Differential-geometric structure and the Lax – Sato integrability of a class of dispersionless heavenly type equations
This short communication is devoted to the study of differential-geometric structure and the Lax – Sato integrability of the reduced Shabat-type, Hirota, and Kupershmidt heavenly equations.
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| author | Hentosh, О. Ye. Pritula, N. N. Prykarpatsky, Ya. A. Гентош, О. Є. Притула, М. М. Прикарпатський, Я. А. |
| author_facet | Hentosh, О. Ye. Pritula, N. N. Prykarpatsky, Ya. A. Гентош, О. Є. Притула, М. М. Прикарпатський, Я. А. |
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| description | This short communication is devoted to the study of differential-geometric structure and the Lax – Sato integrability of the
reduced Shabat-type, Hirota, and Kupershmidt heavenly equations. |
| first_indexed | 2026-03-24T02:08:01Z |
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К О Р О Т К I П О В I Д О М Л Е Н Н Я
UDC 517.9
M. M. Prytula (Ivan Franko Nat. Univ. Lviv),
O. E. Hentosh (Inst. Appl. Problems Mech. and Math. Nat. Acad. Sci. Ukraine, Lviv),
Ya. A. Prykarpatskyy (Univ. Agriculture Krakow, Poland)
DIFFERENTIAL-GEOMETRIC STRUCTURE
AND THE LAX – SATO INTEGRABILITY OF A CLASS
OF DISPERSIONLESS HEAVENLY TYPE EQUATIONS
ДИФЕРЕНЦIАЛЬНО-ГЕОМЕТРИЧНА СТРУКТУРА
ТА IНТЕГРОВНIСТЬ ЛАКСА – САТО ДЛЯ ОДНОГО КЛАСУ
БЕЗДИСПЕРСIЙНИХ РIВНЯНЬ НЕБЕСНОГО ТИПУ
This short communication is devoted to the study of differential-geometric structure and the Lax – Sato integrability of the
reduced Shabat-type, Hirota, and Kupershmidt heavenly equations.
Це коротке повiдомлення присвячено вивченню диференцiально-геометричної структури та iнтегровностi Лакса –
Сато для редукованих небесних рiвнянь типу Шабата, Хiроти та Купершмiдта.
1. Introduction. We study differential-geometric structure and the Lax – Sato integrability of a class
of dispersionless hydrodynamic equations including the reduced Shabat-type, Hirota and Kuper-
shmidt heavenly equations, based on the before devised [3, 6] Lie-algebraic integrability scheme.
It is demonstrated that their compatibility conditions coincide with the corresponding heavenly type
equations under consideration. It is shown that all these equations originate in this way and can be
represented as the Lax – Sato compatibility conditions for specially constructed loop vector fields on
toroidal manifolds.
2. The reduced Shabat-type heavenly equations. 2.1. The first reduced Shabat-type
heavenly equation. The entitled above equation [1] reads as
uyt + utuxy - uxtuy = 0 (1)
for a function u \in C\infty ( \BbbR 2\times \BbbT 1;\BbbR ), where (y, t;x) \in \BbbR 2\times \BbbT 1. To show the Lax – Sato integrability
of the equation (1), take a seed element \~l \in \~\scrG \ast :=\widetilde \mathrm{d}\mathrm{i}ff\ast
(\BbbT 1) in the following form:
\~l =
\Biggl(
u - 2
t
\lambda + 1
+
u2t - u2y
u2yu
2
t
+
u - 2
y
\lambda
\Biggr)
dx,
where \lambda \in \BbbC \setminus \{ 0, - 1\} and \~\scrG is This element generates two independent hierarchies of Casimir
functionals \gamma (1), \gamma (2) \in I( \~\scrG \ast ), whose gradient expansions are given by the following asymptotic
expansions:
\nabla \gamma (1)(l) \sim ut +O(\mu 2),
as \lambda + 1 := \mu \rightarrow 0, and
