On a Nonlocal Ostrovsky-Whitham Type Dynamical System, Its Riemann Type Inhomogeneous Regularizations and Their Integrability

0.2 MB·English

by  Jolanta Golenia

Symmetry, Integrability and Geometry: Methods and Applications SIGMA 6(2010), 002, 13 pages
On a Nonlocal Ostrovsky–Whitham Type
Dynamical System, Its Riemann Type Inhomogeneous
⋆
Regularizations and Their Integrability
Jo lanta GOLENIA †1, Maxim V. PAVLOV †2,
Ziemowit POPOWICZ †3 and Anatoliy K. PRYKARPATSKY †4†5
†1 The Department of Applied Mathematics, AGH University of Science and Technology,
Krak´ow 30059, Poland
0 E-mail: [email protected]
1
†2 Department of Mathematical Physics, P.N. Lebedev Physical Institute,
0
2 53 Leninskij Prospekt, Moscow 119991, Russia

n E-mail: [email protected]
a
J †3 The Institute for Theoretical Physics, University of Wrocl aw, Wrocl aw 50204, Poland

7 E-mail: [email protected]
1
URL: www.ift.uni.wroc.pl/~ziemek/

] †4 The Department of Mining Geodesics, AGH University of Science and Technology,
I
S
Krak´ow 30059, Poland
.
n
†5 Department of Economical Cybernetics, Ivan Franko State Pedagogical University,
i
l
n Drohobych, Lviv Region, Ukraine
[ E-mail: [email protected]

2
Received October 14, 2009, in final form January 03, 2010; Published online January 07, 2010
v
5 doi:10.3842/SIGMA.2010.002
4
1
Abstract. Short-waveperturbationsinarelaxingmedium,governedbyaspecialreduction
1
. of the Ostrovsky evolution equation, and later derived by Whitham, are studied using the
1
gradient-holonomicintegrabilityalgorithm. Thebi-Hamiltonicityandcompleteintegrability
0
0 of the corresponding dynamical system is stated and an infinite hierarchy of commuting to
1 each other conservation laws of dispersive type are found. The well defined regularization
: of the model is constructed and its Lax type integrability is discussed. A generalized hyd-
v
i rodynamical Riemann type system is considered, infinite hierarchies of conservation laws,
X
related compatible Poisson structures and a Lax type representation for the special case
r N =3 are constructed.
a
Key words: generalizedRiemanntypehydrodynamicalequations;Whithamtypedynamical
systems; Hamiltonian systems; Lax type integrability; gradient-holonomic algorithm
2010 Mathematics Subject Classification: 35C05;37K10
1 Introduction
Many importantproblemsofpropagatingwaves innonlinearmediawithdistributedparameters,
for instance, invisiblenon-dissipative darkmatter, playing adecisive role [9,10]in theformation
of large scale structure in the Universe like galaxies, clusters of galaxies, super-clusters, can be
described by means of evolution differential equations of special type. It is also well known [2,
⋆
This paper is a contribution to the Proceedings of the Eighth International Conference “Symmetry in
Nonlinear Mathematical Physics” (June 21–27, 2009, Kyiv, Ukraine). The full collection is available at
http://www.emis.de/journals/SIGMA/symmetry2009.html
2 J. Golenia, M.V. Pavlov, Z. Popowicz and A.K. Prykarpatsky
17, 25, 28] that shortwave perturbations in a relaxing one dimensional medium can be described
by means of some reduction of the Ostrovsky equations, coinciding with the Whitham type
evolution equation
du/dt = 2uu + (x,s)u ds, (1.1)
x s
RK
Z
discussedfirstin[28]. Herethekernel : R R Rdependsonthemediumelasticityproperties
K × →
with spatial memory and can, in general, be a function of the pressure gradient u C∞(R;R),
x
∈
evolving with respect to equation (1.1). In particular, if the nonlinear medium is endowed still
with spatial memory properties, that is the wave amplitude depends on the orbit, swept by its
front, the propagation of the corresponding wave can be modeled by means of the so called
generalized Ostrovsky evolution equations [20]. Namely, if to put (x,s) = 1 x s , x,s R,
K 2| − | ∈
then equation (1.1) can be reduced to
du/dt = 2uu +∂−1u,
x
which was, in particular, studied before in [17, 21, 22].
