The Cartan – Monge geometric approach to the characteristic method for nonlinear partial differential equations of the first and higher orders
We develop the Cartan – Monge geometric approach to the characteristic method for nonlinear part ial differential equations of the first and higher orders. The Hamiltonian structure of characteristic vector fields related with nonlinear partial differential equations of the first order is analyzed,...
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| Zitieren: | The Cartan - Monge geometric approach to the characteristic method for nonlinear partial differential equations of the first and higher orders / D.L. Blackmore, N.К. Prykarpatska, V.Hr. Samoylenko, E. Wachnicki, M. Pytel-Kudela // Нелінійні коливання. — 2007. — Т. 10, № 1. — С. 26-36. — Бібліогр.: 13 назв. — англ. |
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| author | Blackmore, D.L. Prykarpatska, N.K. Samoylenko, V.Hr. Wachnicki, E. Pytel-Kudela, M. |
| author_facet | Blackmore, D.L. Prykarpatska, N.K. Samoylenko, V.Hr. Wachnicki, E. Pytel-Kudela, M. |
| citation_txt | The Cartan - Monge geometric approach to the characteristic method for nonlinear partial differential equations of the first and higher orders / D.L. Blackmore, N.К. Prykarpatska, V.Hr. Samoylenko, E. Wachnicki, M. Pytel-Kudela // Нелінійні коливання. — 2007. — Т. 10, № 1. — С. 26-36. — Бібліогр.: 13 назв. — англ. |
| collection | DSpace DC |
| description | We develop the Cartan – Monge geometric approach to the characteristic method for nonlinear part ial differential equations of the first and higher orders. The Hamiltonian structure of characteristic vector fields related with nonlinear partial differential equations of the first order is analyzed, the tensor fields of special structure are constructed for defining characteristic vector fields naturally related with nonlinear partial differential equations of higher orders.
|
| first_indexed | 2025-12-01T09:06:31Z |
| format | Article |
| fulltext |
UDC 517 . 9
THE CARTAN – MONGE GEOMETRIC APPROACH
TO THE CHARACTERISTIC METHOD FOR NONLINEAR PARTIAL
DIFFERENTIAL EQUATIONS OF THE FIRST AND HIGHER ORDERS
ГЕОМЕТРИЧНИЙ ПIДХIД КАРТАНА – МОНЖА ДО МЕТОДУ
ХАРАКТЕРИСТИК ДЛЯ НЕЛIНIЙНИХ ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ
З ЧАСТИННИМИ ПОХIДНИМИ ПЕРШОГО ТА ВИЩИХ ПОРЯДКIВ
D. L. Blackmore
New Jersey Inst. Technol.
Newark, New Jersey, 07102, USA
e-mail: deblac@njit.edu
N. K. Prykarpatska
AGN Univ. Sci. and Technol.
Krakow, 30059, Poland
V. Hr. Samoylenko
Kyiv. Nat. Taras Shevchenko Univ.
Volodymyrs’ka Str., 64, 01033, Kyiv, Ukraine
e-mail: vsam@univ.kiev.ua
E. Wachnicki
Pedagog. Acad., Inst. Math.
Krakow, 30059, Poland
M. Pytel-Kudela
AGH Univ. Sci. and Technol.
Krakow, 30059, Poland
e-mail: kudela@uci.agh.edu.pl
We develop the Cartan – Monge geometric approach to the characteristic method for nonlinear partial
differential equations of the first and higher orders. The Hamiltonian structure of characteristic vector
fields related with nonlinear partial differential equations of the first order is analyzed, the tensor fields of
special structure are constructed for defining characteristic vector fields naturally related with nonlinear
partial differential equations of higher orders.
Розвинуто геометричний пiдхiд Картана – Монжа до методу характеристик для нелiнiйних
диференцiальних рiвнянь з частинними похiдними першого та вищих порядкiв. Дослiджено га-
мiльтонову структуру характеристичних векторних полiв, пов’язаних iз нелiнiйними диферен-
цiальними рiвняннями з частинними похiдними першого порядку, та побудовано тензорнi поля
зi спецiальною структурою для визначення характеристичних полiв, природно пов’язаних iз
нелiнiйними диференцiальними рiвняннями з частинними похiдними вищих порядкiв.