\nabla \gamma (2)(l) \sim uy +O(\mu 2),
c\bigcirc M. M. PRYTULA, O. E. HENTOSH, YA. A. PRYKARPATSKYY, 2018
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 2 293
294 M. M. PRYTULA, O. E. HENTOSH, YA. A. PRYKARPATSKYY
as \lambda := \mu \rightarrow 0. Having put now, by definition,
\nabla h(t)(l) := \mu (\mu - 2\nabla \gamma (1)(l)) -
\bigm| \bigm|
\mu =\lambda +1
, \nabla h(y)(l) := \mu (\mu - 2\nabla \gamma (2)(l)) -
\bigm| \bigm|
\mu =\lambda
,
one easily ensues from the compatibility condition
\partial \~A(y)/\partial t - \partial \~A(t)/\partial y = [ \~A(y), \~A(t)], (2)
for a set of the vector fields
\~A(t) := \nabla h(t)(l) \partial
\partial x
, \~A(y) := \nabla h(y)(l) \partial
\partial x
(3)
a compatible Lax – Sato representation as the following system of vector field equations:
\partial \psi
\partial t
+
ut
\lambda + 1
\partial \psi
\partial x
= 0,
\partial \psi
\partial y
+
uy
\lambda
\partial \psi
\partial x
= 0, (4)
satisfied for \psi \in C\infty (\BbbR 2 \times \BbbT 1;\BbbC ), any (t, y;x) \in \BbbR 2 \times \BbbT 1 and all \lambda \in \BbbC \setminus \{ 0, - 1\} .
2.2. The second reduced Shabat-type heavenly equation. The entitled above equation [1] reads
as
uyy - uxtuy + utuxy = 0 (5)
for a function u \in C\infty (\BbbR 2 \times \BbbT 1;\BbbR ), where (y, t;x) \in \BbbR 2 \times \BbbT 1. In this case for demonstrating the
Lax – Sato integrability of the equation (5) we will take a seed element \~l \in \~\scrG \ast :=\widetilde \mathrm{d}\mathrm{i}ff\ast
(\BbbT 1) as
\~l = (\lambda u - 2
y + 2(ut + u2y)u
- 3
y + \lambda - 1ut(3ut + 4uy)u
- 4
y )dx,
giving rise to two independent Casimir functionals \gamma (1), \gamma (2) \in I( \~\scrG \ast ), whose gradient expansions are
given by the following asymptotic expansions:
\nabla \gamma (1)(l) \sim - \lambda uy + ut +O(1/\lambda 2),
\nabla \gamma (2)(l) \sim \lambda uy - (ut + uy) +O(1/\lambda 2),
as \lambda \rightarrow \infty . Having put now, by definition,
\nabla \gamma (t)(l) := (\lambda \nabla \gamma (1)(l))| + = \lambda ut - \lambda 2uy,
\nabla \gamma (y)(l) := - (\lambda \nabla \gamma (1)(l) + \lambda \nabla \gamma (2)(l)| + = \lambda uy,
we construct the vector fields
\~A(t) := \nabla h(t)(l) \partial
\partial x
= (\lambda ut - \lambda 2uy)
\partial
\partial x
,
\~A(y) := \nabla h(y)(l) \partial
\partial x
= \lambda uy
\partial
\partial x
,
satisfying the compatibility condition (2), entailing the heavenly equation (1). Moreover, this com-
patibility condition (2) is, evidently, equivalent to the following Lax – Sato vector field representation
\partial \psi
\partial t
+ (\lambda ut - \lambda 2uy)
\partial \psi
\partial x
= 0,
\partial \psi
\partial y
+ \lambda uy
\partial \psi
\partial x
= 0, (6)
satisfied for \psi \in C2(\BbbR 2 \times \BbbT 1;\BbbC ), any (t, y;x) \in \BbbR 2 \times \BbbT 1 and all \lambda \in \BbbC .
The obtained above results can be formulated as the following theorem.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 2
DIFFERENTIAL-GEOMETRIC STRUCTURE AND THE LAX – SATO INTEGRABILITY OF A CLASS. . . 295
Theorem 1. The reduced Shabat-type heavenly equations (1) and (5) are completely integrable
Hamiltonian flows equivalent to the Lax – Sato vector field compatibility conditions (4) and (6),
respectively.
