On the Linearization Covering Technique and Its Application to Integrable Nonlinear Differential Systems
In this letter, I analyze a covering jet manifold scheme, its relation to the invariant theory of the associated vector fields, and applications to the Lax-Sato-type integrability of nonlinear dispersionless differential systems. The related contact geometry linearization covering scheme is also dis...
Збережено в:
| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Дата: | 2018 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2018
|
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/209441 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On the Linearization Covering Technique and its Application to Integrable Nonlinear Differential Systems / A.K. Prykarpatski // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 32 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859705840445947904 |
|---|---|
| author | Prykarpatski, A.K. |
| author_facet | Prykarpatski, A.K. |
| citation_txt | On the Linearization Covering Technique and its Application to Integrable Nonlinear Differential Systems / A.K. Prykarpatski // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 32 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | In this letter, I analyze a covering jet manifold scheme, its relation to the invariant theory of the associated vector fields, and applications to the Lax-Sato-type integrability of nonlinear dispersionless differential systems. The related contact geometry linearization covering scheme is also discussed. The devised techniques are demonstrated for such nonlinear Lax-Sato integrable equations as Gibbons-Tsarev, ABC, Manakov-Santini, and the differential Toda singular manifold equations.
|
| first_indexed | 2025-12-01T02:50:20Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 023, 15 pages
On the Linearization Covering Technique
and its Application to Integrable Nonlinear
Differential Systems
Anatolij K. PRYKARPATSKI †‡
† The Department of Physics, Mathematics and Computer Science,
Cracow University of Technology, Kraków 30-155, Poland
‡ Ivan Franko State Pedagogical University of Drohobych, Lviv Region, Ukraine
E-mail: pryk.anat@cybergal.com
Received January 22, 2018, in final form February 28, 2018; Published online March 16, 2018
https://doi.org/10.3842/SIGMA.2018.023
Abstract. In this letter I analyze a covering jet manifold scheme, its relation to the invari-
ant theory of the associated vector fields, and applications to the Lax–Sato-type integrability
of nonlinear dispersionless differential systems. The related contact geometry linearization
covering scheme is also discussed. The devised techniques are demonstrated for such non-
linear Lax–Sato integrable equations as Gibbons–Tsarev, ABC, Manakov–Santini and the
differential Toda singular manifold equations.
Key words: covering jet manifold; linearization; Hamilton–Jacobi equations; Lax–Sato re-
presentation; ABC equation; Gibbons–Tsarev equation; Manakov–Santini equation; contact
geometry; differential Toda singular manifold equations
2010 Mathematics Subject Classification: 17B68; 17B80; 35Q53; 35G25; 35N10; 37K35;
58J70; 58J72; 34A34; 37K05; 37K10
1 Introductory notions and examples
Some three years ago I. Krasil’shchik in his work [21] has analyzed the so-called Gibbons–Tsarev
equation
zyy + ztzty − zyztt + 1 = 0, (1.1)
and its so-called nonlinear first-order differential covering
∂w
∂t
− 1
zy + ztw − w2
= 0,
∂w
∂y
+
zt − w
uy + ztw − w2
= 0, (1.2)
which for any solution z : R2 → R to equation (1.1) is compatible for all (t, y) ∈ R2. He has
shown that the latter makes it possible to determine for any smooth solution w : R2 → R to
the equation (1.2) suitable smooth functions ψ : R × R2 → R satisfying the associated linear
representation of Lax–Sato type:
∂ψ
∂t
+
1
zy + ztλ− λ2
∂ψ
∂λ
= 0,
∂ψ
∂y
− zt − λ
zy + ztλ− λ2
∂ψ
∂λ
= 0,
which for any solution to the equation (1.1) is also compatible for all (t, y) ∈ R2 and an arbitrary
parameter λ ∈ R.
In his work [21] I. Krasil’shchik also posed an interesting problem of providing a differential-
geometric explanation of the linearization procedure for a given nonlinear differential-geometric
relation J1(Rn × R2;R)|E in the jet manifold J1
(
Rn × R2;R
)
, n ∈ Z+, realizing a compatible
mailto:pryk.anat@cybergal.com
https://doi.org/10.3842/SIGMA.2018.023
2 A.K. Prykarpatski
covering [1] for the associated nonlinear differential equation E [x, τ ;u] = 0, imbedded into some
jet manifold Jk
(
Rn × R2;Rm
)
for some k,m ∈ Z+. His extended version of explanation of
this procedure, presented in [21], appeared abstract enough and held it in a hidden form with
no new example demonstrating its application. From this point of view, our work seeks to
unveil some important points of this linearization procedure in the framework of the classical
inhomogeneous vector field equations and present its new and important applications. We
consider the jet manifold Jk
(
Rn × R2;Rm
)
for some fixed k,m ∈ Z+ and some differential
relation [18] in a general form E [x, τ ;u] = 0, satisfied for all (x; τ) ∈ Rn × R2 and suitable
smooth mappings u : Rn × R2 → Rm.
As a new and interesting example, we can take n = 1 = m, k = 2 and choose a differential
relation E [x; y, t;u] = 0 of the form
utuxy − k1uxuty − k2uyutx = 0, (1.3)
the so-called ABC equation, or the Zakharevich equation, first discussed in [31] for k1+k2−1 = 0,
where u : R× R2 → R and k1, k2 ∈ R are arbitrary parameters satisfying the conditions
k1 + k2 − 1 = 0 ∨ k1 + k2 − 1 6= 0.
The first case k1 + k2 − 1 = 0 was before intensively investigated in [12, 25, 31] and recently
in [19], where there was found its Lax–Sato type linearization and many other its interesting
mathematical properties were revealed. For the second case k1 +k2−1 6= 0 the following crucial
result was stated in equivalent form by P.A. Burovskiy, E.V. Ferapontov and S.P. Tsarev in [8]
and recently by I. Krasil’shchik, A. Sergyeyev and O. Morozov in [22].
Proposition 1.1. The covering system J1
(
R × R2;R
)
|E of quasi-linear first-order differential
relations
∂w
∂t
+
utw
uxk1(k1 + k2 − 1)
∂w
∂x
− w(w + k1 + k2 − 1)utx
uxk1
= 0,
∂w
∂y
+
uyw
uxk1(w + k1 + k2 − 1)
∂w
∂x
− w(k1 + k2 − 1)uyx
uxk1
= 0 (1.4)
on the jet manifold J1
(
R × R2;R
)
is compatible, that is, it holds for any its smooth solution
w : R × R2 → R at all points (x; y, t) ∈ R × R2, iff the function u : R × R2 → R satisfies the
ABC-equation (1.3).
Moreover, this result was recently generalized in [22] to the following “linearizing” proposi-
tion.
Proposition 1.2. A system J1
lin
(
R2 × R2;R
)
|E of linear first-order differential relations
∂ψ
∂t
+
λu
k2−1
k1
x ut
k1(k1 + k2 − 1)
∂ψ
∂x
− λ2u
k2−k1−1
k1
x (utuxx − k1uxxuxt)
k21
∂ψ
∂λ
= 0,
∂ψ
∂y
+
k2λu
k2−1
k1
x uy
k1
(
λu
k2−k1−1
k1
x + k1 + k2 − 1
) ∂ψ∂x − λ2k2(k1 + k2 − 1)u
k2−k1−1
k1
x uyuxx
k21
(
λu
k2−k1−1
k1
x + k1 + k2 − 1
) ∂ψ
∂λ
= 0 (1.5)
on the covering jet manifold J1
(
R2 × R2;R
)
is compatible, that is, it holds for any its smooth
solution ψ : R2 × R2 → R at all points (x, λ; y, t) ∈ R2 × R2, iff the function u : R × R2 → R
satisfies the ABC-equation (1.3).