Since some media possess elasticity properties depending strongly on the spatial pressure
gradient u , x R, the corresponding Whitham type kernel looks like
x
∈
(x,s) := θ(x s)u (1.2)
s
K − −
forx,s R,naturallymodelingtherelaxingspatialmemoryeffects. Theresultingequation(1.1)
∈
with the kernel (1.2) becomes
du/dt = 2uu ∂−1u2 := K[u], (1.3)
x− x
which appears to possess very interesting mathematical properties. The latter will be the main
topic of the next sections below.
Owing to the results, obtained before in [25, 26], the dynamical system (1.3) appeared to be
a Lax type integrable bi-Hamiltonian flow, but with ill posed temporal evolution. As it was
demonstratedin[26], asuitablefinite-dimensionalreductionscheme, ifappliedtothecorrespon-
dinghierarchyofconservationlawsforconstructingexplicitsolutionstotheOstrovsky–Whitham
typenonlineardynamicalsystem (1.3)by meansof quadratures,meets sometechnical problems.
Some of these integrability aspects were before presented in [2], where a suitable well posed
regularization of the equation (1.3) in the form
u = 2uu v
t x − := K[u,v] (1.4)
v = 2uv
t x
(cid:27)
for treating this nonlocality problem was proposed.
Below the well posed integrability problem for the Ostrovsky–Whitham type nonlinear and
nonlocal dynamical system (1.4) will be reanalyzed in detail making use of this regulariza-
tion scheme. The corresponding implectic structures and Lax type representations are found
by means of the differential-geometric tools, devised and extended in [7, 8, 15, 24]. A natu-
ral Riemann type generalization of the dynamical system (1.4) is proposed, owing to a recent
observation by D. Holm and M. Pavlov:
DNu = 0, N Z , (1.5)
t ∈ +
which at N = 2 is exactly equivalent to the system (1.4). The integrability properties of
equation(1.5)atN = 3wereanalyzedindetail,theconservationlaws,correspondingcompatible
implectic structures and Lax type representation are constructed.
On a Nonlocal Ostrovsky–Whitham Type Dynamical System 3
It is worth to mention that the obtained in this work Lax type pair (3.6) for the regularized
dynamical system (1.4) was found first in work [4]. It coincides with those found later in [23],
making use of a very special bi-Lagrangian representation of the dynamical system (1.4). But
the existence of the singular co-implectic structure (3.14) in these references was not stated.
A detailed analysis of the relationships between solutions of dynamical systems (1.3) and (1.4),
based on a reciprocal transformation, suggested by M. Pavlov in [23], was presented recently
in [27]. Mention also work [14], where the geometric aspects of the equation like (2.1) were
studied.
Notealsoherethattheoryofintegrablehomogenoushydrodynamictypesystemswithdistinct
characteristic velocities was constructed by S.P. Tsarev. In this paper we consider the first
example in a literature of nonhomogeneous integrable hydrodynamic type systems with a sole
characteristic velocity. Such a theory does not exist at this moment.
2 A regularization scheme and the geometric
integrability problem
Define a smooth periodic function v C∞(R;R), such that
∈ 2π
v := ∂−1u2
x
for any x,t R, where the function u C∞(R;R) solves equation (1.3). Then it is easy to state
∈ ∈ 2π
that the following regularized nonlinear dynamical system
u = 2uu v
t x − := K[u,v] (2.1)
v = 2uv
t x
(cid:27)
of hydrodynamic type, which was introduced before in [6], studied in [2, 4, 12, 13, 19] and
analyzed as a Gurevich–Zybin system in [23], and is already well defined on the extended
2π-periodic functional space := C∞(R;R2) and equivalent on the functional submanifold
M 2π
:= (u,v) : v u2 = 0 to that given by expression (1.3), as it was mentioned
Mred { ∈ M x − x }
in [2] and discussed recently in [27]. The system (2.1) can be rewritten as the following set of
equations
u = 2uu v, v = 2uv ,
t x t x
−
u = w, v = u w,
x x x
w = v +2uw , (2.2)
t x x
which is equivalent to a set of differential two-forms
α := α(1) = du dx+2udu dt vdx dt, α(2) = dv dx+2udv dt,
{ } ∧ ∧ − ∧ ∧ ∧
(cid:8)α(3) = du dt wdx dt, α(4) = dv dt wdu dt,
∧ − ∧ ∧ − ∧
α(5) = dw dx+dv dt+2udw dt . (2.3)
∧ ∧ ∧
(cid:9)
This set of two-forms generates the closed ideal (α), since
I
d α(1) = α(2) dt, dα(2) = 2du α(4), dα(3) = α(5) dt,
− ∧ ∧ − ∧
dα(4) = dw α(3) wdt α(5), dα(5) = 2dw α(3) 2wdt α(5).