3. The Hirota heavenly equation. The Hirota equation describes [2, 4] three-dimensional
Veronese webs and reads as
\alpha uxuyt + \beta uyuxt + \gamma utuxy = 0 (7)
for any evolution parameters t, y \in \BbbR and the spatial variable x \in \BbbT 1, where \alpha , \beta and \gamma \in \BbbR are
arbitrary constants, satisfying the numerical constraint
\alpha + \beta + \gamma = 0.
To demonstrate the Lax-type integrability of the Hirota equation (7) we choose a seed vector field
\~l \in \~\scrG \ast :=\widetilde \mathrm{d}\mathrm{i}ff\ast
(\BbbT 1) in the rational form
\~l =
\Biggl(
u2x
u2t (\lambda + \alpha )
-
u2x(u
2
y + u2t )
2\alpha u2tu
2
y
+
u2x
u2y(\lambda - \alpha )
\Biggr)
dx.
The corresponding gradients for the Casimir invariants \gamma (j) \in I( \~\scrG \ast ), j = 1, 2, are given by the
following asymptotic expansions:
\nabla \gamma (1)(l) \sim
\sum
j\in \BbbZ +
\nabla \gamma (1)j (l)\mu j , (8)
as \lambda + \alpha := \mu \rightarrow 0, and
\nabla \gamma (2)(l) \sim
\sum
j\in \BbbZ +
\nabla \gamma (2)j (l)\mu j , (9)
as \lambda - \alpha = \mu \rightarrow 0. For the first case (8) one easily obtains that
\nabla \gamma (1)(l) \sim - 2\gamma
ut
ux
+O(\mu 2),
and for the second one (9) one obtains
\nabla \gamma (2)(l) \sim 2\beta
uy
ux
+O(\mu 2),
where we took into account that the two Hamiltonian flows on \~\scrG \ast :
d\~l
dy
= ad\ast \nabla h(y)(\~l)
\~l,
d\~l
dt
= ad\ast \nabla h(t)(\~l)
\~l
with respect to the evolution parameters y, t \in \BbbR hold for the following conservation laws gradients:
\nabla h(t)(l) := \mu (\mu - 2\nabla \gamma (1)(l)) -
\bigm| \bigm|
\mu =\lambda +\alpha
=
- 2\gamma
\lambda + \alpha
ut
ux
,
\nabla h(y)(l) := \mu (\mu - 2\nabla \gamma (2)(l)) -
\bigm| \bigm|
\mu =\lambda - \alpha
=
2\beta
\lambda - \alpha
uy
ux
.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 2
296 M. M. PRYTULA, O. E. HENTOSH, YA. A. PRYKARPATSKYY
It is easy now to check that the compatibility condition (2) for the set of vector fields (3) gives rise
to the Hirota heavenly equation (7), whose equivalent Lax – Sato vector field representation reads as
a system of the linear vector field equations
\partial \psi
\partial t
- 2\gamma ut
ux(\lambda + \alpha )
\partial \psi
\partial x
= 0,
\partial \psi
\partial y
+
2\beta uy
ux(\lambda - \alpha )
\partial \psi
\partial x
= 0, (10)
satisfied for \psi \in C\infty (\BbbR 2 \times \BbbT 1;\BbbC ) for all (y, t;x) \in \BbbR 2 \times \BbbT 1 and \lambda \in \BbbC \setminus \{ \pm \alpha \} . Thus the obtained
result can be formulated as the next theorem.
Theorem 2. The Hirota heavenly equation (7) is a completely integrable Hamiltonian flow
equivalent to the Lax – Sato vector field compatibility condition (10).