On the Linearization Covering Technique 3
A similar result, when n = 1, m, k = 2, we have stated for the Manakov–Santini equations
utx + uyy + (uux)x + vxuxy − vyuxx = 0,
vxt + vyy + uvxx + vxuxy − vyvxx = 0, (1.6)
whose Lax–Sato integrability was extensively studied earlier in [6, 11, 13, 19, 23, 26]. The
system (1.6) as a jet submanifold J1
(
R×R2;R
)
|E ⊂ J1(R×R2;R) allows the following nonlinear
first-order differential Lax-type representation
∂w
∂t
+
(
w2 − wvx + u− vy
)∂w
∂x
+ uxw − uy + vyy + vx(vy − u)x = 0,
∂w
∂y
+ w
∂w
∂x
− vxxw + (u− vy)x = 0, (1.7)
compatible on solutions to the nonlinear differential relation E [x, τ ;u] = 0 (1.6) on J2
(
R ×
R2;R2
)
. The existence of the compatible representation (1.7) makes it possible to state the
following proposition.
Proposition 1.3. A covering system J1
lin
(
R2×R2;R
)
|E of linear first-order differential relations
∂ψ
∂t
+
(
λ2 + λvx + u− vy
)∂ψ
∂x
+ (uy − λux)
∂ψ
∂λ
= 0,
∂ψ
∂y
+ (vx + λ)
∂ψ
∂x
− ux
∂ψ
∂λ
= 0
on the jet manifold J1
(
R2 × R2;R
)
is compatible, that is, it holds for any its smooth solution
ψ : R2 ×R2 → R at all points (x, λ; y, t) ∈ R2 ×R2, iff the function u : R×R2 → R satisfies the
Manakov–Santini equation (1.6).
From the point of view, based on the propositions above, the substance of our present analysis,
similarly to that suggested in [21], is to recover the intrinsic mathematical structure responsible
for the existence of the “linearizing” covering jet manifold mappings
J1
lin
(
Rn+1 × R2;R
)
|E
J1
(
Rn × R2;R
)
|E (1.8)
for any dimension n ∈ Z+, compatible with our differential relation E [x, τ ;u] = 0, as it was
presented above in the form (1.4) and (1.5) for the differential relation (1.3). Thus, for a given
nonlinear differential relation E [x, τ ;u] = 0 on the jet manifold Jk
(
Rn×R2;Rm
)
for some k ∈ Z+
one can pose the following problem:
Suppose there holds a compatible system J1
(
Rn×R2;Rm
)
|E ⊂ J1
(
Rn×R2;Rm
)
of quasi-linear
first-order differential relations. How then can one construct a linearizing first-order differential
system J1
lin
(
R(1+n)+2;R
)
|E ⊂ J1
(
R(1+n)+2;R
)
in the form of equations involving vector fields on
the covering space Rn+1 × R2, implementing the implications (1.8)?
The latter is interpreted as the associated Lax–Sato representation [32], recently developed
in [3, 4, 5, 13, 14, 29, 30], for the given differential relation E [x, τ ;u] = 0 on the jet manifold
Jk
(
Rn × R2;Rm
)
.
As a dual approach to this linearization covering scheme we present also the so-called contact
geometry linearization, proposed recently in [27] and generalizing from three to four dimensions
the well-known, see, e.g., [28] and reference therein, Hamiltonian linearization covering method.
As an example, we state the following proposition.
Proposition 1.4. The following [8] nonlinear singular manifold Toda differential relation
uxy sinh2 ut = uxuyutt (1.9)
4 A.K. Prykarpatski
on the jet manifold J2
(
R2 × R2;R
)
allows the Lax–Sato type linearization covering
∂ψ
∂t
+
(
e−2ut − 1
)
2ux
∂ψ
∂x
−
[
λ
(
e−2ut − 1
2ux
)
x
+ λ2
(
e−2ut − 1
2ux
)
z
]
∂ψ
∂λ
= 0,
∂ψ
∂y
− uye
−2ut
ux
∂ψ
∂x
+
[
λ
(
uye
−2ut
ux
)
x
+ λ2
(
uye
−2ut
ux
)
z
]
∂ψ
∂λ
= 0
for smooth invariant functions ψ ∈ C2
(
R3 × R2;R
)
, all (x, z, λ; τ) ∈ R3 × R2 and any smooth
solution u : R2 × R2→ R to the differential relation (1.9).
2 The linearization covering scheme
Implementation of the scheme (1.8) is based on the notion of invariants of suitably specified
vector fields on the extended base space Rn+1×R2, whose definition suitable for our needs is as
follows: a smooth mapping ψ : Rn+1 × R2 → R is an invariant of a set of vector fields
X(k) :=
∂
∂τk
+
n∑
j=1
a
(k)
j (x, λ; τ)
∂
∂xj
+ b(k)(x, λ; τ)
∂
∂λ
on Rn+1 × R2 involving parameters τ ∈ R2 with smooth coefficients
(
a(k), b(k)
)
: Rn+1 × R2 →
En × R, k = 1, 2, if the conditions
X(k)ψ = 0 (2.1)
hold for k = 1, 2 and all (x, λ; τ) ∈ Rn+1×R2. The system of linear equations (2.1) is equivalently
representable as a jet submanifold J1
lin
(
Rn+1 × R2;R
)
|E ⊂ J1
(
Rn+1 × R2;R
)
. It is also well
known [9] that simultaneously the following vector field flows
∂xj
∂τk
= a
(k)
j (x, λ; τ),
∂λ
∂τk
= b(k)(x, λ; τ) (2.2)
are compatible for any j = 1, . . . , n, k = 1, 2 and all (x, λ; τ) ∈ Rn+1 ×R2 too. Taking now into
account that there is such an invariant function ψ : Rn+1 × R2 → R, which can be represented
as ψ(x, λ; τ) = w(x; τ) − λ := 0 for some smooth mapping w : Rn × R2 → R, it gives upon its
substitution into (2.1) the following a priori compatible reduced system of quasi-linear first-order
differential relations
∂w
∂τk
+
n∑
j=1
a
(k)
j (x,w; τ)
∂w
∂xj
− b(k)(x,w; τ) = 0 (2.3)
for k = 1, 2 on the jet manifold J0
(
Rn × R2;R
)
. Moreover, subject to the system (2.3) one can
also observe [9, 17] that, modulo solutions to the vector field flow evolution equations (2.2), the
expression w(x; τ) = ψ(x, λ(τ); τ) + λ(τ) for all (x; τ) ∈ Rn × R2, where ψ : Rn+1 × R2 → R is
a first integral of the vector field flows (2.2). Thus, the reduction scheme just described above
provides the algorithm
J1
lin
(
Rn+1 × R2;R
)
|E → J1
(
Rn × R2;R
)
|E
from the implications (1.8) formulated above. The corresponding inverse implication
J1
lin
(
Rn+1 × R2;R
)
|E ← J1
(
Rn × R2;R
)
|E (2.4)
can be algorithmically described as follows.
