Integrability Analysis of a Two-Component Burgers-Type Hierarchy
The Lax integrability of a two-component polynomial Burgers-type dynamical system is analyzed by using a differential-algebraic approach. Its linear adjoint matrix Lax representation is constructed. A related recursive operator and an infinite hierarchy of nonlinear Lax integrable dynamical systems...
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| Date: | 2015 |
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Institute of Mathematics, NAS of Ukraine
2015
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| author | Blackmore, D. Özçağ, E. Prykarpatsky, A. K. Soltanov, K. N. Блекмор, Д. Озчаг, Е. Прикарпатський, А. К. Солтанов, К. Н. |
| author_facet | Blackmore, D. Özçağ, E. Prykarpatsky, A. K. Soltanov, K. N. Блекмор, Д. Озчаг, Е. Прикарпатський, А. К. Солтанов, К. Н. |
| author_sort | Blackmore, D. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:47:54Z |
| description | The Lax integrability of a two-component polynomial Burgers-type dynamical system is analyzed by using a differential-algebraic approach. Its linear adjoint matrix Lax representation is constructed. A related recursive operator and an infinite hierarchy of nonlinear Lax integrable dynamical systems of the Burgers–Korteweg–de-Vries type are obtained by the gradient-holonomic technique. The corresponding Lax representations are presented. |
| first_indexed | 2026-03-24T02:16:13Z |
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UDC 517.9
D. Blackmore (NJIT, Newark, USA),
A. K. Prykarpatsky (AGH Univ. Sci. and Technology Krakow, Poland
and Ivan Franko State Ped. Univ. Drohobych, Ukraine),
E. Özçağ, K. Soltanov (Hacettepe Univ. Ankara, Turkey)
INTEGRABILITY ANALYSIS
OF A TWO-COMPONENT BURGERS TYPE HIERARCHY
АНАЛIЗ IНТЕГРОВНОСТI ДВОКОМПОНЕНТНОЇ IЄРАРХIЇ
ТИПУ БЮРГЕРСА
The Lax integrability of a two-component polynomial Burgers type dynamical system is analyzed using a differential-
algebraic approach and its linear adjoint matrix Lax representation is constructed. A related recursive operator and an
infinite hierarchy of nonlinear Lax integrable dynamical systems of the Burgers – Korteweg – de-Vries type are obtained by
the gradient-holonomic technique. The corresponding Lax representations are presented.
Лаксiвську iнтегровнiсть двокомпонентної полiномiальної динамiчної системи типу Бюргерса проаналiзовано за
допомогою диференцiально-алгебраїчного пiдходу. Побудовано її лiнiйне спряжене матричне лаксiвське зображен-
ня. Вiдповiдний рекурсивний оператор та нескiнченну iєрархiю нелiнiйних динамiчних систем типу Бюргерса –
Кортевега – де Фрiза, iнтегровних за Лаксом, отримано за допомогою градiєнтно-голономного методу. Наведено
вiдповiднi лаксiвськi зображення.
1. Introduction. Recently a great deal of research [16, 23, 24, 34, 37] has been devoted to
classifying polynomial integrable dynamical systems on smooth functional manifolds. For example,
in [37] one can find an extensive list of two-component polynomial dynamical systems of Burgers
and Korteweg – de Vries type, which either reduce by means of a (in general nonlocal) change of
variables to separable triangle Lax integrable forms or transform to completely linearizable flows.
Amongst these systems, the authors of [37] singled out the following two-component Burgers type
dynamical system ut
vt
= K[u, v] :=
uxx + 2uux + vx
uxv + uvx
(1.1)
on a smooth Schwartz type functional manifold M ⊂ C∞(R;R2), where (u, v)ᵀ ∈M, the subscripts
x and t denote, respectively, the partial derivatives with respect to the variables x ∈ R and t ∈ R+
and t is the evolution parameter. Hereafter we will a priori assume that the dynamical system (1.1)
possesses sufficiently smooth solutions in t ∈ R+, as it follows from the standard functional-analytic
compactness principle considerations in [21].
It is mentioned in [37, p. 7706] that the Burgers type dynamical system (1.1) was extensively
studied in [8, 22], where “. . . the symmetry integrability of (1.1) as well as the existence of a
recursive operator has already been demonstrated . . . ” for it. The dynamical system (1.1) appears to
have interesting applications, as its long-wave limit reduces to the well-known Leroux system [9, 38],
describing dynamical processes in two-component hydro- and lattice gas dynamics. As claimed in
c© D. BLACKMORE, A. K. PRYKARPATSKY, E. ÖZÇAĞ, K. SOLTANOV, 2015
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 2 147
148 D. BLACKMORE, A. K. PRYKARPATSKY, E. ÖZÇAĞ, K. SOLTANOV
[37, p. 7726], the integrability of (1.1) still remains unproven, and having found no relevant new
result in the literature, we have decided to resolve this open problem by means of a combination of
the gradient-holonomic [1, 31] method, linear adjoint mapping [30] techniques and recently devised
[2, 27, 28] differential-algebraic integrability tools. The following is our main result.
Theorem 1.1. The two-component polynomial Burgers dynamical system (1.1) possesses only
two local conserved quantities
∫
dxu and
∫
dxv and no other infinite affine ordered local conserved
quantities. Moreover, on the functional manifold M the system (1.1) is linearizable by means of a
Hopf – Cole type transformation and a dual adjoint mapping to the matrix Lax representation
Dx
f
f̂
=
(u+ λ−1v − λ)/2 0
1 (λ− λ−1v − u )/2]
f
f̂
, (1.2)
and
Dt
f
f̂
=
Dx(u+ λ)]/2+
+(u+ λ)(u+ λ−1v − λ)/2
0
(u+ λ)
−Dx(u+ λ)]/2+
+(u+ λ)(λ− u− λ−1v)/2
f
f̂
, (1.3)
compatible for all λ ∈ C\{0} with (f, f̂)ᵀ ∈ Λ0(K̄{u, v;D−1x σ|N})2, where K̄{u, v;D−1x σ|N} de-
notes a [35] finitely extended differential ring K̄{u, v}. The related with the Lax operator (1.2) infinite
hierarchy of generalized Burgers dynamical systems allows the following compact representation:
Dtn(u+ λ−1v − λ) = Dx
[
Dx αn(x;λ) + (u+ λ−1v − λ)αn(x;λ)
]
,
where the evolution parameters tn ∈ R+ and
αn(x;λ) := (λnαn(x;λ))+ (1.4)
for all natural n ∈ N is the corresponding nonnegative degree polynomial part generated by the
asymptotic local functional solution αn(x;λ) ∼
∑
j∈Z+
λ−jαj [u, v] as |λ| → ∞ to the differential
functional equation
D2
xαn(x;λ) +Dx((u+ λ−1v − λ)αn(x;λ)) = 0.