4. The Kupershmidt hydrodynamic system. These two compatible to each other hydrodynamic
systems [5, 7] read as
3vy - 6uvx + 6uxv + 6uuy - 6u2ux - 2ut = 0,
- 12vx + 6uy - 12uux = 0,
6uvxx - 3vxy - 6uxxv - 6uxuy + 6u2uxx - 6uuxy + 12uu2x + 2uxt = 0,
6vxx + 6uuxx - 3uxy + 6u2x = 0
(11)
for smooth functions (u, v) \in C\infty (\BbbR ;\BbbR 2) with respect to evolution parameters t, y \in \BbbR and the
spatial variable x \in \BbbT 1. Its Lax-type integrability stems from a seed vector field \=l \in \=\scrG \ast , where \=\scrG
denotes the holomorphic in \lambda \in \BbbS 1\pm Lie algebra \=\scrG := \mathrm{d}\mathrm{i}ffhol( \BbbC \times \BbbT 1) \subset \mathrm{d}\mathrm{i}ff( \BbbC \times \BbbT 1) of the
diffeomorphism group \mathrm{D}\mathrm{i}ff(\BbbC \times \BbbT 1), and
\=l = [\lambda (vx + 2uux) + \lambda 2ux]dx+ [(v + u2) + 2\lambda u+ \lambda 2]d\lambda .
The corresponding gradients for the Casimir invariants \gamma (j) \in I( \=\scrG \ast ), j = 1, 2, are easily constructed
from the determining conditions ad\ast \nabla h(j)(\=l)
\=l = 0, j = 1, 2, as the following asymptotic expansions:
\nabla \gamma (j)(l) \sim
\sum
j\in \BbbZ +
\nabla \gamma (j)j (l)\lambda - j ,
giving rise to the expressions
\nabla \gamma (1)(l) \sim (2(\lambda + u), - 2\lambda ux)
\intercal +O(\lambda - 1),
\nabla \gamma (2)(l) \sim (3(\lambda 2 + 2\lambda u++u2 + v), - 3\lambda (\lambda ux + 2uux + vx))
\intercal +O(\lambda - 1),
as | \lambda | \rightarrow \infty . Now taking into account the following Hamiltonian flows on \~\scrG \ast :
d\=l/dy = - ad\ast \nabla h(y)(\=l)
\=l, d\=l/dy = - ad\ast \nabla h(t)(\=l)
\=l (12)
with respect to the evolution parameters y, t \in \BbbR , where, by definition,
\nabla h(y)(\=l) := \nabla \gamma (1)(\=l))+ = 2(\lambda + u)\partial /\partial x - 2\lambda ux\partial /\partial \lambda ,
\nabla h(t)(\=l) := \nabla \gamma (2)(\=l))+ = 3(\lambda 2 + 2\lambda u+ u2 + v)\partial /\partial x - 3\lambda (\lambda ux + 2uux + vx)\partial /\partial \lambda
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 2
are holomorphic vector fields on \BbbC \times \BbbT 1, we can easily derive the corresponding compatible Kuper-
shmidt hydrodynamic systems (11).
It is also easy to check that the compatibility condition for a set of the vector fields (12) gives
rise to the equivalent Lax – Sato vector field representation
\partial \psi
\partial t
- 3(\lambda 2 + 2\lambda u+ u2 + v)
\partial \psi
\partial x
+ 3\lambda (\lambda ux + 2uux + vx)
\partial \psi
\partial \lambda
= 0,
\partial \psi
\partial y
- 2(\lambda + u)
\partial \psi
\partial x
+ 2\lambda ux
\partial \psi
\partial \lambda
= 0,
(13)
satisfied for \psi \in C2(\BbbR 2 \times \BbbT 1;\BbbC ) for all (y, t;x) \in \BbbR 2 \times \BbbT 1 and \lambda \in \BbbC . The result obtained we
formulate as the following theorem.
Theorem 3. The Kupershmidt hydrodynamic heavenly type system (11) is representable as
commuting Hamiltonian flows (12) on orbits of the coadjoint action of the holomorphic Lie algebra
\=\scrG = \mathrm{d}\mathrm{i}ffhol( \BbbC \times \BbbT 1) and are equivalent to the Lax – Sato vector field compatibility condition (13).
5. Conclusion. As we have demonstrated above, the devised in [3, 6] Lie-algebraic scheme of
studying Lax – Sato-type integrability proved to be both natural and anlytically effective, reducing
the problem to describing an infinite hierarchy of commuting to each other Hamiltonian flows as the
corresponding Lie – Poisson orbits of loop diffeomorphism groups on torus.