c© D. L. Blackmore, N. K. Prykarpatska, V. Hr. Samoylenko, E. Wachnicki, M. Pytel-Kudela, 2007
26 ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
THE CARTAN – MONGE GEOMETRIC APPROACH TO THE CHARACTERISTIC METHOD . . . 27
1. Introduction: geometric backgrounds of the classical characteristic method. The characteris-
tic method [1 – 4] proposed in XIX century by A. Cauchy was very nontrivially developed by
G. Monge, having introduced the geometric notion of a characteristic surface, related with parti-
al differential equations of the first order. The latter, being augmented with a very important
notion of characteristic vector fields, appeared to be fundamental [4, 5 – 7] for the characteristic
method, whose main essence consists in bringing about the problem of studying solutions to our
partial differential equation to an equivalent one of studying some set of ordinary differential
equations. This way of reasoning succeeded later in a development of the Hamilton – Jacobi
theory, making it possible to describe a wide class of solutions to first order partial differential
equations of the form
H(x;u, ux) = 0, (1.1)
where H ∈ C2(Rn+1 ×Rn; R), ||Hx|| 6= 0, is called a Hamiltonian function and u ∈ C2(Rn; R)
is an unknown function to be found. The equation (1.1) ) is endowed still with a boundary value
condition,
u|Γϕ = u0, (1.2)
with u0 ∈ C1(Γϕ; R) defined on some smooth almost everywhere hypersurface
Γϕ := {x ∈ Rn : ϕ(x) = 0, ||ϕx|| 6= 0}, (1.3)
where ϕ ∈ C1(Rn; R) is some smooth function on Rn.
Following Monge’s ideas, let us introduce the characteristic surface SH ⊂ Rn+1 × Rn as
SH := {(x;u, p) ∈ Rn+1 × Rn : H(x;u, p) = 0}, (1.4)
where we put, by definition, p := ux ∈ Rn for all x ∈ Rn. The characteristic surface (1.4) was
effectively described by Monge within his geometric approach by means of the so-called Monge
cones K ⊂ T (Rn+1) and their duals K∗ ⊂ T ∗(Rn+1) [4, 6]. The corresponding differential-
geometric analysis of this Monge scenario was later done by E. Cartan, who reformulated [4,
8] the geometric picture drown by Monge by means of the related compatibility conditions on
dual Monge cones and the notion of an integral submanifold ΣH ⊂ SH naturally assigned to
special vector fields on the characteristic surface SH . In particular, Cartan had introduced on
SH the differential 1-form
α(1) := du− < p, dx >, (1.5)
where < ·, · > is the usual scalar product in Rn, and demanded its vanishing along the dual
Monge cones K∗ ⊂ T ∗(Rn+1), concerning the corresponding integral submanifold imbedding
mapping
π : ΣH :→ SH . (1.6)
This means that the 1-form
π∗α
(1)
1 := du− < p, dx > |ΣH
⇒ 0 (1.7)
for all points (x;u, p) ∈ ΣH of a solution surface ΣH defined in such a way that K∗ = T ∗(ΣH).
The obvious corollary from the condition (1.7) is the second Cartan condition
dπ∗α
(1)
1 = π∗dα
(1)
1 = < dp,∧dx > |ΣH
⇒ 0. (1.8)
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
28 D. L. BLACKMORE, N. K. PRYKARPATSKA, V. HR. SAMOYLENKO, E. WACHNICKI, M. PYTEL-KUDELA
These two Cartan’s conditions (1.7) and (1.8) should be still augmented with the characteristic
surface SH invariance condition for the differential 1-form α
(1)
2 ∈ Λ1(SH),
α
(1)
2 := dH|SH
⇒ 0. (1.9)
The conditions (1.7), (1.8) and (1.9), when imposed on the characteristic surface SH ⊂ Rn+1×
×Rn, make it possible to construct the proper characteristic vector fields on SH , whose sui-
table characteristic strips [4, 6] generate the sought solution surface ΣH . Thereby, having solved
the corresponding Cauchy problem related with the boundary value conditions (1.2) and (1.3)
for these characteristic vector fields, considered as ordinary differential equations on SH , one
can construct a solution to our partial differential equation (1.1). And what is interesting, this
solution in many cases can be represented [1, 9] in exact functional-analytic Hopf – Lax type
form. The latter is a natural consequence from the related Hamilton – Jacobi theory, whose
main ingredient consists in proving the fact that the solution to our equation (1.1) is exactly
the extremal value of some Lagrangian functional, naturally associated [2, 7, 10] with a given
Hamiltonian function.
Below we will construct the proper characteristic vector fields for partial differential equati-
ons of the first order (1.1) on the characteristic surface SH , generating the solution surface
ΣH as suitable characteristic strips related to the boundary conditions (1.2) and (1.3), and
next generalize the Cartan – Monge geometric approach to partial differential equations of the
second and higher orders.