As a simple consequence of the Theorem 1.1 one finds that the Burgers type dynamical sys-
tem (1.1) does not admit a Hamiltonian formulation on the functional manifold M, and the corre-
sponding recursive operator
Φ :=
Dx +Dx uD
−1
x 1
Dx vD
−1
x 0
,
satisfying the determining commutator equation
DtΦ = [K ′,Φ],
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 2
INTEGRABILITY ANALYSIS OF A TWO-COMPONENT BURGERS TYPE HIERARCHY 149
for the two-component Burgers type dynamical system (1.1), introduced in [8, 22], cannot be factor-
ized by compatible Poissonian structures, as they generally do not exist.
The scalar Lax representation (1.2) can be reduced by means of the nonlocal change of variables
g̃ = f2 exp[λ2t−D−1x (λ−1v − λ)] ∈ Λ0(K̄{u, v;D−1x σ|N}) to the simpler form
Dtg̃ = Dxxg̃ + vg̃ = 0, Dxg̃ = ug̃, (1.5)
compatible for g̃ ∈ Λ0(K̄{u, v;D−1x σ|N}) and not depending on the parameter λ ∈ C\{0}. The
above representation (1.5) can be easily generalized to the following higher order evolution equation
case:
Dtg̃ = Dn
x g̃ + vg̃ = 0, Dxg̃ = ug̃, (1.6)
where n ∈ Z+. Making use of the above nonlocal change of variables ĝ = g̃ exp(λ−1D−1x v) ∈
∈ Λ0
(
K̄{u, v;D−1x σ|N}
)
, one can obtain a new infinite hierarchy of two-component Lax integrable
polynomial Burgers type dynamical systems, generalizing those discussed in [35, 36]. For instance,
when n = 3 we find the following dynamical Burgers type system of the third order:
ut = u3x + 3 (uux)x + 3u2ux + vx,
vt = (ur[u, v])x,
where r : J [u, v] → C∞(R2;R2) is a polynomial mapping on the jet-space J(R2;R2) of elements
(x, t;u, v,Dxu,Dxv,Dtu,Dtv,D
2
xu,D
2
xv, . . .) ∈ J(R2;R2), suitably determined by the relation-
ship (1.4) at n = 2. Its scalar Lax representation is easily obtained from (1.2), (1.3) or from (1.6).
The latter for n = 3 easily gives rise to the scalar Lax representation
Dxĝ = (u+ λ−1v)ĝ,
Dtĝ = Dxxxĝ + 3λ−1vDxxĝ + 3(vx/λ+ v2/λ2)Dxĝ+
+(vxx/λ+ 3vvx/λ
2 + v3/λ3 − (uη[u, v])/λ)ĝ = 0,
compatible for all λ ∈ C\{0} and ĝ ∈ Λ0
(
K̄{u, v;D−1x σ|N}
)2
, which can be suitably extended by
means of the related adjoint mapping to the matrix representation.
2. Differential-algebraic preliminaries. As our consideration of the integrability problem,
discussed above, will be based on some differential-algebraic techniques, we need to include some
additional differential-algebraic preliminaries [1, 11 – 15].
Take the ring K := R{{x, t}}, (x, t) ∈ R×(0, T ), of convergent germs of real-valued smooth
functions from C∞(R2;R) and construct the associated differential quotient ring K{u, v} :=
:= Quot
(
K[Θu,Θv]
)
with respect to the functional variables u, v ∈ K, where Θ denotes [10,
11, 14, 17, 32] the standard monoid of all commuting differentiations Dx and Dt, satisfying the
standard Leibnitz condition, and defined by the natural conditions
Dx(x) = 1 = Dt(t), Dt(x) = 0 = Dx(t).
The ideal I{u, v} ⊂ K{u, v} is called differential if the condition I{u, v} = ΘI{u, v} holds. In the
differential ring K{u, v}, interpreted as an invariant differential ideal in K, there are two naturally
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 2
150 D. BLACKMORE, A. K. PRYKARPATSKY, E. ÖZÇAĞ, K. SOLTANOV
defined differentiations
Dt, Dx : K{u, v} → K{u, v},
satisfying the commuting relationship
[Dt, Dx] = 0.
Consider the ring K{u, v}, u, v ∈ K, and the exterior differentiation d : K{u, v} → Λ1(K{u, v}),
d : Λp(K{u, v}) → Λp+1(K{u, v}) for p ∈ Z+, acting in the freely generated Grassmann algebras
Λ(K{u, v}) = ⊕p∈Z+Λp(K{u, v}) over the field C, where
Λ1(K{u, v}) := K{u, v}dx+K{u, v}dt+
∑
j,k∈Z+
K{u, v}du(j,k) +
∑
j,k∈Z+
K{u, v}dv(j,k),
u(j,k) := Dj
tD
k
xu, v(j,k) := Dj
tD
k
xv,
Λ2(K{u, v}) := K{u, v}dΛ1(K{u, v}), . . . , Λp+1(K{u, v}) := K{u, v}dΛp(K{u, v}).
The triple A : =(K{u, v},Λ(K{u, v}); d) will be called the Grassmann differential algebra with
generatrices u, v ∈ K. In the algebra A, generated by u, v ∈ K, one naturally defines the action of
differentiations Dt, Dx and ∂/∂u(j,k), ∂/∂v(j,k) : A → A, j, k ∈ Z+, as follows:
Dtu
(j,k) = u(j+1,k), Dxu
(j,k) = u(j,k+1),
Dt v
(j,k) = v(j+1,k), Dxv
(j,k) = v(j,k+1),
Dtdu
(j,k) = du(j+1,k), Dxdu
(j,k) = du(j,k+1),
Dtdv
(j,k) = dv(j+1,k), Dxdv
(j,k) = dv(j,k+1), (2.1)
dP [u, v] =
∑
j,k∈Z+
du(j,k) ∧ ∂P [u, v]/∂u(j,k) +
∑
j,k∈Z+
dv(j,k) ∧ ∂P [u, v]/∂v(j,k) =
=
∑
j,k∈Z+
(±)∂P [u, v]/∂u(j,k) ∧ du(j,k)+
+
∑
j,k∈Z+
(±)∂P [u, v]/∂v(j,k) ∧ dv(j,k) := 〈P ′[u, v],∧(du, dv)ᵀ〉R2 ,
where the sign ∧ denotes the standard [15] exterior multiplication in Λ(K{u, v}), and for any
P [u, v] ∈ Λ(K{u, v}) the mapping
P ′[u, v]∧ : Λ0(K{u, v})2 → Λ(K{u, v}), (2.2)
is linear. Moreover, the commutation relationships
Dxd = dDx, Dtd = dDt
hold in the Grassmann differential algebra A. The following remark [14] is also important.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 2
INTEGRABILITY ANALYSIS OF A TWO-COMPONENT BURGERS TYPE HIERARCHY 151
Remark 2.1. Any Lie derivative LV : K{u, v}→ K{u, v}, satisfying the condition LV : K⊂K,
can be uniquely extended to the differentiation LV : A → A, satisfying the commutation condition
LV d = d LV .