On the Linearization Covering Technique 5
Consider now a compatible system J1
(
Rn+2;R
)
|E ⊂ J1
(
Rn+2;R
)
of the first-order nonlinear
differential relations
∂w
∂τk
+
n∑
j=1
a
(k)
j (x,w; τ)
∂w
∂xj
− b(k)(x,w; τ) = 0 (2.5)
with smooth coefficients
(
a(k), b(k)
)
: Rn+1 × R2 → En × R, k = 1, 2. As the first step it is
necessary to check whether the associated system of vector field flows
∂xj
∂τk
= a
(k)
j (x,w; τ) (2.6)
on Rn+1 modulo the flows (2.5) for all j = 1, . . . , n and k = 1, 2 is compatible too. If the answer is
“yes”, then it just means [9] that any solution to (2.5) as a complex function w : Rn×R2 → R is
representable as w(x; τ)−λ = α(ψ(x, λ; τ)) for any λ ∈ R and some smooth mapping α : R→ R,
where the mapping ψ : Rn+1 × R2 → R is a first integral of the vector field equations
∂ψ
∂τk
+
n∑
j=1
a
(k)
j (x, λ; τ)
∂ψ
∂xj
+ b(k)(x, λ; τ)
∂ψ
∂λ
= 0 (2.7)
on the extended space Rn+1 ×R for all (x, λ; τ) ∈ Rn+1 ×R2. Moreover, the value w(x(τ); τ) =
λ ∈ R for all τ ∈ R2 is constant as it follows from the condition α(ψ(x(τ), w; τ)) = 0 for the
τ ∈ R2. Thus, we derived that the vector field equations (2.7) realize the covering linear first-
order differential relations J1
lin
(
Rn+1×R2;R
)
|E ⊂ J1
(
Rn+1×R2;R
)
for k = 1, 2 and all (x, λ; τ) ∈
Rn+1 ×R2, linearizing the first-order nonlinear differential relations (2.5) and interpreting it as
the corresponding Lax–Sato representation.
On the contrary, if the associated system of vector field flows (2.6) is not compatible, it is
necessary to recover a hidden isomorphism transformation
J1
(
Rn+1 × R2;R
)
3 (x,w; τ)→ (x, w̃; τ) ∈ J1
(
Rn+1 × R2;R
)
, (2.8)
for which the resulting a priori compatible first-order nonlinear differential relations
∂w̃
∂τk
+
n∑
j=1
ã
(k)
j (x, w̃; τ)
∂w̃
∂xj
− b̃(k)(x, w̃; τ) = 0 (2.9)
possess already a compatible associated system of the corresponding vector field flows
∂xj
∂τk
= ã
(k)
j (x, w̃; τ) (2.10)
on the space Rn × R, for which any solution w̃ : Rn+2 → R generates a first integral ψ̃ : Rn+1 ×
R2→ R of an associated compatible system of the linear vector field equations
∂ψ̃
∂τk
+
n∑
j=1
ã
(k)
j (x, λ; τ)
∂ψ̃
∂xj
+ b̃(k)(x, λ; τ)
∂ψ̃
∂λ
= 0 (2.11)
on the space Rn+1 × R2→ R for k = 1, 2, where ψ̃(x, λ; τ) := α̃(w̃(x; τ) − λ) for all (x, λ; τ) ∈
Rn+1×R and some smooth mapping α̃ : R→ R. From this one easily infers, as it was done above,
a linearized covering jet submanifold J1
lin
(
Rn+1×R2;R
)
|E ⊂ J1
(
Rn+1×R2;R
)
, as a compatible
system of the vector field equations (2.11), generated by the nonlinear first-order differential
system J1
(
Rn×R2;R
)
|E ⊂ J1
(
Rn×R2;R
)
on the space Rn×R2. The latter exactly determines
the inverse implication (2.4) as applied to general compatible first-order nonlinear differential
relations (2.7), providing for the nonlinear first-order differential system J1
(
Rn × R2;R
)
|E ⊂
J1
(
Rn × R2;R
)
its corresponding Lax–Sato representation.
6 A.K. Prykarpatski
Remark 2.1. The existence of the mapping (2.8) can be inferred from the following reasoning.
Assume that the mapping (2.8) exists and is equivalent to the relation
w(x; τ) := ρ(x, w̃(x; τ); τ) (2.12)
for some smooth function ρ : Rn+1 × R2→ R and all (x; τ) ∈ Rn × R2, where, by definition, the
corresponding mapping w̃ : Rn ×R2 → R satisfies the following system of first-order differential
relations:
∂w̃
∂τk
+
n∑
j=1
a
(k)
j (x, ρ(x, w̃; τ); τ)
∂w̃
∂xj
= b̃(k)(x, w̃; τ), (2.13)
compatible for all τk ∈ R, k = 1, 2, and x ∈ Rn. Here the functions b̃(k) : Rn+1×R2→ R, k = 1, 2,
defined as
b̃(k)(x, w̃; τ) :=
b(k) − n∑
j=1
(
∂ρ
∂τk
+ a
(k)
j
∂ρ
∂xj
) ( ∂ρ
∂xj
)−1∣∣∣∣∣
w=ρ(x,w̃;τ)
,
should depend on the mapping (2.12) in such a way that the vector fields
∂xj
∂τk
= a
(k)
j (x, ρ(x, w̃; τ); τ) := ã
(k)
j (x, w̃; τ) (2.14)
be also compatible for all j = 1, . . . , n and k = 1, 2 modulo the flows (2.13). The latter means
that the equation (2.13) can be equivalently represented as a compatible system of the following
vector field equations
∂ψ̃
∂τk
+
n∑
j=1
ã
(k)
j (x, λ; τ)
∂ψ̃
∂xj
+ b̃(k)(x, ρ(x, λ; τ); τ)
∂ψ̃
∂λ
= 0 (2.15)
on its first integral ψ̃ : Rn+1×R2→ R, where ψ̃(x, λ; τ) = α(w̃(x; τ)−λ) for an arbitrarily chosen
smooth mapping α : R→ R, any parameter λ ∈ R and all (x; τ) ∈ Rn × R2. The resulting
system (2.15) means exactly a suitable Lax–Sato type linearization of the compatible quasi-
linear first-order differential relations (2.3). Regarding the mapping (2.12) and the functions
b̃(k) : Rn+1 × R2→ R, k = 1, 2, that depend on it, one can easily observe that the compatibility
condition for the vector fields (2.14) reduces to the a priori compatible first-order quasi-linear
differential relations
∂a
(k)
j
∂ρ
∂ρ
∂τs
−
∂a
(s)
j
∂ρ
∂ρ
∂τk
+
(
∂a
(k)
j
∂ρ
b̃(s) −
∂a
(s)
j
∂ρ
b̃(k)
)
∂ρ
∂w̃
+
n∑
m=1
(
∂a
(k)
j
∂ρ
a(s)m −
∂a
(s)
j
∂ρ
a(k)m
)
∂ρ
∂xm
= 0, (2.16)
where j = 1, . . . , n and k 6= s = 1, 2, and whose solution is exactly the searched-for map-
ping (2.12). Taking here into account that we have only two functional parameters b(s) : Rn+1×
R2→ R, s = 1, 2, the system of 2n differential relations (2.16) can be, in general, compatible
only for the case n = 1. For all other cases n ≥ 2 the compatibility condition for (2.16) should
be checked separately by direct calculations.