− ∧ − ∧ − ∧ − ∧
The set of differential forms (2.3), being integrable, defines the integral submanifold M¯ by
means of the condition (α) = 0. Making now use of the differential-geometric method devised
I
4 J. Golenia, M.V. Pavlov, Z. Popowicz and A.K. Prykarpatsky
in [11, 18, 24] and extending algorithmically the approach of [15], we will look for a reduced
upon the integral submanifold M¯ connection one-form Γ, belonging to some not yet determined
its holonomy Lie algebra . This 1-form can be represented as follows:
G
Γ = (u,v,w)dx+ (u,v,w)dt, (2.4)
A B
where the elements , satisfy determining equations
A B ∈ G
∂ ∂ ∂ ∂
Ω = Adu dx+ Adv dx+ Adw dx+ Bdu dt
∂u ∧ ∂v ∧ ∂w ∧ ∂u ∧
∂ ∂
+ B dv dt+ B dw dt+[ , ]dx dt
∂v ∧ ∂w ∧ A B ∧
g (du dx+2udu dt vdx dt)+g (dv dx+2udv dt)
1 2
⇒ ∧ ∧ − ∧ ∧ ∧
+g (du dt wdx dt)+g (dv dt wdu dt)
3 4
∧ − ∧ ∧ − ∧
+g (dw dx+2udw dt+dv dt) (α) (2.5)
5
∧ ∧ ∧ ∈ I ⊗G
for some -valued functions g ,...,g on M. From (2.5) one finds that
1 5
G ∈ G
∂ ∂ ∂
A = g , A =g , A = g ,
1 2 5
∂u ∂v ∂w
∂ ∂
B = 2ug +g wg , B = 2ug +g +g ,
1 3 4 2 4 5
∂u − ∂v
∂
B = 2ug , [ , ]= vg wg . (2.6)
5 1 3
∂w A B − −
Thereby, from the obtained set of relationships (2.6) one can find that
∂ ∂ ∂ ∂ ∂
= 2u + (u,v), g = C A, g = 2 + C +w C w A,
4 3
B A C ∂v − ∂w A ∂u ∂v − ∂w
∂ ∂ ∂ ∂
[ , ] = v A 2w w C w2 C +w2 A,
A C − ∂u − A− ∂u − ∂v ∂w
serving for final searching for connection (2.4).
3 The bi-Hamiltonian structure and Lax-type representation
Consider the following polynomial expansion of the element (u,v;w) with respect to the
A ∈ G
variable w:
= (u,v)+ (u,v)w+ (u,v)w2
0 1 2
A A A A
and substitute it into the last equation of (2.6). As a result we obtain:
∂ ∂ ∂C
0 1
[ ,C]= v A , [ ,C]= v A 2 ,
0 1 0
A − ∂u A − ∂u − A − ∂u
∂ ∂C
2
[ ,C]= v A , (3.1)
2 1
A − ∂u − ∂v −A
or
∂ ∂C
2
= [C, ] v A . (3.2)
1 2
A A − ∂u − ∂v
which can be substituted into the second equation of (3.1):
∂ ∂C ∂C ∂2 ∂2C ∂C
[[C, ],C] 2v A2,C ,C = v , v2 A2 v 2 .
A2 − ∂u − ∂v − ∂u A2 − ∂u2 − ∂u∂v − A0− ∂u
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
On a Nonlocal Ostrovsky–Whitham Type Dynamical System 5
Thus, recalling (3.1) and (3.2), we have that
∂ ∂C ∂C ∂2 ∂2C ∂C
2 = [C,[C, ]]+2v A2,C + ,C v , v2 A2 v ,
A0 A2 ∂u ∂v − ∂u A2 − ∂u2 − ∂u∂v − ∂u
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
∂ ∂ ∂C
0 2
[ ,C]= v A , = [C, ] v A . (3.3)
0 1 2
A − ∂u A A − ∂u − ∂v
Now we willassumethat theelement C := C is constant and theelements and arelinear
0 0 2
A A
with respect to variables u and v, that is
(0) (1) (2) (0) (1) (2)
= + u+ v, = + u+ v.