The variational derivative, or the functional gradient ∇P [u, v] ∈ Λ(K{u, v})2 with respect to
the variables u, v ∈ K, is defined for any P [u, v] ∈ Λ(K{u, v}) by means of the following expression:
gradP [u, v] = P ′,∗[u, v](1),
where a mapping P ′,∗[u, v] : Λ0(K{u, v}) → Λ0(K{u, v})2 is the formal adjoint mapping for that
of (2.2). This is intrinsically based on the following important lemma, stated for a special case in
[10 – 14, 26].
Lemma 2.1. Let the differentiations Dx and Dt : Λ(K{u, v}) → Λ(K{u, v}) satisfy the condi-
tions (2.1). Then the mapping
Ker grad/(Imd⊕ C) 'H1(A) :=
= Ker{d : Λ1(K{u, v})→ Λ2(K{u, v})}/dΛ0(K{u, v})
is a canonical isomorphism, where H1(A) is the corresponding cohomology class of the Grassmann
complex Λ(K{u, v}).
It is well known [32] that for the differential ring K{u, v} not all of the cohomology classes
Hj(A), j ∈ Z+, are trivial. Nonetheless, one can impose on the functions u, v ∈ K some additional
restrictions, which will give rise to the condition H1(A) = 0, or equivalently to the relationship
Ker∇ = ImDx ⊕ ImDt ⊕ C. In addition, the following simple relationship holds:
grad(ImDx ⊕ ImDt) = 0.
Using Lemma 2.1 one can define the equivalence class à := A/{ImDx ⊕ ImDt ⊕ R} :=
:= D(A; dxdt), whose elements will be called functionals; that is, any element γ ∈ D(A; dxdt)
can be represented as a suitably defined integral γ :=
∫ ∫
dxdtγ[u, v] ∈ D(A; dxdt) for some
γ[u, v] ∈ Λ(K{u, v}) with respect to the Lebesgue measure dxdt on R2.
Consider now our two-component dynamical system (1.1) as a polynomial differential constraint
Dt(u, v)ᵀ = K[u, v], (2.3)
imposed on the ring K{u, v}. The following definitions will be useful for our further analysis.
Definition 2.1. Let the reduced ring K̄{u, v} := K{u, v}|Dt(u,v)ᵀ=K[u,v]. Then the triple A :=
:= (K̄{u, v},Λ(K̄{u, v}), d) will be called a reduced Grassmann differential algebra over the reduced
ring K̄{u, v}.
Definition 2.2. Any pair of elements (γ[u, v], ρ[u, v])ᵀ ∈ Λ0(K̄{u, v})2, satisfying the relation-
ship
Dtγ[u, v] +Dxρ[u, v] = 0, (2.4)
is called a scalar conserved quantity with respect to the differentiations Dx and Dt.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 2
152 D. BLACKMORE, A. K. PRYKARPATSKY, E. ÖZÇAĞ, K. SOLTANOV
Using the differential-algebraic setting described above, one can naturally define the spaces of
functionals D(A; dx) := A/{DxA} and D(A; dt) = A/{DtA} on the the reduced Grassmann
differential algebra A. From the functional point of view these factor spaces D(A; dx) and D(A; dt)
can be understood more classically as the corresponding spaces of suitably defined integral expressions
subject to the measures dx and dt, respectively. Then the relationship (2.4) means that the functional
γ :=
∫
dxγ[u, v] ∈ D(A; dx) is a conserved quantity for the differentiation Dt, and the functional
Υ :=
∫
dtρ[u, v] ∈ D(A; dt) is a conserved quantity for the differentiation Dx.
Since the differential relationship (2.3) naturally defines [14, 15] on the reduced ring K{u, v} a
smooth vector field K : K{u, v} →T (K{u, v}), one can construct the corresponding Lie derivative
LK : A → A along this vector field and calculate the differential Lax [20] expression
∂ϕ[u, v]/∂t+ LKϕ[u, v] = 0 (2.5)
for the element ϕ[u, v] := gradγ[u, v] ∈ Λ0
(
K̄{u, v;D−1x σ|N}
)2
, where K̄{u, v;D−1x σ|N} denotes
some finitely extended differential ring K̄{u, v} and γ ∈ D(A; dx) is an arbitrary scalar conserved
quantity with respect to the differentiation Dt. The following classical Noether – Lax lemma [1, 20,
26, 31], inverse to the Lax relationship (2.5), holds.
Lemma 2.2 (E. Noether – P. Lax). Let a quantity ϕ[u, v] ∈ Λ0(K̄{u, v;D−1x σ|N})2 be such that
the following equation:
Dtϕ[u, v] +K ′,∗[u, v]ϕ[u, v] = 0, (2.6)
equivalent to (2.5), holds in the ring K̄{u, v;D−1x σ|N} satisfying the differential constraint (2.3).
Then, if the Volterra condition ϕ′,∗[u, v] = ϕ′[u, v] is satisfied in the ring K̄{u, v;D−1x σ|N}, the
constructed homology functional
γ :=
1∫
0
dλ
∫
dx
〈
ϕ[λu, λv], (u, v)ᵀ
〉
C2 ∈ D(A; dx)
is a scalar conserved quantity with respect to the differentiation Dt.
Assume now that the nonlinear two-component polynomial dynamical system (2.3) has a nontrivial
compatible differential Lax representation in the form
Dxf(x, t;λ) = l[u, v;λ]f(x, t;λ), Dtf(x, t;λ) = p[u, v;λ]f(x, t;λ) (2.7)
for some matrices l[u, v;λ], p[u, v;λ] ∈ End Λ0(K̄{u, v})q, f(x, t;λ) ∈ Λ0
(
K̄{u, v;D−1x σ|N}
)q
,
analytically depending on a parameter λ ∈ C, where q ∈ Z+\{0, 1} is finite. Then the following
important proposition, based on the gradient-holonomic approach, devised in [1, 31], holds.