On the Linearization Covering Technique 7
2.1 Example: the Gibbons–Tsarev equation
As a first degenerate case of the above scheme (2.4) we consider a compatible nonlinear first-order
system J1
(
R2;R
)
|E ⊂ J1
(
R2;R
)
for n = 0, discussed before in [21]:
∂w
∂t
− 1
zy + ztw − w2
= 0,
∂w
∂y
+
zt − w
uy + ztw − w2
= 0, (2.17)
where (t, y;w) ∈ R2 × R and a mapping u : R2 → R satisfies the Gibbons–Tsarev equation
E [y, t;u] = 0 in the form
zyy + ztzty − zyztt + 1 = 0, (2.18)
first derived in [15]. As the nonlinear system (2.17) is compatible and the associated system of
vector field flows (2.10) is empty, one easily infers [26] that any solution w : R2 → R to (2.17)
generates a first integral ψ : R× R2 → R of a system of the vector field equations
∂ψ
∂t
+
1
zy + ztλ− λ2
∂ψ
∂λ
= 0,
∂ψ
∂y
− zt − λ
zy + ztλ− λ2
∂ψ
∂λ
= 0, (2.19)
where, by definition, ψ(λ; y, t) := α(w(t, y) − λ) for all (λ; t, y) ∈ R × R2 and some smooth
mapping α : R→ R. The compatible system (2.19) considered as the jet submanifold J1
lin
(
R1 ×
R2;R
)
|E ⊂ J1
(
R1 × R2;R
)
solves the problem of constructing the linearizing implication (2.4).
As it was demonstrated in [16, 24], the substitution
u :=
1
2
(
−zt +
√
z2t + 4zy
)
, v :=
1
2
(
−zt −
√
z2t + 4zy
)
gives rise to the next equivalent dynamical system
uy = vut − (u− v)−1, vy = uvt + (u− v)−1 (2.20)
on a functional space M ⊂ C∞
(
R;R2
)
subject to the evolution parameter y ∈ R modulo
evolution with respect to the joint evolution parameter t ∈ R. Taking into account the Lax–Sato
representation (2.19), one easily infers [26] the corresponding linearizing Lax–Sato representation
for the dynamical system (2.20):
∂ψ
∂t
− 1
(λ+ u)(λ+ v)
∂ψ
∂λ
= 0,
∂ψ
∂y
− u+ v + λ
(λ+ u)(λ+ v)
∂ψ
∂λ
= 0. (2.21)
The above Lax–Sato representation can be now reanalyzed more deeply within the Lie-
algebraic scheme devised recently in [19]. Namely, we define the complex torus diffeomorphism
Lie group G̃ := Diff
(
T1
C
)
, holomorphically extended to the interior D1
+ ⊂ C and to the ex-
terior D1
− ⊂ C regions of the unit disc D1 ⊂ C1, such that for any g(λ) ∈ G̃|D1
−
, λ ∈ D1
−,
g(∞) = 1 ∈ Diff
(
T1
)
, and study its specially chosen coadjoint orbits related to the compatible
system of linear vector field equations (2.21).
As a first step for solving this problem one needs to consider the corresponding Lie algebra
G̃ := Diff
(
T1
C
)
and its decomposition into the direct sum of subalgebras
G̃ = G̃+ ⊕ G̃−
of Laurent series with positive as |λ| → 0 and strongly negative as λ| → ∞ degrees, respectively.
Then, owing to the classical Adler–Kostant–Symes theory, for any element l̃ ∈ G̃∗ ' Λ1
(
T1
C
)
the
following formally constructed flows
dl̃/dy = − ad∗∇h(y)(l̃) l̃, dl̃/dt = − ad∗∇h(t)(l̃) l̃ (2.22)
8 A.K. Prykarpatski
with the evolution parameters y, t ∈ R2 are always compatible, if h(py) and h(pt) ∈ I
(
G̃∗
)
are arbitrarily chosen functionally independent Casimir functionals on the adjoint space G̃∗,
and by definition, ∇h(y)(l̃) := ∇h(py)(l̃)−, ∇h(t)(l̃) := ∇h(pt)(l̃)− are their gradients, suitably
projected onto the subalgebra G̃−. Bearing in mind the above classical result, consider the
Casimir functional h(py) on G̃∗, whose gradient ∇h(py)(l̃) := ∇h(py)(l)∂/∂λ∈ G̃ as |λ| → ∞ is
taken, for simplicity, in the asymptotic form
∇h(py)(l̃) ∼
(
λ+ u+ v
(λ+ u)(λ+ v)
+ α0 + α1λ
)
∂
∂λ
,
where λ ∈ T1
C, |λ| → ∞, and coefficients αj ∈ C∞
(
R2;R
)
, j = 0, 1, are arbitrarily chosen
nontrivial functional parameters, giving rise to the gradient projection
∇h(y)(l̃) := ∇h(py)(l̃)|− =
λ+ u+ v
(λ+ u)(λ+ v)
, (2.23)
generating the first flow of (2.22). As the differential 1-form l̃ = l(λ; y, t)dλ ∈ Λ1
(
T1
C
)
' G̃
satisfies, by definition, the condition
ad∗∇h(py)(l̃)
l̃ = 0, (2.24)
equivalent to the differential equation
d
dλ
[
l(λ; y, t)
(
∇h(py)(l)
)2]
= 0, (2.25)
one easily infers from (2.23) and (2.25) that the coefficient
l(λ; y, t) =
(
∇h(py)(l)
)−2
=
(λ+ u)2(λ+ v)2
[λ+ u+ v + (α0 + α1λ)(λ+ u)(λ+ v)]2
,
satisfies the relation l(λ; y, t)
(
∇h(py)(l)
)2
= 1 for all (t, y) ∈ R2.
Now we formulate the following useful observation.
Lemma 2.2. The set I(G̃∗) of the functionally independent Casimir invariants is one-dimen-
sional.
Proof. Any asymptotic solution to the determining equation (2.26) is invariant under the mul-
tiplication ∇h(pt)(l̃)→ σ(t, y)∇h(pt)(l̃) by an arbitrary smooth function σ : R2 → R. This proves
the lemma. �
Consider now the gradient ∇h(pt)(l̃) ∈ G̃ of the Casimir functional h(pt) ∈ D(G̃∗), that satisfies
the condition, which is identical to (2.24), that is,
ad∗∇h(pt)(l̃) l̃ = 0,
which is equivalent to the following linear differential equation
2l(λ; y, t)
∂
∂λ
∇h(pt)(l̃) +∇h(pt)(l̃) ∂
∂λ
l(λ; y, t) = 0, (2.26)
and whose solution can be naturally represented as the asymptotic series
∇h(pt)(l̃) ∼ 1
(λ+ u)(λ+ v)
+
∑
j∈Z+
βjλ
j (2.27)
for some nontrivial coefficients βj ∈ C∞
(
R2;R
)
successively determined from the equation (2.26).
On the Linearization Covering Technique 9
As follows from Lemma 2.2, the asymptotic solution (2.27) to the determining equation (2.26)
exists exclusively owing to its natural invariance under the multiplication
∇h(pt)(l̃)→ σ(t, y)∇h(pt)(l̃)
by arbitrary smooth function σ : R2 → R. In particular, this means that asymptotically as
λ → ∞ the product l(λ; y, t)(∇h(pt)(l))2 ∼ 0, for otherwise, if l(λ; y, t)(∇h(pt)(l))2 9 0, this
product becomes, owing to (2.25), a nonzero constant involving the parameter λ ∈ C. The
latter, evidently, means that ∇h(pt)(l) = ∇h(py)(l̃)σ(t, y) for any smooth arbitrary function
σ ∈ C∞
(
R2;R
)
, producing no new flow with respect to the evolution parameter t ∈ R.