2. The characteristic vector field method: first order partial differential equations. Consider
on the surface SH ⊂ Rn+1 × Rn a characteristic vector field KH : SH → T (SH) in the form
dx
dτ
= aH(x;u, p)
dp
dτ
= bH(x;u, p)
du
dτ
= cH(x;u, p)
:= KH(x;u, p), (2.1)
where τ ∈ R is a suitable evolution parameter and (x;u, p) ∈ SH . Since, owing to the Cartan –
Monge geometric approach, conditions (1.7), (1.8) and (1.9) hold along the solution surface ΣH ,
we can satisfy them, applying the interior differentiation iKH
: Λ(SH) → Λ(SH) [10 – 12] to the
corresponding differential forms α(1)
1 and dα(1)
1 ,
iKH
α
(1)
1 = 0, iKH
dα
(1)
1 = 0.
As a result of simple calculations one finds that
cH = < p, aH >,
(2.2)
β(1) := < bH , dx > − < aH , dp > |SH
= 0
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
THE CARTAN – MONGE GEOMETRIC APPROACH TO THE CHARACTERISTIC METHOD . . . 29
for all points (x;u, p) ∈ SH . The obtained 1-form β(1) ∈ Λ1(SH) must be, evidently, compatible
with the defining invariance condition (1.9) on SH . This means that there exists a scalar function
µ ∈ C1(SH ; R) such that the condition
µα
(1)
2 = β(1)
holds on SH . This gives rise to the following final relationships:
aH = µ
∂H
∂p
, bH = −µ
(
∂H
∂x
+ p
∂H
∂u
)
,
which, together with the first equality of (2.2) complete the search for the structure of the
characteristic vector fields KH : SH → T (SH),
KH =
(
µ
∂H
∂p
;
〈
p, µ
∂H
∂p
〉
,−µ
(
∂H
∂x
+ p
∂H
∂u
))ᵀ
.
Now we can pose a suitable Cauchy problem for the equivalent set of ordinary differential
equations (2.1) on SH as follows:
dx
dτ
= µ
∂H
∂p
: x
∣∣
τ0
= x0(x) ∈ Γϕ, x
∣∣
τ=t(x)
= x ∈ Rn\Γϕ,
du
dτ
=
〈
p, µ
∂H
∂p
〉
: u|τ=0 = u0(x0(x)), u
∣∣
τ=t(x)
= u(x), (2.3)
dp
dτ
= −µ
(
∂H
∂x
+ p
∂H
∂u
)
: p
∣∣
τ=0
=
∂u0(x0(x))
∂x0
,
where x0(x) ∈ Γϕ is the intersection point of the corresponding vector field orbit, starting
at a fixed point x ∈ Rn\Γϕ, with the boundary hypersurface Γϕ ⊂ Rn at the moment of
"time"τ = t(x) ∈ R. As a result of solving the corresponding "inverse"Cauchy problem (2.3)
one finds the following exact functional-analytic expression for a solution u ∈ C2(Rn; R) to the
boundary-value problem (1.2) and (1.3):
u(x) = u0(x0(x)) +
t(x)∫
0
L̄(x;u, p)dτ, (2.4)
where, by definition,
L̄(x;u, p) :=
〈
p, µ
∂H
∂p
〉
for all (x;u, p) ∈ SH . If the Hamiltonian function H : Rn+1 × Rn → R is nondegenerate, that
is HessH := det(∂2H/∂p∂p} 6= 0 for all (x;u, p) ∈ SH , then the first equation of (2.3) can be
solved with respect to the variable p ∈ Rn as
p = ψ(x, ẋ;u)
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
30 D. L. BLACKMORE, N. K. PRYKARPATSKA, V. HR. SAMOYLENKO, E. WACHNICKI, M. PYTEL-KUDELA
for (x, ẋ) ∈ T (Rn), where ψ : T (Rn)×R → Rn is some smooth mapping. This gives rise to the
following canonical expression of the Lagrangian function
L(x, ẋ;u) := L̄(x;u, p)|p=ψ(x,ẋ;u),
and to the resulting solution (2.4),
u(x) = u0(x0(x)) +
t(x)∫
0
L(x, ẋ;u)dτ. (2.5)
The functional-analytic form (2.5) is already proper for constructing its equivalent Hopf – Lax
type form, being very important for finding so called generalized solutions [1, 5, 13] to the
partial differential equation (1.1). This aspect of the Cartan – Monge geometric approach we
suppose to analyze in detail elsewhere.