Proposition 2.1. The Lax integrable dynamical system (2.3) possesses a set (either finite or
infinite) of naturally ordered functionally independent scalar conserved differential quantities
Dtσj [u, v] +Dxρj [u, v] = 0, (2.8)
where the pairs (σj [u, v], ρj [u, v])ᵀ ∈ Λ0(K̄{u, v})2, j ∈ Z+.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 2
INTEGRABILITY ANALYSIS OF A TWO-COMPONENT BURGERS TYPE HIERARCHY 153
Proof. Assume that the Lax integrable dynamical system (2.3) has a set (either finite or infinite)
of naturally ordered functionally independent scalar conserved differential quantities (2.8). Let
K̄{u, v;D−1x σ|N} denote the finitely extended differential ring K{u, v; {D−1x σj [u, v] : 0 ≤ j ≤ N}}
for an arbitrary integerN ∈ Z+ under the constraints (2.3). Then the Lax equation (2.6), if considered
on the invariant functional submanifold
MN :=
{
(u, v)ᵀ ∈M : grad
〈
c(N),
∫
dxΣ(N)
〉
CN+1
= 0,
c(N) ∈ CN+1\{0}, Σ(N) := (σ0, σ1, . . . , σN )ᵀ ∈ Λ0(K̄
{
u, v})N+1
}
,
allows [1, 31], as |λ| → ∞, an asymptotic solution ϕ(x;λ) ∈ Λ0
(
K̄{u, v;D−1x σ|N}
)2
in the form
ϕ(x;λ) ∼ ψ(x, t;λ) exp
{
ω(x, t;λ) +D−1x σ(x, t;λ)
}
, (2.9)
with a scalar analytical “dispersion” function ω(x, t; ·) : C→ C determined for all (x, t) ∈ R× [0, T ),
and the compatible local functional expansions
Λ0(K̄{u, v})2 3 σ(x, t;λ) ∼
∑
j∈Z+
σj [u, v]λ−j+|σ|,
Λ0(K̄{u, v})2 3 ψ(x, t;λ) ∼
∑
j∈Z+
ψj [u, v]λ−j+|ψ|
(2.10)
for some fixed integers |σ|, |ψ| ∈ Z+. Moreover, owing to the Lax equation (2.6), all of the scalar
functionals
γj :=
∫
dxσj [u, v] (2.11)
for j ∈ Z+ are conserved quantities with respect to the differentiation Dt. Now conversely, if the
Lax equation (2.6) possesses an asymptotic, as |λ| → ∞, solution in the form (2.9) ϕ[u, v;λ] ∈
∈ Λ0(K̄{u, v;D−1x σ|N})2 with compatible expansions (2.10), then all of the scalar functionals (2.11)
are, a priori, the conserved quantities with respect to the differentiation Dt. That is, there exist scalar
quantities ρj [u, v] ∈ Λ0(K̄{u, v}), j ∈ Z+, satisfying the relationships (2.8).
Proposition 2.1 is proved.
The analytical expressions for representation (2.9) and asymptotic expansions (2.10) for a Lax
integrable dynamical system (2.3) follow readily from the general theory of asymptotic solutions [6,
33] to linear differential equations, applied to a linear differential system (2.7) and from the following
important fact [1, 7, 25, 31]: The trace functional ∆[u, v;λ] := tr(F (x, t;λ)C(λ)F̄ (x, t;λ)) ∈
∈ Λ0
(
K̄{u, v;D−1x σ|N}
)
with any constant matrix C(λ) ∈ EndCq is for almost all λ ∈ C a
conserved quantity with respect to both differentiations Dt and Dx, where F (x, t;λ) and F̄ (x, t;λ),
(x, t) ∈ R× R+, are, respectively, the fundamental solutions to the linear Lax equation
Dxf(x, t;λ) = l[u, v;λ]f(x, t;λ) (2.12)
and its adjoint
Dxf̄(x, t;λ) = −f̄(x, t;λ)l[u, v;λ], (2.13)
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 2
154 D. BLACKMORE, A. K. PRYKARPATSKY, E. ÖZÇAĞ, K. SOLTANOV
where f(x, t;λ), f̄ᵀ(x, t;λ) ∈ Λ0
(
K̄{u, v;D−1x σ|N}
)q
. Consequently, the corresponding gradient
grad∆[u, v;λ] := ϕ[u, v;λ] ∈ Λ0
(
K̄{u, v;D−1x σ|N}
)2
,
owing to Lemma 2.2, a priori satisfies the Lax equation (2.6). Having assumed that |λ| → ∞, from
the asymptotic properties of linear equations (2.12) and (2.13) one obtains the result of Proposition 2.1.
3. Asymptotic, differential-algebraic and symplectic integrability analysis. We shall now
analyze the Lax integrability of the two-component polynomial Burgers type dynamical system (1.1).
In view of the approach described above, it is necessary to prove to the Lax equation (2.6) has
an asymptotic solution of the form (2.9) in Λ0
(
K̄{u, v;D−1x σ|N}
)2
. Concerning the dynamical
system (1.1), the following proposition holds.
Proposition 3.1. The Lax equation (2.6) with the differential matrix operator
K ′,∗[u, v] =
D2
x − 2uDx −vDx
−Dx −uDx
,
possesses the asymptotic solution
ϕ(x;λ) = (1, 1/λ)ᵀg(x;λ) exp[−λ2t− λx+D−1x (u+ λ−1v)], (3.1)
as |λ| → ∞, where the scalar invertible local functional element
g(x;λ) := exp
−u+
∑
j∈Z+\{0,1}
D−1x σj [u, v]/λj
∈ Λ0(K̄{u, v}). (3.2)
The solution (3.1) corresponds to the local conserved quantity ∆(λ) :=
∫
dx(u+λ−1v) ∈ D(A; dx)
in the extended ring K̄{u, v;D−1x σ|N}2 :
grad∆(λ)[u, v] = ϕ(x;λ) ∈ Λ0
(
K̄{u, v;D−1x σ|N}
)2
.
Proof. Assume that the Lax equation (2.6) possesses the asymptotic solution (2.9) as |λ| → ∞,
where ω(x, t;λ) = −λx− λ2t,
Λ0
(
K̄{u, v;D−1x σ|N}
)2 3 ϕ(x;λ) = ψ(x, t;λ) exp
{
−λ2t− λx+D−1x σ(x, t;λ)
}
(3.3)
and
Λ0(K̄{u, v})2 3 ψ(x, t;λ) = (1, a(x, t;λ))ᵀ
which reduces to an equivalent system of the differential-functional relationships
D−1x σt − λ2 + σx + (−λ+ σ)2 − (2u+ va)(−λ+ σ)− vax = 0,
at + a(−λ2 +D−1x σt)− uax − (au+ 1)(−λ+ σ) = 0.