A0 A0 A0 A0 A2 A2 A2 A2
Whence and from (3.3) one gets:
(0) (0) (1) (2) (1)
2 = [C ,[C , ]], [ ,C ]= 0, [ ,C ]= ,
A0 0 0 A2 A0 0 A0 0 −A0
(1) (1) (2) (2) (1)
2 = [C ,[C , ]], 2 = [C ,[C , ]]+2[ ,C ]. (3.4)
A0 0 0 A2 A0 0 0 A2 A2 0
To solve the algebraic system (3.4) we need to calculate [24] the corresponding holonomy Lie
algebra of the connection (2.4). As a result of simple, but slightly cumbersome calculations,
we derive that elements (j), j = 0,...,2, and C belong to the Lie algebra sl(2;C), whose
A2 0
basis L , L and L can be taken to satisfy the following canonical commutation relations:
0 + −
[L ,L ] = L , [L ,L ] =2L .
0 ± ± + − 0
±
Thereby, making use of the standard determining expansions
(j) (j) (j)
= c L +c L , C = k L +k L , (3.5)
A2 ± ± 0 0 0 ± ± 0 0
± ±
X X
where j = 0,...,2, and substituting (3.5) into (3.4), we obtain some relationships on values
c(j),c(j) C, j = 0,...,2, and k ,k C. Resolving by means of simple but slightly cumber-
± 0 ∈ ± 0 ∈
some calculations these relationships, we find the searched for basic elements and of the
A B
connection Γ, depending on a spectral parameter λ C, thereby giving rise to the correspon-
∈
ding Lax type commutative spectral representation for dynamical system (2.1) in the following
(2 2)-matrix form:
×
df df
= ℓ[u,v;λ]f, = p(ℓ)f, p(ℓ) := 2uℓ[u,v;λ]+q, (3.6)
dx dt
λu v 0 0 2λu u 2v u
ℓ[u,v;λ] := − x − x , q := , p(ℓ) = − x − x ,
λ2 λu λ 0 λ+2λ2u 2λu u
x x
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
defining the generalized time-independent spectrum Spec(ℓ) C: λ Spec(ℓ), if the correspon-
⊂ ∈
ding solution f L (R;C2). It is worth to remark here that the Lax type representation (3.6),
∞
∈
found for the dynamical system (2.1), is not unique. Moreover, making use of other imbeddings
of the connection form (2.4) into a suitable holonomy Lie algebra , one can construct different
G
Lax type representations, which could appear to be more useful for finding exact solutions to
dynamical system (2.1) by means of, for instance, the inverse spectral transform method.
The standard Riccati equation, derived from (3.6), allows to obtain right away an infinite
hierarchy of local conservation laws:
2π 2π (u v v u )
γˆ := u2 v dx, γˆ := x xx− x xx dx, ..., (3.7)
−1 x− x 0 2v u2 v
Z0 p Z0 x x− x
p
6 J. Golenia, M.V. Pavlov, Z. Popowicz and A.K. Prykarpatsky
and so on. All of conservation laws (3.7) except γ , are singular at the Cauchy condition (2.2).
−1
This means that we need to construct other hierarchy of polynomial conservation laws regular
on the functional submanifold
:= (u,v) : u2 v = 0, x R/2πZ . (3.8)
Mred ∈ M x− x ∈
The latter exis(cid:8)ts owing to the results of [23, 24]. The(cid:9)simplest way to search for them consists
in determining the bi-Hamiltonian structure of flow (2.1). As it is easy to check, dynamical
system (2.1) is canonically Hamiltonian, that is
d
(u,v)⊺ := ϑˆgradHˆ = Kˆ[u,v],
ϑ
dt −
where the corresponding co-symplectic structure ϑˆ: T∗( ) T( ) is canonical, equals
M → M
0 1
ϑˆ= (3.9)
1 0
(cid:18) − (cid:19)
and satisfies the Noether equation
L ϑˆ= 0= dϑˆ/dt ϑˆKˆ′,∗ Kˆ′ϑˆ.