Construct now the gradient projection
∇h(t)(l̃) := ∇h(pt)(l̃)|− =
1
(λ+ u)(λ+ v)
generating the second flow of (2.22). As a consequence of the above results we can easily derive
the following compatibility condition for the flows (2.22):[
∂
∂y
−∇h(y)(l̃), ∂
∂y
−∇h(ty)(l̃)
]
= 0,
which is equivalent modulo the dynamical system (2.20) to the following system of two a priori
compatible linear vector field equations:
∂ψ
∂y
− λ+ u+ v
(λ+ u)(λ+ v)
∂ψ
∂λ
= 0,
∂ψ
∂t
− 1
(λ+ u)(λ+ v)
∂ψ
∂λ
= 0 (2.28)
for ψ ∈ C2(R;R) and all (λ; t, y) ∈ C× R2. Thus, we can formulate the results, obtained above,
as the following proposition.
Proposition 2.3. A system J1
lin
(
R×R2;R
)
|E of the linear first-order differential relations (2.28)
on the covering jet manifold J1
(
R × R2;R
)
is compatible, that is, it holds for any its smooth
solution ψ : R× R2 → R at all points (λ; y, t) ∈ R× R2 iff the function u : R2 → R satisfies the
Gibbons–Tsarev equation (2.18).
Moreover, taking into account that the flows (2.22) are, by construction, Hamiltonian sys-
tems on the coadjoint space G̃∗, their reduction on the space of functional variables (u, v) ∈
C∞
(
R2;R2
)
will be, respectively, Hamiltonian too. This reduction scheme is now under study
and is expected to be presented elsewhere.
2.2 Example: the ABC equation
For the second example we consider a compatible system J1
(
R×R2;R
)
|E of the nonlinear first-
order differential relations (1.4) on the jet manifold J1
(
R× R2;R
)
. It is easy to check that the
associated system of vector field flows
∂x
∂t
=
utw
uxk1(k1 + k2 − 1)
,
∂x
∂y
=
uyw
uxk1(w + k1 + k2 − 1)
,
modulo differential relations (1.4), is not compatible for all (w; t, y) ∈ R×R2. Thus, we need to
construct such a mapping (2.8) that the resulting system of nonlinear differential relations (2.9)
will possess an associated system of vector field flows already compatible for all (w̃; t, y) ∈ R×R2.
To do this let us get rid from the very beginning in the equations (1.4) of the strictly linear
part, giving rise to the representation its solution as w(x; t, y) = (ux(x; t, y))αw̃(x; t, y) for α =
10 A.K. Prykarpatski
(k1+k2−1)/k1 and all (x; t, y) ∈ R×R2. Substituting this into (1.4) yields the following a priori
compatible system J1
(
R× R2;R
)
|E of the first-order differential relations, cf. [22],
∂w̃
∂t
+
w̃uα−1x ut
k1α
∂w̃
∂x
+
w̃2uαx(utuxx − k1uxxuxt)
k21
= 0,
∂w̃
∂y
+
k2w̃u
α−1
x uy
k1(w̃uαx + α)
∂w̃
∂x
+
k2αw̃
2uαxuyuxx
k21(w̃uαx + α)
= 0 (2.29)
on the jet manifold J1
(
R×R2;R
)
. We can now easily check that the above expression w̃(x; t, y) =
(ux(x; t, y))−αw(x; t, y) for all (x; t, y) ∈ R × R2 determining the mapping (2.8), is exactly the
one we searched for, inasmuch as the associated system of vector field flows
∂x
∂t
=
w̃uα−1x ut
k1α
,
∂x
∂y
=
k2w̃u
α−1
x uy
k1(w̃uαx + α)
on R× R2 proves to be compatible modulo the system J1
(
R× R2;R
)
|E of a priori compatible
differential relations (2.29) on J1
(
R× R2;R
)
. Based on this compatibility result one can easily
construct the corresponding linearizing system J1
lin
(
R2 × R2;R
)
|E on the covering jet manifold
J1
(
R2 × R2;R
)
by implemening the inverse implication (2.4) as the Lax–Sato representation
∂ψ
∂t
+
λuα−1x ut
k1α
∂ψ
∂x
− λ2uαx(utuxx − k1uxxuxt)
k21
∂ψ
∂λ
= 0,
∂ψ
∂y
+
k2λu
α−1
x uy
k1(λuαx + α)
∂ψ
∂x
− λ2k2αu
α−1
x uyuxx
k21(λuαx + α)
∂ψ
∂λ
= 0, (2.30)
exactly coinciding with (1.5), where, by definition, ψ(x, λ; t, y) := α(w̃(x; t, y) − λ) for all
(x, λ; t, y) ∈ R2 × R2 and any smooth mapping α : R → R. Thus, the linear differential system
(2.30) solves the problem, which was posed above, of constructing the inverse implication (2.4)
for the compatible nonlinear differential system J1
(
R × R2;R
)
|E (1.4), thereby proving Propo-
sition 1.3.
2.3 Example: the Manakov–Santini equations
The Manakov–Santini equations
utx + uyy + (uux)x + vxuxy − vyuxx = 0,
vxt + vyy + uvxx + vxvxy − vyvxx = 0, (2.31)
where (u, v) ∈ C∞
(
R × R2;R2
)
, as is well known [23], are obtained as some generalization of
the dispersionless limit for the Kadomtsev–Petviashvili equation. Equations (2.31) possess the
following compatible nonlinear first-order differential covering J1
(
R×R2;R
)
|E ⊂ J1
(
R×R2;R
)
:
∂w
∂t
+
(
w2 − wvx + u− vy
)∂w
∂x
+ (w − vx)(ux − wvxx)− uy + vyy + vxvxy = 0,
∂w
∂y
+ w
∂w
∂x
− vxxw + (u− vy)x = 0, (2.32)
giving rise for all (x; t, y) ∈ R×R2 to the Manakov–Santini differential relation E [x; y, t;u, v] = 0
(2.31) as a submanifold on the associated jet manifold J2
(
R× R2;R2
)
. It is now easy to check
that the system of vector field flows
∂x
∂t
= w2 − wvx + u− vy,
∂x
∂y
= w,
On the Linearization Covering Technique 11
naturally related to (2.32), is for all (t, y) ∈ R2 not compatible modulo these differential rela-
tions (2.32). Thus, one needs to construct such an isomorphic transformation
J1
(
R2 × R2;R
)
3 (x,w; t, y)→ (x, w̃; t, y) ∈ J1
(
R2 × R2;R
)
that the expression w(x; y, t) = β(x, w̃; y, t) for some, in general nonlinear, smooth mapping
β : R2 × R2 → R transforms the first-order differential covering (2.32) into an equivalent com-
patible first-order differential covering (2.9), for which the corresponding vector field flows (2.10)
already become compatible. This problem is solved easily enough giving rise to the simple map-
ping
w = w̃ + vx
using which one obtains the following compatible first-order differential covering:
∂w̃
∂t
+
(
w̃2 + w̃vx + u− vy
)∂w̃
∂x
− uy + w̃ux = 0,
∂w̃
∂y
+ (vx + w̃)
∂w̃
∂x
+ ux = 0. (2.33)
It is easy to check that the system of vector field flows
∂x
∂t
= w̃2 + w̃vx + u− vy,
∂x
∂y
= vx + w̃,
naturally related to (2.33), is already compatible for all (t, y) ∈ R2 modulo the differential
relations (2.33). Based on this compatibility result stated above, one can easily construct the
associated linearizing first-order differential system J1
lin
(
R2×R2;R
)
|E on the covering jet manifold
J1
(
R2 × R2;R
)
, realizing the inverse implication (2.4) as the Lax–Sato representation
∂ψ
∂t
+
(
λ2 + λvx + u− vy
)∂ψ
∂x
+ (uy − λux)
∂ψ
∂λ
= 0,
∂ψ
∂y
+ (vx + λ)
∂ψ
∂x
− ux
∂ψ
∂λ
= 0,
thus proving Proposition 1.3.