The coefficients of the corresponding asymptotic expansions
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INTEGRABILITY ANALYSIS OF A TWO-COMPONENT BURGERS TYPE HIERARCHY 155
cΛ0(K̄{u, v}) 3 a(x, t;λ) ∼
∑
j∈Z+
aj [u, v]λ−j ,
Λ0(K̄{u, v}) 3 σ(x, t;λ) ∼
∑
j∈Z+
σj [u, v]λ−j ,
should satisfy two infinite hierarchies of recurrent relationships
D−1x σj−1,t + σj−1,x + 2σj +
∑
k∈Z+
σj−1−kσk − 2uδj−1,−1 − 2uσj−1−
−aj v − v
∑
k∈Z+
σj−1−kak − vaj−1,x = 0,
(3.4)
aj−2,t − aj +
∑
k∈Z+
aj−2−kD
−1
x σk,t − uaj−2,x − δj−2,−1−
−σj−2 − uaj−1 − u
∑
k∈Z+
σj−2−kak = 0,
compatible for all j ∈ Z+. It is easy to calculate from (3.4) the corresponding coefficients
σ0 = u, σ1 = ux + v, σ2 = vx + uxx + uux,
σ3 = Dx(u3/3 + uv + uxx + 2uux + vx), . . . , σj = Dx(. . .), . . . ,
a0 = 0, a1 = 1, a2 = 0, a3 = 0, . . . , aj = 0, . . .
and thereby conclude that only two functionals
γ0 :=
∫
dxσ0[u, v] =
∫
dxu, γ1 :=
∫
dxσ1[u, v] =
∫
dxv,
are nontrivial conservation laws with respect to the differentiation Dt, since all other functionals
γj :=
∫
dxσj [u, v] =
∫
dxDx(. . .) = 0
are trivial in the ring K̄{u, v}. This also means that the gradient ϕ(x;λ) := grad
∫
dx(u+ λ−1v) =
= (1, 1/λ)ᵀ ∈ Λ0
(
K̄{u, v}
)2
satisfies the Lax equation (2.6) in the ring K̄{u, v} and thus it should
coincide with the expression (3.3). As a result it is easy to show that
(1, 1/λ)ᵀ = (1, 1/λ)ᵀg(x;λ) exp
[
−λ2t− λx+D−1x (u+ λ−1v)
]
,
where the scalar invertible element
g(x;λ) := exp
−u+
∑
j∈Z+\{0,1}
D−1x σj [u, v]/λj
∈ Λ0(K̄{u, v}),
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156 D. BLACKMORE, A. K. PRYKARPATSKY, E. ÖZÇAĞ, K. SOLTANOV
giving rise to the expressions (3.1) and (3.2).
Proposition 3.1 is proved.
As a consequence of Proposition 3.1, we can formulate the following result.
Theorem 3.1. The two-component polynomial Burgers type dynamical system (1.1) possesses
only two local conserved quantities
∫
dxu and
∫
dxv and no other infinite affinely ordered conserved
quantities (either local or nonlocal). Moreover, on the functional manifold M the Burgers type
dynamical system (1.1) is linearizable by means of a Hopf – Cole type transformation and the related
linear adjoint mapping to the matrix Lax representation
Dx
f
f̂
=
(u+ λ−1v − λ)/2 0
1 (λ− λ−1v − u )/2]
f
f̂
, (3.5)
and
f
f̂
=
Dx(u+ λ)]/2+
+(u+ λ)(u+ λ−1v − λ)/2
0
(u+ λ)
−Dx(u+ λ)]/2+
+(u+ λ)(λ− u− λ−1v)/2
f
f̂
,
compatible for all λ ∈ C\{0}, where vector function (f, f̂)ᵀ ∈ Λ0
(
K̄{u, v;D−1x σ|N}
)2
, N = 2.
Proof. It follows from Proposition 3.1 and recent results of [35] that one can apply to the two-
component polynomial Burgers type dynamical system (1.1) the following generalized Hopf – Cole
type linearizing transformation:
u := Dx ln g̃(x;λ), (3.6)
where
g̃(x;λ) := g(x;λ)−1 exp
(
λ2t+ λx − λ−1D−1x v
)
∈ Λ0
(
K̄{u, v;D−1x σ|N}
)
and, by construction, it should be set N = 2. Substituting (3.6) into (1.1), we obtain the system of
linear equations
Dtg̃ = D2
xg̃ + vg̃, Dxg̃ = ug̃, (3.7)
which clearly reduces to the following system of differential relationships:
Dtf̃ = (ux + u2 + v)f̃/2 = 0, Dxf̃ = (u/2)f̃ , (3.8)
upon making the change of variables f̃ := g̃1/2 ∈ Λ0
(
K̄{u, v;D−1x σ|N}
)
. The system (3.8) can be
specified further by making use of the substitution f̃ := f exp
(
λx − λ−1D−1x v
)
, which giving rise
to the scalar operator Lax representation
Dtf = Dx(u+ λ)f/2 + (u+ λ)Dxf, Dxf = (u+ λ−1v − λ)f/2, (3.9)
compatible for all λ ∈ C\{0}.
Now we proceed to the construction of a suitably linearly extended adjoint differential relationships
[30] for the system of equations (3.9). Using the standard method, it is easy to show that the following
linearly adjoint relationship, compatible with the second equation of (3.9),
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INTEGRABILITY ANALYSIS OF A TWO-COMPONENT BURGERS TYPE HIERARCHY 157
Dxf̂ = Dx
(
f̂f
f
)
= −f−1f̂Dxf + f−1Dx(f̂f) =
= −
[
(λ− λ−1v − u )/2
]
f̂ + f−1Dx(f̂ f) = −
[
(λ− λ−1v − u )/2
]
f̂ + χ[u, v;λ]f, (3.10)
holds, where Dx(f̂f) := χ[u, v;λ]f2 for some arbitrarily chosen element χ[u, v;λ] ∈ K̄{u, v}. The
compatibility of (3.10) with the first equation of (3.9) and its suitable extension gives rise to the
condition χ[u, v;λ] = 1. Thus, we have shown that the linearly adjoint relationship compatible with
the second equation of (3.9) is
Dxf̂ = −
(
λ− λ−1v − u
)
f̂/2 + f. (3.11)
The adjoint linear relationship compatible with the first equation of (3.9) is easily seen to be
Dtf̂ = −Dx(u+ λ)f̂/2 + (u+ λ)Dxf̂ , (3.12)
and it is compatible with (3.11) for all λ ∈ C\{0}. It is now useful to rewrite equations (3.9), (3.11)
and (3.12) as the following two equivalent matrix systems:
Dx
f
f̂
=
(u+ λ−1v − λ)/2 0
1 (λ− λ−1v − u )/2]
f
f̂
, (3.13)
and
Dt
f
f̂
=
Dx(u+ λ)]/2+
+(u+ λ)(u+ λ−1v − λ)/2
0
(u+ λ)
−Dx(u+ λ)]/2+
+(u+ λ)(λ− u− λ−1v)/2
f
f̂
,
(3.14)
where (f, f̂)ᵀ ∈ Λ0
(
K̄{u, v;D−1x σ|N}
)2
, N = 2. It follows from systems (3.13) and (3.14) that
Dx(f̂ f) = f2, Dt(f̂f) = (u+ λ)f2. (3.15)
Since the mutual compatibility condition of relationships (3.15) reduces to the expression
Dtf = [Dx(u+ λ)/2]f + (u+ λ)Dxf,
exactly coinciding with the first equation of the system (3.14), we now can interpret both sys-
tems (3.13) and (3.14) as the corresponding matrix Lax representation for the Burgers system (1.1).