Kˆ − −
To prove this, it is enough to find by means of the small parameter method, devised before
in [24] a non-symmetric (ϕ′ = ϕ′,∗) solution ϕ T( ) to the following Lie–Lax equation:
6 ∈ M
dϕ/dt+Kˆ′,∗ϕ= gradL (3.10)
for some suitably chosen smooth functional L (M). As a result of easy calculations one
∈ D
obtains that
2π
ϕ = ( v,0)⊺, L = uvdx. (3.11)
− −
Z0
Making use of (3.11) and the classical Legendrian relationship for the suitable Hamiltonian
function
H := (ϕ,Kˆ) L, (3.12)
−
and the corresponding symplectic structure
0 1
ϑˆ−1 := ϕ′ ϕ′,∗ = − (3.13)
− 1 0
(cid:18) (cid:19)
one obtains the implectic structure (3.9) and the corresponding non-singular Hamilton function
2π
Hˆ := (v2/2+v u2)dx.
ϑ x
Z0
It is here worth to mention that the determining Lie–Lax equation (3.10) possesses still another
solution
u u2 1 2π
ϕ = x, x , L = uv dx,
x
2 −2v 4
(cid:18) x(cid:19) Z0
giving rise, owing to formulas (3.13) and (3.12) to the new co-implectic (singular “symplectic”)
structure
∂ ∂u v−1
ηˆ−1 := ϕ′ ϕ′,∗ = − x x (3.14)
− −uxvx−1∂ 21(u2xvx−2∂+∂u2xvx−2) !
On a Nonlocal Ostrovsky–Whitham Type Dynamical System 7
and the Hamiltonian functional
1 2π
Hˆ := (uv vu )dx.
η x x
2 −
Z0
The co-implectic structure (3.14) is, evidently, singular since ηˆ−1(u ,v )⊺ = 0. Remark also
x x
that, owing to the general symplectic theory results [1, 8, 11, 15, 16, 18, 24] for nonlinear
dynamical systems on smooth functional manifolds, operator (3.14) defines on the manifold
M
acloseddifferentialtwo-form. Therebyitisa priori co-symplectic, satisfyingon thestandard
M
Jacobi brackets condition. Moreover, the implectic structure ηˆ: T∗( ) T∗( ) satisfies the
M → M
determining Noether equation
L ηˆ= 0 = dηˆ/dt ηˆKˆ′,∗ Kˆ′ηˆ,
Kˆ − −
whose solutions can also be obtained by means of the small parameter method, devised before
in [16, 24]. As a result, the second implectic operator has the form
∂−1 2u ∂−1
ηˆ:= x , (3.15)
2∂−1u 2v ∂−1+2∂−1v
x x x
(cid:18) (cid:19)
giving rise to a new infinite hierarchy of polynomial conservation laws
1
γˆ := dλ (ϑˆ−1ηˆ)ngradHˆ [uλ ,u (3.16)
n ϑ
h } i
Z0
for all n Z .
+
∈
In particular, one can easily observe that there hold representations
d d
(u,v)⊺ = ηˆgradHˆ , (u,v)⊺ = ϑˆgradHˆ ,
η η
dt − dx −
where
1 2π
Hˆ := (uv vu )dx.
η x x
2 −
Z0
Thereby, one can formulate the following proposition.
Proposition 1. The Riemann type hydrodynamical system (2.1) is a Lax type integrable bi-
Hamiltonian flow on the functional manifold . The corresponding implectic pairs are compa-
M
tible and given by matrix operators (3.9) and (3.15), the Lax type representation is presented in
the differential matrix form (3.6).
Now, making use of (3.16), one can apply the standard reduction procedure upon the corre-
sponding finite dimensional functional subspaces 2n , n Z , and obtain a large set of
+
M ⊂ M ∈
exact solutions of special quasi-periodic and solitonic type to dynamical system (2.1) upon the
functional submanifold , if the Cauchy data are taken to satisfy constraint (3.8). Here we
red
M
need to mention that a general solution to the system (2.1), obtained in [23, 27], is presented in
an unwieldy involved form, almost completely not feasible for practical applications.