3 The contact geometry linearization covering scheme
3.1 The setup
We consider two Hamilton–Jacobi-type compatible for all (x; τ) := (x; t, y) ∈ R× R2 in general
nonlinear first-order differential relations:
∂z
∂t
+ H̃(t)(x, z, ∂z/∂x; t, y) = 0,
∂z
∂y
+ H̃(y)(x, z, ∂z/∂x; t, y) = 0, (3.1)
where z : R3→ R is the so-called “action function” and H̃(t), H̃(y) : R3×R2→ R are some smooth
generalized Hamiltonian functions. The relations (3.1) follow from the contact geometry [7, 10]
interpretation of some mechanical systems generated by some vector fields. Namely, a differential
one-form α(1) ∈ Λ1
(
R3 × R
)
, defined by the expression
α(1) := dz − λdx,
12 A.K. Prykarpatski
is contact, as α(1)∧dα(1) defines a volume form on R3, and vector fields XH(t) , XH(y) ∈ Γ
(
T
(
R3×
R
))
are called, respectively, contact vector fields (with respect to α(1)), if there exist some
functions µ(t), µ(y) : R3 × R→ R such that for all (x, z; τ) ∈ R2 × R2 the following equalities
−iX
H(t)
α(1) = H(t) := H̃(t)|∂z/∂x=λ, −iX
H(y)
α(1) = H(y) := H̃(y)|∂z/∂x=λ,
LH(t)α(t) = µ(t)α(t), LH(y)α(y) = µ(y)α(y) (3.2)
hold, where LH(t) , LH(y) are the Lie derivatives [9, 10, 17] with respect to the vector fields
XH(t) , XH(y) ∈ Γ
(
T
(
R3 × R
))
. From the conditions (3.2) one easily finds [2, 20] that
XH(t) =
∂H(t)
∂λ
∂
∂x
−
(
∂H(t)
∂x
+ λ
∂H(t)
∂z
)
∂
∂λ
+
(
−H(t) + λ
∂H(t)
∂λ
)
∂
∂z
,
XH(y) =
∂H(y)
∂λ
∂
∂x
−
(
∂H(y)
∂x
+ λ
∂H(y)
∂z
)
∂
∂λ
+
(
−H(y) + λ
∂H(y)
∂λ
)
∂
∂z
, (3.3)
where we have putH(t) := H̃(t)|∂z/∂x=λ, H(y) := H̃(y)|∂z/∂x=λ, and compatibility of the nonlinear
relations (3.1) is equivalent to the commutativity condition for the following vector fields:[
∂
∂t
+XH(t) ,
∂
∂y
+XH(y)
]
= 0
for all (x, z, λ; τ) ∈ R3 × R2, depending parametrically on λ ∈ R. The latter condition can be
rewritten as a compatible Lax–Sato representation for the vector field equations
∂ψ
∂t
+XH(t)ψ = 0,
∂ψ
∂y
+XH(y)ψ = 0
for smooth invariant functions ψ ∈ C2
(
R3 × R2;R
)
and all (x, z, λ; τ) ∈ R3 × R2.
Remark 3.1. It is worth mentioning here that in the case when the Hamiltonian functions
in (3.1) do not depend on the “action function” z : R3→ R, the contact vector fields naturally
reduce to the classical Hamiltonian ones:
XH(t) =
∂H(t)
∂λ
∂
∂x
− ∂H(t)
∂x
∂
∂λ
, XH(y) =
∂H(y)
∂λ
∂
∂x
− ∂H(y)
∂x
∂
∂λ
,
well known [10] from the symplectic geometry.
3.2 Example: the differential Toda singular manifold equation
The above general contact geometry linearization scheme is a modification of the one originally
discovered and applied to integrable 4D dispersionless equations by A. Sergyeyev in [27]. Below
we will apply this scheme to a degenerate case when the system (3.1) is given by the following
linear first-order differential relations:
∂z
∂t
+
(
e−2ut − 1
)
2ux
∂z
∂x
= 0,
∂z
∂y
− uyu−1x e−2ut
∂z
∂x
= 0 (3.4)
for a smooth mapping z : R × R2→ R, whose compatibility condition is the interesting [8] dif-
ferential Toda singular manifold equation (1.9) for a smooth function u : R × R2→ R for all
(x; y, t) ∈ R× R2, defining a differential relation J2
(
R2 × R2;R
)
|E ⊂ J2
(
R2 × R2;R
)
on the jet
manifold J2
(
R2 × R2;R
)
uxy sinh2 ut = uxuyutt. (3.5)
On the Linearization Covering Technique 13
Although the differential relations (3.4) are linear, they contain no “spectral” parameter λ ∈ R
using which one can construct the conservation laws for (2.3) and apply the modified inverse
scattering transform for constructing its particular exact solutions.
However, the above contact geometry linearization scheme makes it possible to represent the
system in question as a set of the compatible Hamilton–Jacobi equations
∂z
∂t
+
(
e−2ut − 1
)
2ux
λ = 0,
∂z
∂y
− uye
−2ut
ux
λ = 0
with the contact Hamiltonians
H(t) :=
(
e−2ut − 1
)
2ux
λ, H(y) := −uye
−2ut
ux
λ,
where the function u : R2 × R2→ R depends here on the additional variable z ∈ R. Taking into
account (3.3), one can construct the corresponding extended contact vector fields
X̃H(t) :=
∂
∂t
+XH(t) =
∂
∂t
+
(e−2ut − 1)
2ux
∂
∂x
−
[
λ
(
e−2ut − 1
2ux
)
x
+ λ2
(
e−2ut − 1
2ux
)
z
]
∂
∂λ
,
X̃H(y) :=
∂
∂y
+XH(y) =
∂
∂y
− uye
−2ut
ux
∂
∂x
+
[
λ
(
uye
−2ut
ux
)
x
+ λ2
(
uye
−2ut
ux
)
z
]
∂
∂λ
,
compatible on the solution set to the nonlinear differential relation (3.5) on the jet manifold
J2
(
R2 × R2;R
)
. The latter makes it possible to state our final proposition on the contact
geometry linearization covering of the system (3.5).
Proposition 3.2. The linear first-order differential relations (3.4) on the jet manifold J2
(
R2×
R2;R
)
allow the following quadratic in the parameter λ ∈ R Lax–Sato type linearization covering
∂ψ
∂t
+
(
e−2ut − 1
)
2ux
∂ψ
∂x
−
[
λ
(
e−2ut − 1
2ux
)
x
+ λ2
(
e−2ut − 1
2ux
)
z
]
∂ψ
∂λ
= 0,
∂ψ
∂y
− uye
−2ut
ux
∂ψ
∂x
+
[
λ
(
uye
−2ut
ux
)
x
+ λ2
(
uye
−2ut
ux
)
z
]
∂ψ
∂λ
= 0
for smooth invariant functions ψ ∈ C2
(
R3 × R2;R
)
, all (x, z, λ; τ) ∈ R3 × R2 and any smooth
solution u : R2 × R2→ R to the differential relation (3.5).