Theorem 3.1 is proved.
The scalar representation (3.7) can be easily generalized to the following higher order evolution
equation:
Dtg̃ = Dn
x g̃ + vg̃ = 0, Dxg̃ = ug̃, (3.16)
where n ∈ N\{1, 2}, g̃ ∈ Λ0
(
K̄{u, v;D−1x σ|N}
)
and no a priori constraint is imposed on the function
v ∈ K̄{u, v}, except the functional
∫
dxv ∈ D(A; dx) has to be a conserved quantity with respect to
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158 D. BLACKMORE, A. K. PRYKARPATSKY, E. ÖZÇAĞ, K. SOLTANOV
the differentiationDt. Upon applying to (3.16) the nonlocal change of variables u := 2Dx ln f̃ [u, v;λ]
for f̃ ∈ Λ0
(
K̄{u, v;D−1x σ|N}
)
, N = 2, one can obtain a new infinite hierarchy of two-component
integrable polynomial Burgers type dynamical systems, generalizing the systems studied in [35, 36].
For instance, when n = 3, we find the following dynamical Burgers – Korteweg – de Vries type
dynamical system of the third order:
ut = u3x + 3 (uux)x + 3u2ux + vx,
vt = (u r[u, v])x,
(3.17)
where r[u, v] ∈ K̄{u, v} is for the present an arbitrary element. To choose those for which the
dynamical systems of type (3.17) will possess suitably extended matrix Lax representations, it is
natural to take the first pair of Lax type equations (3.5)
Dxf = (u+ λ−1v − λ)f/2,
Dxf̂ = −(u+ λ−1v − λ)f̂/2 + f
for (f, f̂)ᵀ ∈ Λ0(K̄{u, v;D−1x σ|N})2, N = 2, and to supplement it by means of the following
systems of evolution equations, naturally generalizing that of (3.14) with respect to the temporal
parameters tn ∈ R:
Dtnf = Dxαn(x;λ)f/2 + αn(x;λ)Dxf,
Dtn f̂ = −Dxαn(x;λ)f̂/2 + αn(x;λ)Dxf̂ ,
which are, by construction, compatible for all λ ∈ C\{0} for a polynomial in λ, αn(x;λ) ∈ K̄{u, v},
n ∈ N, satisfying the standard determining relationship
Dtn(u+ λ−1v − λ) = Dx
[
Dxαn(x;λ) + (u+ λ−1v − λ)αn(x;λ)
]
. (3.18)
It is also easy to check that for n = 1 the choice
α1(x;λ) = u+ λ
yields the Burgers dynamical system (1.1).
The general algebraic structure of the whole infinite hierarchy of resulting dynamical sys-
tems (3.18) can be easily extracted from the matrix spectral Lax pair (3.13)
Dx
f
f̂
=
(u+ λ−1v − λ)/2 0
1 (λ− u− λ−1v)/2
f
f̂
:= l[u, v;λ]
f
f̂
, (3.19)
which allows, by means of the gradient-holonomic scheme [1, 31], to obtain the commutator equation
DxS = [l, S], S =
S11 S12
S21 S22
,
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INTEGRABILITY ANALYSIS OF A TWO-COMPONENT BURGERS TYPE HIERARCHY 159
for the related “monodromy” matrix S := S(x;λ) ∈ sl(2;C). This leads to the resulting [25, 31]
canonical differential relationships for the gradient
(ϕ1, ϕ2)
ᵀ := ϕ := grad(trS) ∈ Λ0
(
K̄{u, v;D−1x σ|N}
)2
of the dynamical Dx and Dt-invariant trace functional tr S(x;λ) ∈ D(A; dx) :
−Dxϕ1 +D−1x uDxϕ1 +D−1x vDxϕ2 = λϕ1,
ϕ1 = λϕ2,
(3.20)
where the component ϕ1 ∈ K̄{u, v} has the following differential-algebraic representation:
ϕ1 = (u+ λ−1v − λ)S21 +DxS21 (3.21)
for some polynomial expression S21 := S21(x;λ) ∈ K̄{u, v}. The differential expressions (3.20) can
be rewritten in the useful matrix form:
Λϕ = λϕ, Λ :=
−Dx +D−1x uDx D−1x vDx
1 0
, (3.22)
where the recursive operator Λ : T ∗(K̄{u, v}) → T (K̄{u, v}) satisfies the determining operator
equation
DtΛ = [Λ,K ′,∗],
which follows directly from the Noether – Lax condition (2.6) and the adjoint linear spectral relation-
ship (3.22).
Recall now that our Burgers type dynamical system (1.1) possesses only two conservations laws:
γ0 =
∫
dxu and γ1 =
∫
dxv ∈ D(A; dx). This means that the expression (3.21) exactly equals
ϕ1 = gradu γ0[u, v] = 1, or equivalently the condition
Dx
[
DxS21(x;λ) + (u+ λ−1v − λ)S21(x;λ)
]
= 0 (3.23)
should be satisfied for some element S21(x;λ) ∈ Λ0(K̄{u, v}) and all λ ∈ C\{0}. The following
proposition characterizes asymptotic solutions to (3.23) as |λ| → ∞ and their relationships to the
generalized dynamical systems (3.18).
Proposition 3.2. The nonnegative degree polynomial part of the asymptotic solution S21(x;λ) ∼
∼
∑
j∈Z+
λ−jS
(j)
21 [u, v;λ] (as |λ| → ∞) to the differential relationship (3.23) makes it possible
to represent the generating elements αn(x;λ) ∈ Λ0(K̄{u, v}) of the generalized dynamical sys-
tems (3.18) as
αn(x;λ) = (λnS21(x;λ))+
for any n ∈ Z+.
Proof. Taking into account that the whole hierarchy of the generalized Burgers dynamical
systems (3.18) can be represented in the recursive form
Dtn(u, v)ᵀ = Φn(Dxu,Dxv)ᵀ, Φ := Λ∗ =
Dx +Dx uD
−1
x 1
Dx vD
−1
x 0
, (3.24)
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160 D. BLACKMORE, A. K. PRYKARPATSKY, E. ÖZÇAĞ, K. SOLTANOV
we can rewrite it as
Dtn(u+ λ−1v − λ) = Dx
[
Dxαn(x;λ) + (u+ λ−1v − λ)αn(x;λ)
]
.
Here
αn(x;λ) := (λnα(x;λ))+
is the corresponding nonnegative degree polynomial part generated by the asymptotic solution
α(x;λ) ∼
∑
j∈Z+
λ−jαj [u, v;λ] as |λ| → ∞ to the differential functional equation
D2
xα(x;λ) +Dx((u+ λ−1v − λ)α(x;λ)) = 0, (3.25)
which is equivalent to the dual to (3.22) symmetry relationship
Φ(Dxα,Dxβ)ᵀ = λ(Dxα,Dxβ)ᵀ
with the generalized symmetry of the flow (1.1)
(Dxα,Dxβ)ᵀ :=
∑
j∈Z+
λ−jΦj(Dxu,Dxv)ᵀ.