4 A Riemann type hydrodynamical generalization
It is here interesting to mention (owing to recent observations by D. Holm for N = 2 and for
arbitrary N Z by M. Pavlov) that the dynamical system (2.1) can be equivalently rewritten
+
∈
up to the time rescaling as
D2u= 0, D := ∂/∂t+u∂, (4.1)
t t
8 J. Golenia, M.V. Pavlov, Z. Popowicz and A.K. Prykarpatsky
under the flow velocity condition dx/dt := u, which is a partial case [5] of the generalized
Riemann type hydrodynamic system
DNu = 0 (4.2)
t
for any integer N Z . If N = 3, having defined the new variables v := D u, z := D v, one
+ t t
∈
easily obtains the new dynamical system
u = v uu
t x
−
v = z uv := K[u,v,z] (4.3)
t x
− 
z = uz
t x 
−
of hydrodynamical type, which proves also to possess infinite hierarchies of polynomial conser-
vation laws.
As we are interested first in the conservation laws for the system (4.3), the following propo-
sition holds.
Proposition 2. Let H(λ) := 2πh(x;λ)dx D( ) be an almost everywhere smooth func-
0 ∈ M
tional on the manifold , depending parametrically on λ C, and whose density satisfies the
M R ∈
differential condition
h = λ(uh) (4.4)
t x
for all t R and λ C on the solution set of equation (4.1). Then the following iterative
∈ ∈
differential relationship
(f/h) = λ(uf/h) (4.5)
t x
holds, if a smooth function f C∞(R;R) (parametrically depending on λ C) satisfies for all
∈ ∈
t R the linear equation
∈
f = 2λu f +λuf . (4.6)
t x x
Proof. We have from (4.4)–(4.6) that
(f/h) = f /h fh /h2 = f /h λfu /h λfuh /h2 = f /h+λfu(1/h) λu f/h
t t t t x x t x x
− − − −
= λ(uf) /h+λuf(1/h) = λ(uf/h) ,
x x x
proving the proposition. (cid:4)
The obvious generalization of the previous proposition is read as follows.
Proposition 3. If a smooth function h C∞(R;R) satisfies the relationships
∈
h = ku h+uh ,
t x x
where k R, then
∈
2π
H = h1/kdx
Z0
is a conservation law for the Riemann type hydrodynamical system (2.1).
On a Nonlocal Ostrovsky–Whitham Type Dynamical System 9
The following polynomial dispersionless functionals, constructed by means of Proposition 3,
are conserved with respect to the flow (4.3):
2π n+2
H(1) := dxzn vu v u z ,
n x − x − n+1
Z0 (cid:18) (cid:19)
2π
H(4) := dx 7v v2u+z 6zu+2v u2 3v2 4vuu ,
x x x
− − −
Z0
2π (cid:2) (cid:0) 2π (cid:1)(cid:3)
H(5) := dx z2u 2zvv , H(6) := dx z v3+3z2v u+z3 ,
x x z x
−
Z0 Z0
2π (cid:0) (cid:1) (cid:0) (cid:1)
H(7) := dx z v3+3z2vu 3z3 ,
x x
−
Z0
2π (cid:0) (cid:1)
H(8) := dxz 6z2u+3zv u2 3zv2 4zvu 2v v2u+2v3u ,
x x x x
− − −
Z0
2π (cid:0) (cid:1)
H(9) := dx 1001v v4u+ 1092z2u2+364zv u3
x x
−
Z0
1092z(cid:2)v2u 728zvu(cid:0)u2 364v v2u2+273v4 +728v3u u ,
x x x
− − −
2π 2π
H(2) := dxz vzn, H(3) := dxz v2 2zu n, (cid:1)(cid:3)
n x n x −
Z0 Z0
(cid:0) (cid:1)
where n Z . In particular, as n = 1,2,..., from (4.3) one obtains that
+
∈
2π 2π
(2) (2)
H := dxz v, H := dxz zv, ...,
0 x 1 x
Z0 Z0
2π 2π
H(3) := dxz v2 2uz , H(3) := dxz v4+4z2u2 4zv2u , ...,
1 x − 2 x −
Z0 Z0
(cid:0) (cid:1) (cid:0) (cid:1)