4 Conclusions
We have studied an interesting problem posed by I. Krasil’shchik in [21] concerning a differential-
geometric explanation of the linearization procedure for a given nonlinear differential-geometric
relation J1
(
Rn × R2;R
)
|E in the jet manifold J1
(
Rn × R2;R
)
, n ∈ Z+, realizing a compatible
covering [1, 26] for the corresponding nonlinear differential equation E [x, τ ;u] = 0, imbedded
into some associated jet manifold Jk
(
Rn ×R2;Rm
)
for some k,m ∈ Z+. We have analyzed this
covering jet manifold scheme, its relation to the invariant theory of the quasi-linear associated
vector fields and applications to the Lax–Sato type integrability of nonlinear dispersionless
differential systems. In particular, we discussed the compatible Jacobi type nonlinear relations,
the related contact geometry and its application to the linearization covering scheme. The
devised techniques were applied to such nonlinear Lax–Sato integrable equations as Gibbons–
Tsarev, ABC, Manakov–Santini and the differential Toda singular manifold equations.
14 A.K. Prykarpatski
Acknowledgements
The author cordially thanks Professors M. B laszak, B. Szablikowski and J. Cieślinski for useful
discussions of the results during the International Conference in Functional Analysis dedicated
to the 125th anniversary of Stefan Banach held on 18–23 September, 2017 in Lviv, Ukraine.
He is also greatly indebted to Professors V.E. Zakharov (University of Arizona, Tucson) and
J. Szmigelski (University of Saskatchewan, Saskatoon) for their interest in the work and in-
structive discussions during the XXXV Workshop on Geometric Methods in Physics, held in
Bia lowieża, Poland. The author is grateful to Professor B. Kruglikov (University of Tromsø,
Norway) for interest in the work and mentioning some misprints and important references,
which were very helpful in preparing the manuscript. He is also indebted to the referees for
their remarks and instrumental suggestions. Last but not least, thanks go to the Department
of Mathematical Sciences of the NJIT (Newark NJ, USA) for the invitation to visit the NJIT
during the Summer Semester of 2017, where an essential part of this paper was completed. Local
support from the Institute of Mathematics at the Cracow University of Technology is also much
appreciated.
References
[1] Baran H., Krasil’shchik I.S., Morozov O.I., Vojčák P., Integrability properties of some equations obtained
by symmetry reductions, J. Nonlinear Math. Phys. 22 (2015), 210–232, arXiv:1412.6461.
[2] Blackmore D.L., Prykarpatska N.K., Samoylenko V.H., Wachnicki E., Pytel-Kudela M., The Cartan–Monge
geometric approach to the characteristic method for nonlinear partial differential equations of the first and
higher orders, Nonlinear Oscil. 10 (2007), 22–31.
[3] Bogdanov L.V., Interpolating differential reductions of multidimensional integrable hierarchies, Theoret. and
Math. Phys. 167 (2011), 705–713, arXiv:1011.0631.
[4] Bogdanov L.V., Dryuma V.S., Manakov S.V., Dunajski generalization of the second heavenly equation:
dressing method and the hierarchy, J. Phys. A: Math. Theor. 40 (2007), 14383–14393, arXiv:0707.1675.
[5] Bogdanov L.V., Konopelchenko B.G., On the heavenly equation hierarchy and its reductions, J. Phys. A:
Math. Gen. 39 (2006), 11793–11802, nlin.SI/0512074.
[6] Bogdanov L.V., Pavlov M.V., Linearly degenerate hierarchies of quasiclassical SDYM type, J. Math. Phys.
58 (2017), 093505, 13 pages, arXiv:1603.00238.
[7] Bruce A.J., Grabowska K., Grabowski J., Remarks on contact and Jacobi geometry, SIGMA 13 (2017), 059,
22 pages, arXiv:1507.05405.
[8] Burovskiy P.A., Ferapontov E.V., Tsarev S.P., Second-order quasilinear PDEs and conformal structures in
projective space, Internat. J. Math. 21 (2010), 799–841, arXiv:0802.2626.
[9] Cartan H., Differential forms, Houghton Mifflin Co., Boston, Mass, 1970.
[10] Dubrovin B.A., Fomenko A.T., Novikov S.P., Modern geometry — methods and applications. Part I. The
geometry of surfaces, transformation groups, and fields, Graduate Texts in Mathematics, Vol. 93, 2nd ed.,
Springer-Verlag, New York, 1992.
[11] Dunajski M., Ferapontov E.V., Kruglikov B., On the Einstein–Weyl and conformal self-duality equations,
J. Math. Phys. 56 (2015), 083501, 10 pages, arXiv:1406.0018.
[12] Dunajski M., Kryński W., Einstein–Weyl geometry, dispersionless Hirota equation and Veronese webs,
Math. Proc. Cambridge Philos. Soc. 157 (2014), 139–150, arXiv:1301.0621.
[13] Ferapontov E.V., Kruglikov B.S., Dispersionless integrable systems in 3D and Einstein–Weyl geometry,
J. Differential Geom. 97 (2014), 215–254, arXiv:1208.2728.
[14] Ferapontov E.V., Kruglikov B.S., Novikov V., Integrability of dispersionless Hirota type equations in 4D
and the symplectic Monge–Ampère property, arXiv:1707.08070.
[15] Gibbons J., Tsarev S.P., Reductions of the Benney equations, Phys. Lett. A 211 (1996), 19–24.
[16] Gibbons J., Tsarev S.P., Conformal maps and reductions of the Benney equations, Phys. Lett. A 258 (1999),
263–271.
https://doi.org/10.1080/14029251.2015.1023582
https://arxiv.org/abs/1412.6461
https://doi.org/10.1007/s11072-007-0003-3
https://doi.org/10.1007/s11232-011-0055-5
https://doi.org/10.1007/s11232-011-0055-5
https://arxiv.org/abs/1011.0631
https://doi.org/10.1088/1751-8113/40/48/005
https://arxiv.org/abs/0707.1675
https://doi.org/10.1088/0305-4470/39/38/006
https://doi.org/10.1088/0305-4470/39/38/006
https://arxiv.org/abs/nlin.SI/0512074
https://doi.org/10.1063/1.5004258
https://arxiv.org/abs/1603.00238
https://doi.org/10.3842/SIGMA.2017.059
https://arxiv.org/abs/1507.05405
https://doi.org/10.1142/S0129167X10006215
https://arxiv.org/abs/0802.2626
https://doi.org/10.1007/978-1-4612-4398-4
https://doi.org/10.1063/1.4927251
https://arxiv.org/abs/1406.0018
https://doi.org/10.1017/S0305004114000164
https://arxiv.org/abs/1301.0621
http://projecteuclid.org/euclid.jdg/1405447805
https://arxiv.org/abs/1208.2728
https://arxiv.org/abs/1707.08070
https://doi.org/10.1016/0375-9601(95)00954-X
https://doi.org/10.1016/S0375-9601(99)00389-8
On the Linearization Covering Technique 15
[17] Godbillon C., Géométrie différentielle et mécanique analytique, Hermann, Paris, 1969.
[18] Gromov M., Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Vol. 9,
Springer-Verlag, Berlin, 1986.