The observation that the differential functional equation (3.25) coincides with (3.23) proves the
proposition.
The linear spectral problem (3.19) can be a priori generalized to the form
Dx
f
f̂
=
(u+ λ−1v + λ) h+ µux
1 (−λ− u− λ−1v)
f
f̂
,
which at the constraint v = 0 coincides with that for the Boussinesq – Burgers (at µ = 0)
ut = −2uux + hx/2,
ht = −2(uh)x + uxxx/2,
(3.26)
and with the Broer – Kaup – Kupershmidt (at µ = 1)
ut = −uux + uxx − hx,
ht = −(uh)x − hxx
(3.27)
hydrodynamic systems, whose integrability and soliton-like solutions were studied in [4, 18, 19, 31].
The related multicomponent “dark”-type extensions of (3.26) and (3.27) were recently constructed
in [5].
Theorem 1.1 also implies that the Burgers type dynamical system (1.1) does not admit a Hamil-
tonian formulation on the functional manifold M. This means that the recursive operator (3.22)
constructed above and earlier in [8, 22] for the two-component Burgers type dynamical system (1.1)
is not factorizable by means of suitably defined compatible Poissonian structures, as they on the
whole, do not exist. This is strongly related to the fact that the dynamical system (1.1) does not
possess an infinite hierarchy of local conservation laws, whose existence yields such a factorization.
Nonetheless, similar to the situation in [35], if one shows that the Burgers type dynamical system (1.1)
does possess another infinite hierarchy of nonlocal conservation laws, then to some degree the sym-
metry recursive operator (3.24) will already be factorized by means of the constructed Poissonian
structures. But to our knowledge, this remains an open problem.
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INTEGRABILITY ANALYSIS OF A TWO-COMPONENT BURGERS TYPE HIERARCHY 161
4. Conclusion. Employing mainly the differential-algebraic approach [1, 27, 28, 31, 35] for
testing the Lax integrability of nonlinear dynamical systems on functional manifolds, we proved
that the two-component polynomial Burgers type dynamical system (1.1) has an adjoint matrix
Lax representation and a corresponding recursive operator, which does not allow a bi-Poissonian
factorization and so only makes it possible to construct two local conserved quantities. The problem of
constructing a generalized bi-Poissonian factorization of a suitably powered recursive operator, similar
to that in [35], is left for future analysis. Thus, the differential-algebraic approach, together with
considerations based on the symplectic geometry, can serve as a simple, effective tool for analyzing the
Lax integrability of a wide class of polynomial nonlinear dynamical systems on functional manifolds.
Moreover, as was recently demonstrated in [29], this approach also appears to be useful in the case
of nonlocal polynomial dynamical systems.
Acknowledgements. D. Blackmore acknowledges the support of the National Science Founda-
tion (Grant CMMI-1029809), A. Prykarpatsky cordially thanks Prof. J. Cieślińskiemu (Białystok Uni-
versity, Poland), Prof. I. Mykytyuk (Pedagogical University of Krakow, Poland) and Prof. A. Augusty-
nowicz (Gdansk University, Poland) for useful discussions of the results obtained. A. Prykarpatsky,
E. Özçağ and K. Soltanov gratefully acknowledge partial support of the research in this paper from
the Turkey-Ukrainian: TUBITAK-NASU Grant 110T558.
1. Blackmore D., Prykarpatsky A. K., Samoylenko V. H. Nonlinear dynamical systems of mathematical physics. – New
York: World Sci. Publ., 2011.
2. Blackmore D., Prykarpatsky Y. A., Bogolubov N., Prykarpatski A. Integrability of and differential-algebraic structures
for spatially 1D hydrodynamical systems of Riemann type // Chaos, Solitons and Fractals. – 2014. – 59. – P. 59 – 81.
3. Błaszak M. Bi-Hamiltonian dynamical systems. – New York: Springer, 1998.
4. Bogolubov N. N. (Jr. )., Prykarpatsky A. K. Complete integrability of the nonlinear Ito and Benney – Kaup systems:
gradient algorithm and Lax representation // Theor. and Math. Phys. – 1986. – 67, № 3. – P. 586 – 596.
5. Chen Tao, Zhu Li-Li, Zhang Lei. The generalized Broer – Kaup – Kupershmidt system and its Hamiltonian extension
// Appl. Math. Sci. – 2011. – 5, № 76. – P. 3767 – 3780.
6. Coddington E. A., Levinson N. Theory of differential equations. – New York: McGraw-Hill, 1955.
7. Faddeev L. D., Takhtadjan L. A Hamiltonian methods in the theory of solitons. – New York; Berlin: Springer, 2000.
8. Foursov M. V. On integrable coupled Burgers-type equation // Phys. Lett. A. – 2000. – 272. – P. 57 – 64.
9. Fritz AJ., Troth B. Derivation of the Leroux system as the hydrodynamic limit of a two-component lattice gas //
Communs Math. Phys. – 2004. – 249, Issue 1. – P. 1 – 27.
10. Gelfand I. M., Dickey L. A. Integrable nonlinear equations and Liouville theorem // Funct. Anal. and Appl. – 1979. –
13. – P. 8 – 20.
11. Gelfand I. M., Dickey L. A. The calculus of jets and nonlinear Hamiltonian systems // Funct. Anal. and Appl. –
1978. – 12, Issue 2. – P. 81 – 94.
12. Gelfand I. M., Dickey L. A. A Lie algebra structure in a formal variational calculation // Funct. Anal. and Appl. –
1976. – 10, Issue 1. – P. 16 – 22.
13. Gelfand I. M., Dickey L. A. The resolvent and Hamiltonian systems // Funct. Anal. and Appl. – 1977. – 11, Issue 2. –
P. 93 – 105.
14. Gelfand I. M., Manin Yu. I., Shubin M. A. Poisson brackets and the kernel of the variational derivative in the formal
calculus of variations // Funct. Anal. and Appl. – 1976. – 10, Issue 4. – P. 274 – 278.
15. Godbillon C. Geometri differentielle et mecanique analytique. – Paris: Hermann, 1969.
16. Ibragimov N. H., Shabat A. B. Infinite Lie – Backlund algebras // Funct. Anal. and Appl. – 1980. – 14. – P. 313 – 315.