and so on. Similarly one can construct also infinite hierarchies of conservation laws for the
hydrodynamical system (4.3), which are both non-polynomial and dispersive:
2π
H(1/4) = dx 2u u z +u v2 +2u2z u v v +3v z 3v z 1/4,
1 − xx x x xx x x xx− x xx x xx x− x xx
Z0
2π (cid:0) (cid:1)
H(1/3) = dx( v z +v z )1/3,
2 − xx x x xx
Z0
2π
H(1/3) = dx(v u v u z )1/3,
3 xx x− x xx− xx
Z0
2π
H(1/2) = dx 2vu z +v2 +z( u v +3z ) 1/2,
1 − x x x − x x x
Z0
2π (cid:2) (cid:3)
H(1/2) = dx 8u3z 3u2v2 18u v z +6v3 +9z 1/2,
2 x x − x x− x x x x x
Z0
2π (cid:0) (cid:1)
H(1/5) = dx 2u u z +u v2 +6u2 z 6u u z
1 − xxx x x xxx x xx x − xx x xx
Z0
3u v(cid:0) v +2u2z u v v +3u v2 +3v z 3v z 1/5,
− xx xx x x xxx− x xxx x x xx xxx x− x xxx
2π
(1/3) (cid:1)
H = dx k u( v z +v z )+k v(u z u z )
3 1 − xx x x xx 1 xx x − x xx
Z0
+z(k u(cid:2) v k u v +k z +k z )+k ( 3u v z +v3 +3z2) 1/3, ...,
2 xx x− 2 x xx 1 xx 2 xx 3 − x x x x x
andsoon,wherek R,j = 1,2,3, arearbitraryrealnumbers. Theproblemwhich(cid:3)remainsstill
j
∈
open consists in proving, if any, that the generalized hydrodynamical system (4.3) is a Lax type
10 J. Golenia, M.V. Pavlov, Z. Popowicz and A.K. Prykarpatsky
integrable bi-Hamiltonian flow on the periodic functional manifold := C(∞)(R/2πZ;R3), as
M
it was proved above for the system (4.2) at N = 2. This problem will be analyzed in the Section
below.
5 The Hamiltonian analysis
Consider the system (4.3) as a nonlinear dynamical system
u = v uu
t x
−
v = z uv := K[u,v,z], (5.1)
t x
− 
z = uz
t x 
−
on the 2π-periodic smooth functional manifold and analyze it from the Hamiltonian point
M
of view. To tackle with this problem, it is enough to construct [7, 11, 24] exact non-symmetric
solutions to the Lie–Lax equation
dϕ/dt+K′,∗ϕ= gradL, ϕ′ = ϕ′,∗, (5.2)
6
for some functional L D( ), where ϕ T∗( ) is, in general, a quasi-local vector, such that
∈ M ∈ M
the system (4.3) allows the following Hamiltonian representation:
K[u,v,z] = ηgradH[u,v,z], H = (ϕ,K) L, η−1 = ϕ′ ϕ′,∗.
− − −
As a test solution to (5.2) one can take the one
1 2π
ϕ = u /2,0, z−1u2/2+z−1v ⊺, L = (2z +vu )dx,
x − x x x x 2 x
Z0
(cid:0) (cid:1)
which gives rise to the following co-implectic operator:
∂ 0 ∂u z−1
− x x
0 0 ∂z
η−1 := ϕ′ ϕ′,∗ =  x . (5.3)
− 1(u2z−2∂+∂u2z−2)
 u z−1∂ z ∂ 2 x x x x 
 − x x x −(vxzx−2∂+∂vxzx−2) 
 
This expression is not strictly invertible, as its kernel possesses the translation vector field
d/dx : T( ) with components (u ,v ,z )⊺ T( ), that is η−1(u ,v ,z )⊺ = 0.
x x x x x x
M→ M ∈ M
Nonetheless, upon formal inverting the operator expression (5.3), we obtain by means of
simple enough, but slightly cumbersome, direct calculations, that the Hamiltonian function
equals
2π
H := dx(u v z). (5.4)
x
−
Z0
and the implectic η-operator looks as
∂−1 u ∂−1 0
x
η := ∂−1u v ∂−1+∂−1v ∂−1z . (5.5)
 x x x x 
0 z ∂−1 0
x
 
 
The same way, representing the Hamiltonian function (5.4) in the scalar form
⊺
1 1
H = (ψ,(u ,v ,z )⊺), ψ = v,u+ , ∂−1√z , (5.6)
x x x
2 − ··· −√z
(cid:18) (cid:19)