[19] Hentosh O.E., Prykarpatsky Y.A., Blackmore D., Prykarpatski A.K., Lie-algebraic structure of Lax–Sato
integrable heavenly equations and the Lagrange–d’Alembert principle, J. Geom. Phys. 120 (2017), 208–227,
arXiv:1612.07760.
[20] Kirillov A.A., Local Lie algebras, Russian Math. Surveys 31 (1976), no. 4, 55–75.
[21] Krasil’shchik I.S., A natural geometric construction underlying a class of Lax pairs, Lobachevskii J. Math.
37 (2016), 60–65, arXiv:1401.0612.
[22] Krasil’shchik I.S., Sergyeyev A., Morozov O.I., Infinitely many nonlocal conservation laws for the ABC equa-
tion with A+B+C 6= 0, Calc. Var. Partial Differential Equations 55 (2016), 123, 12 pages, arXiv:1511.09430.
[23] Manakov S.V., Santini P.M., Cauchy problem on the plane for the dispersionless Kadomtsev–Petviashvili
equation, JETP Lett. 83 (2006), 462–466, nlin.SI/0604023.
[24] Mokhov O.I., Symplectic and Poisson geometry on loop spaces of smooth manifolds and integrable equations,
Contemporary Mathematics, Institute for Computer Studies Publ., Izhevsk, 2004.
[25] Morozov O.I., Sergyeyev A., The four-dimensional Mart́ınez Alonso–Shabat equation: reductions and non-
local symmetries, J. Geom. Phys. 85 (2014), 40–45, arXiv:1401.7942.
[26] Odesskii A.V., Sokolov V.V., Non-homogeneous systems of hydrodynamic type possessing Lax representa-
tions, Comm. Math. Phys. 324 (2013), 47–62, arXiv:1206.5230.
[27] Sergyeyev A., New integrable (3 + 1)-dimensional systems and contact geometry, Lett. Math. Phys. 108
(2018), 359–376, arXiv:1401.2122.
[28] Szablikowski B.M., B laszak M., Meromorphic Lax representations of (1 + 1)-dimensional multi-Hamiltonian
dispersionless systems, J. Math. Phys. 47 (2006), 092701, 23 pages, nlin.SI/0510068.
[29] Takasaki K., Takebe T., SDiff(2) Toda equation – hierarchy, tau function, and symmetries, Lett. Math.
Phys. 23 (1991), 205–214, hep-th/9112042.
[30] Takasaki K., Takebe T., Integrable hierarchies and dispersionless limit, Rev. Math. Phys. 7 (1995), 743–808,
hep-th/9405096.
[31] Zakharevich I., Nonlinear wave equation, nonlinear Riemann problem, and the twistor transform of Veronese
webs, math-ph/0006001.
[32] Zakharov V.E., Shabat A.B., Integration of nonlinear equations of mathematical physics by the method of
inverse scattering. II, Funct. Anal. Appl. 13 (1979), 166–174.
https://doi.org/10.1007/978-3-662-02267-2
https://doi.org/10.1016/j.geomphys.2017.06.003
https://arxiv.org/abs/1612.07760
https://doi.org/10.1070/RM1976v031n04ABEH001556
https://doi.org/10.1134/S1995080216010054
https://arxiv.org/abs/1401.0612
https://doi.org/10.1007/s00526-016-1061-0
https://arxiv.org/abs/1511.09430
https://doi.org/10.1134/S0021364006100080
https://arxiv.org/abs/nlin.SI/0604023
https://doi.org/10.1016/j.geomphys.2014.05.025
https://arxiv.org/abs/1401.7942
https://doi.org/10.1007/s00220-013-1787-x
https://arxiv.org/abs/1206.5230
https://doi.org/10.1007/s11005-017-1013-4
https://arxiv.org/abs/1401.2122
https://doi.org/10.1063/1.2344853
https://arxiv.org/abs/nlin.SI/0510068
https://doi.org/10.1007/BF01885498
https://doi.org/10.1007/BF01885498
https://arxiv.org/abs/hep-th/9112042
https://doi.org/10.1142/S0129055X9500030X
https://arxiv.org/abs/hep-th/9405096
https://arxiv.org/abs/math-ph/0006001
https://doi.org/10.1007/BF01077483
1 Introductory notions and examples
2 The linearization covering scheme
2.1 Example: the Gibbons–Tsarev equation
2.2 Example: the ABC equation
2.3 Example: the Manakov–Santini equations
3 The contact geometry linearization covering scheme
3.1 The setup
3.2 Example: the differential Toda singular manifold equation
4 Conclusions
References
|
| id | nasplib_isofts_kiev_ua-123456789-209441 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-01T02:50:20Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Prykarpatski, A.K. 2025-11-21T18:54:41Z 2018 On the Linearization Covering Technique and its Application to Integrable Nonlinear Differential Systems / A.K. Prykarpatski // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 32 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 17B68; 17B80; 35Q53; 35G25; 35N10; 37K35; 58J70; 58J72; 34A34; 37K05; 37K10 arXiv: 1801.07312 https://nasplib.isofts.kiev.ua/handle/123456789/209441 https://doi.org/10.3842/SIGMA.2018.023 In this letter, I analyze a covering jet manifold scheme, its relation to the invariant theory of the associated vector fields, and applications to the Lax-Sato-type integrability of nonlinear dispersionless differential systems. The related contact geometry linearization covering scheme is also discussed. The devised techniques are demonstrated for such nonlinear Lax-Sato integrable equations as Gibbons-Tsarev, ABC, Manakov-Santini, and the differential Toda singular manifold equations. The author cordially thanks Professors M. Blaszak, B. Szablikowski, and J. Cieślinski for useful discussions of the results during the International Conference in Functional Analysis dedicated to the 125th anniversary of Stefan Banach held on 18–23 September 2017 in Lviv, Ukraine. He is also greatly indebted to Professors V.E. Zakharov (University of Arizona, Tucson) and J. Szmigelski University of Saskatchewan, Saskatoon) for their interest in the work and instructive discussions during the XXXV Workshop on Geometric Methods in Physics, held in Białowieża, Poland. The author is grateful to Professor B. Kruglikov (University of Tromsø, Norway) for his interest in the work and for mentioning some misprints and important references, which were very helpful in preparing the manuscript. He is also indebted to the referees for their remarks and instrumental suggestions. Last but not least, thanks go to the Department of Mathematical Sciences of NJIT (Newark, NJ, USA) for the invitation to visit NJIT during the Summer Semester of 2017, where an essential part of this paper was completed. Local support from the Institute of Mathematics at the Kraków University of Technology is also much appreciated. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On the Linearization Covering Technique and Its Application to Integrable Nonlinear Differential Systems Article published earlier |
| spellingShingle | On the Linearization Covering Technique and Its Application to Integrable Nonlinear Differential Systems Prykarpatski, A.K. |
| title | On the Linearization Covering Technique and Its Application to Integrable Nonlinear Differential Systems |
| title_full | On the Linearization Covering Technique and Its Application to Integrable Nonlinear Differential Systems |
| title_fullStr | On the Linearization Covering Technique and Its Application to Integrable Nonlinear Differential Systems |
| title_full_unstemmed | On the Linearization Covering Technique and Its Application to Integrable Nonlinear Differential Systems |
| title_short | On the Linearization Covering Technique and Its Application to Integrable Nonlinear Differential Systems |
| title_sort | on the linearization covering technique and its application to integrable nonlinear differential systems |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209441 |
| work_keys_str_mv | AT prykarpatskiak onthelinearizationcoveringtechniqueanditsapplicationtointegrablenonlineardifferentialsystems |