17. Kaplanski I. Introduction to differential algebra. – New York, 1957.
18. Kupershmidt B. A. Dark equations // J. Nonlinear Math. Phys. – 2001. – 8, № 3. – P. 363 – 445.
19. Kupershmidt B. A. Mathematics of dispersive water waves // Communs Math. Phys. – 1985. – 99. – P. 51 – 73.
20. Lax Peter D. Almost periodic solutions of the KdV equation // Source: SIAM Rev. – 1976. – 18, № 3. – P. 351 – 375.
21. Lions J. L. Quelgues methodes de resolution des problemes aux limites non lineaires. – Paris: Dunod, 1969.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 2
162 D. BLACKMORE, A. K. PRYKARPATSKY, E. ÖZÇAĞ, K. SOLTANOV
22. Ma W. X. A hierarchy of of coupled Burgers systems possessing a hereditary structure // J. Phys. A: Math. and Gen. –
1993. – 26. – P. L1169 – L1174.
23. Mikhailov A. V., Shabat A. B., Yamilov R. I. The symmetry approach to the classification of nonlinear equations.
Complete list of integrable systems // Russ. Math. Surv. – 1987. – 42, № 4. – P. 1 – 63.
24. Mikhailov A. V., Shabat A. B., Yamilov R. I. Extension of the modul of invertible transformations. Classification of
integrable systems // Communs Math. Phys. – 1988. – 115. – P. 1 – 19.
25. Theory of solitons: the inverse scattering method / Ed. S. P. Novikov. – Springer, 1984.
26. Olver P. Applications of Lie groups to differential equations. – Second ed. – New York: Springer-Verlag, 1993.
27. Prykarpatsky A. K., Artemovych O. D., Popowicz Z., Pavlov M. V. Differential-algebraic integrability analysis of the
generalized Riemann type and Korteweg – de Vries hydrodynamical equations // J. Phys. A: Math. Theor. – 2010. –
43. – P. 295 – 205.
28. Prykarpatsky Y. A., Artemovych O. D., Pavlov M., Prykarpatsky A. K. The differential-algebraic and bi-Hamiltonian
integrability analysis of the Riemann type hierarchy revisited // J. Math. Phys. – 2012. – 53; doi 10.1063/1.4761821.
29. Prykarpatsky Y. A., Artemovych O. D., Pavlov M., Prykarpatsky A. K. The differential-algebraic integrability analysis
of symplectic and Lax type structures related with the hydrodynamic Riemann type systems // Repts Math. Phys. –
2013. – 71, № 3. – P. 305 – 351.
30. Prykarpatsky Y. A. Finite dimensional local and nonlocal reductions of one type of hydrodynamic systems // Repts
Math. Phys. – 2002. – 50, № 3. – P. 349 – 360.
31. Prykarpatsky A., Mykytyuk I. Algebraic integrability of nonlinear dynamical systems on manifolds: classical and
quantum aspects. – Netherlands: Kluwer Acad. Publ., 1998.
32. Ritt J. F. Differential algebra // AMS-Colloq. Publ. – New York: Dover Publ., 1966. – 33.
33. Shubin M. Lectures on mathematical physics. – Moscow: Moscow State Univ., 2001.
34. Sokolov V. V., Wolf T. Classification of integrable polynomial vector evolution equations // J. Phys. A: Math. and
Gen. – 2001. – 34. – P. 11139 – 11148.
35. Prykarpatski A., Soltanov K., Özçağ E. Differential-algebraic approach to constructing representations of commuti-
ng differentiations in functional spaces and its application to nonlinear integrable dynamical systems // Commun.
Nonlinear Sci. Numer Simulat. – 2014. – 19. – P. 1644 – 1649.
36. Tasso H. Hamiltonian formulation of odd Burgers hierarchy // J. Phys. A: Math. and Gen. – 1996. – 29. – P. 7779 –
7784.
37. Tsuchida T., Wolf T. Classification of polynomial integrable systems of mixed scalar and vector evolution equations.
I // J. Phys. A: Math. and Gen. – 2005. – 38. – P. 7691 – 7733.
38. Whitham G. B. Linear and nonlinear waves. – John Wiley & Sons Inc., 1974.
39. Wilson G. On the quasi-Hamiltonian formalism of the KdV equation // Phys. Lett. – 1988. – 132, № 8/9. – P. 445 – 450.
Received 06.01.14
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 2
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| id | umjimathkievua-article-1970 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:16:13Z |
| publishDate | 2015 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/a3/eef2a8009aca937e344cd72c3cc114a3.pdf |
| spelling | umjimathkievua-article-19702019-12-05T09:47:54Z Integrability Analysis of a Two-Component Burgers-Type Hierarchy Аналіз інтегровності двокомпонентної ієрархії типу Бюргерса Blackmore, D. Özçağ, E. Prykarpatsky, A. K. Soltanov, K. N. Блекмор, Д. Озчаг, Е. Прикарпатський, А. К. Солтанов, К. Н. The Lax integrability of a two-component polynomial Burgers-type dynamical system is analyzed by using a differential-algebraic approach. Its linear adjoint matrix Lax representation is constructed. A related recursive operator and an infinite hierarchy of nonlinear Lax integrable dynamical systems of the Burgers–Korteweg–de-Vries type are obtained by the gradient-holonomic technique. The corresponding Lax representations are presented. Лаксівську інтегровність двокомпонентної полiномiальної динамiчної системи типу Бюргерса проаналiзовано за допомогою диференціально-алгебраїчного підходу. Побудовано її лінійнє спряжене матричне лаксівське зображення. Відповідний рекурсивний оператор та нескінченну ієрархію нелінійних динамічних систем типу Бюргерса - Кортевега - де Фріза, інтегровних за Лаксом, отримано за допомогою градієнтно-голономного методу. Наведено відповідні лаксівські зображення. Institute of Mathematics, NAS of Ukraine 2015-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1970 Ukrains’kyi Matematychnyi Zhurnal; Vol. 67 No. 2 (2015); 147-162 Український математичний журнал; Том 67 № 2 (2015); 147-162 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1970/961 https://umj.imath.kiev.ua/index.php/umj/article/view/1970/962 Copyright (c) 2015 Blackmore D.; Özçağ E.; Prykarpatsky A. K.; Soltanov K. N. |
| spellingShingle | Blackmore, D. Özçağ, E. Prykarpatsky, A. K. Soltanov, K. N. Блекмор, Д. Озчаг, Е. Прикарпатський, А. К. Солтанов, К. Н. Integrability Analysis of a Two-Component Burgers-Type Hierarchy |
| title | Integrability Analysis of a Two-Component Burgers-Type Hierarchy |
| title_alt | Аналіз інтегровності двокомпонентної ієрархії типу Бюргерса |
| title_full | Integrability Analysis of a Two-Component Burgers-Type Hierarchy |
| title_fullStr | Integrability Analysis of a Two-Component Burgers-Type Hierarchy |
| title_full_unstemmed | Integrability Analysis of a Two-Component Burgers-Type Hierarchy |
| title_short | Integrability Analysis of a Two-Component Burgers-Type Hierarchy |
| title_sort | integrability analysis of a two-component burgers-type hierarchy |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1970 |
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