derived from the Vlasov kinetic equation (which can be derived from the Liouville equation for the distribution function). He also would like to thank Prof. E. Ferapontov and Prof. G. El for their interest and valuable discussions. Especially M.V.P. is grateful to the SISSA in Trieste (Italy) where some part of this work has been done. This research was in part supported by the RFBR grant 08–01–00464–a and by the grant of Presidium of RAS “Fundamental Problems of Nonlinear Dynamics”. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics A generalized hydrodynamical Gurevich-Zybin equation of Riemann type and its Lax type integrability Узагальнене гідродинамічне рівняння Гуревича - Зибіна типу Рімана і його інтегровність типу Лакса Article published earlier |
| spellingShingle | A generalized hydrodynamical Gurevich-Zybin equation of Riemann type and its Lax type integrability Pavlov, M.V. Prykarpatsky, A.K. |
| title | A generalized hydrodynamical Gurevich-Zybin equation of Riemann type and its Lax type integrability |
| title_alt | Узагальнене гідродинамічне рівняння Гуревича - Зибіна типу Рімана і його інтегровність типу Лакса |
| title_full | A generalized hydrodynamical Gurevich-Zybin equation of Riemann type and its Lax type integrability |
| title_fullStr | A generalized hydrodynamical Gurevich-Zybin equation of Riemann type and its Lax type integrability |
| title_full_unstemmed | A generalized hydrodynamical Gurevich-Zybin equation of Riemann type and its Lax type integrability |
| title_short | A generalized hydrodynamical Gurevich-Zybin equation of Riemann type and its Lax type integrability |
| title_sort | generalized hydrodynamical gurevich-zybin equation of riemann type and its lax type integrability |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/32119 |
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