6. Acknowledgements. The authors cordially thank Prof. Anatolij M. Samoilenko, Prof. Maciej
Błaszak and Prof. Anatolij K. Prykarpatski for their cooperation and useful discussions of the results
in this paper during the International Conference in Functional Analysis dedicated to the 125’th
anniversary of Stefan Banach held on 18 – 23 September, 2017 in Lviv.
References
1. Alonso L. M., Shabat A. B. Hydrodynamic reductions and solutions of a universal hierarchy // Theor. and Math.
Phys. – 2004. – 104. – P. 1073 – 1085.
2. Dunajski M., Kryński W. Einstein – Weyl geometry, dispersionless Hirota equation and Veronese webs,
arXiv:1301.0621.
3. 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. and Phys. 2017. – 120. – P. 208 – 227.
4. Morozov O. I., Sergyeyev A. The four-dimensional Martinez – Alonso – Shabat equation: reductions, nonlocal sym-
metries, and a four-dimensional integrable generalization of the ABC equation. – 2014. – 11 p. – (Preprint submitted
to JGP).
5. Pavlov M. Kupershmidt hydrodynamic chains and lattices // Intern. Math. Res. Not. – 2006. – P. 1 – 43.
6. Prykarpatskyy Ya. A., Samoilenko A. M. The classical M. A. Buhl problem, its Pfeiffer – Sato solutions and the classical
Lagrange – d’Alembert principle for the integrable heavenly type nonlinear equations // Ukr. Mat. Zh. – 2017. –
69, № 12. – P. 1652 – 1689.
7. Szablikowski B., Błaszak M. Meromorphic Lax representations of (1+1)-dimensional multi-Hamiltonian dispersionless
systems // J. Math. Phys. – 2006. – 47, № 9.
Received 20.09.17
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| id | umjimathkievua-article-1557 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:08:01Z |
| publishDate | 2018 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ce/929dcfa4d96e1fc29010e040be698fce.pdf |
| spelling | umjimathkievua-article-15572019-12-05T09:18:03Z Differential-geometric structure and the Lax – Sato integrability of a class of dispersionless heavenly type equations Диференцiально-геометрична структура та iнтегровнiсть лакса–сато для одного класу бездисперсiйних рiвнянь небесного типу Hentosh, О. Ye. Pritula, N. N. Prykarpatsky, Ya. A. Гентош, О. Є. Притула, М. М. Прикарпатський, Я. А. This short communication is devoted to the study of differential-geometric structure and the Lax – Sato integrability of the reduced Shabat-type, Hirota, and Kupershmidt heavenly equations. Це коротке повiдомлення присвячено вивченню диференцiально-геометричної структури та iнтегровностi Лакса – Сато для редукованих небесних рiвнянь типу Шабата, Хiроти та Купершмiдта. Institute of Mathematics, NAS of Ukraine 2018-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1557 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 2 (2018); 293-297 Український математичний журнал; Том 70 № 2 (2018); 293-297 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1557/539 Copyright (c) 2018 Hentosh О. Ye.; Pritula N. N.; Prykarpatsky Ya. A. |
| spellingShingle | Hentosh, О. Ye. Pritula, N. N. Prykarpatsky, Ya. A. Гентош, О. Є. Притула, М. М. Прикарпатський, Я. А. Differential-geometric structure and the Lax – Sato integrability of a class of dispersionless heavenly type equations |
| title | Differential-geometric structure and the
Lax – Sato integrability of a class of dispersionless heavenly type equations |
| title_alt | Диференцiально-геометрична структура
та iнтегровнiсть лакса–сато для одного класу
бездисперсiйних рiвнянь небесного типу |
| title_full | Differential-geometric structure and the
Lax – Sato integrability of a class of dispersionless heavenly type equations |
| title_fullStr | Differential-geometric structure and the
Lax – Sato integrability of a class of dispersionless heavenly type equations |
| title_full_unstemmed | Differential-geometric structure and the
Lax – Sato integrability of a class of dispersionless heavenly type equations |
| title_short | Differential-geometric structure and the
Lax – Sato integrability of a class of dispersionless heavenly type equations |
| title_sort | differential-geometric structure and the
lax – sato integrability of a class of dispersionless heavenly type equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1557 |
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