3. The characteristic vector field method: second order partial differential equations. Assume
we are given a second order partial differential equation
H(x;u, ux, uxx) = 0, (3.1)
where the solution is u ∈ C2(Rn; R) and the generalized "Hamiltonian"function satisfies H ∈
c ∈ C2(Rn+1 ×Rn ×(Rn ⊗Rn); R). Putting p(1) := ux, p
(2) := uxx, x ∈ Rn, one can construct
within the Cartan – Monge generalized geometric approach the characteristic surface
SH :=
{
(x;u, p(1), p(2)) ∈ Rn+1 × Rn × (Rn ⊗ Rn) : H(x;u, p(1), p(2)) = 0
}
(3.2)
and a suitable Cartan’s set of differential one- and two-forms:
α
(1)
1 := du− < p(1), dx > |ΣH
⇒ 0,
dα
(1)
1 := < dx,∧dp(1) > |ΣH
⇒ 0,
(3.3)
α
(1)
2 := dp(1)− < p(2), dx > |ΣH
⇒ 0,
dα
(1)
2 := < dx,∧dp(2) > |ΣH
⇒ 0,
vanishing on a corresponding solution submanifold ΣH ⊂ SH . The set of differential forms
(3.3) should be augmented with the characteristic surface SH invariance differential 1-form
α
(1)
3 := dH|SH
⇒ 0, (3.4)
vanishing, respectively, an the characteristic surface SH .
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
THE CARTAN – MONGE GEOMETRIC APPROACH TO THE CHARACTERISTIC METHOD . . . 31
Let the characteristic vector field KH : SH → T (SH) on SH be given by the expressions
dx
dτ
= aH(x;u, p(1), p(2))
du
dτ
= cH(x;u, p(1), p(2))
dp(1)
dτ
= b
(1)
H (x;u, p(1), p(2))
dp(2)
dτ
= b
(2)
H (x;u, p(1), p(2))
:= KH(x;u, p(1), p(2)), (3.5)
for all (x;u, p(1), p(2)) ∈ SH . To find the vector field (3.5) it is necessary to satisfy the Cartan
compatibility conditions in the following geometric form:
iKH
α
(1)
1 |ΣH
⇒ 0, iKH
dα
(1)
1 |ΣH
⇒ 0,
(3.6)
iKH
α
(1)
2 |ΣH
⇒ 0, iKH
dα
(1)
2 |ΣH
⇒ 0,
where, as above, iKH
: Λ(SH) → Λ(SH) is the internal differentiation of differential forms
along the vector field KH : SH → T (SH). As a result of conditions (3.6) one finds that
cH = < p(1), aH >, b
(1)
H = < p(2), aH >,
β
(1)
1 := < aH , dp
(1) > − < b
(1)
H , dx > |SH
⇒ 0, (3.7)
β
(1)
2 := < aH , dp
(2) > − < b
(2)
H , dx > |SH
⇒ 0,
being satisfied on SH identically. The conditions (3.7) must be augmented still with the characteri-
stic surface invariance condition (3.4). Notice now that β(1)
1 = 0 owing to the second condition
of (3.7) and the third condition of (3.3). Thus, we need now to make compatible the basic scalar
1-form (3.4) with the vector-valued 1-form β
(1)
2 ∈ Λ(SH) ⊗ Rn. To do this let us construct,
making use of the β(1)
2 , the following parametrized set of, respectively, scalar 1-forms:
β
(1)
2 [µ] := < µ̄(1|0) ⊗ aH , dp
(2) > − < b
(2)
H , µ̄(1|0) ⊗ dx > |SH
⇒ 0, (3.8)
where µ̄(1|0) ∈ C1(SH ; Rn) is any smooth vector-valued function on SH . The compatibility
condition for (3.8) and (3.4) gives rise to the next relationships:
µ̄(1|0) ⊗ aH =
∂H
∂p(2)
,
(3.9)
< µ̄(1|0), b
(2)
H > = −
(
∂H
∂x
+ p(1)∂H
∂u
+
〈
∂H
∂p(1)
, p(2)
〉)
,
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
32 D. L. BLACKMORE, N. K. PRYKARPATSKA, V. HR. SAMOYLENKO, E. WACHNICKI, M. PYTEL-KUDELA
holding on SH . Take now a dual vector function µ(1|0) ∈ C1(SH ; Rn) such that < µ(1|0),
µ̄(1|0) > = 1 for all points of SH . Then from (3.9) one easily finds that
aH =
〈
µ(1|0),
∂H
∂p(2)
〉
,
(3.10)
b
(2)
H = −µ(1|0),∗ ⊗
(
∂H
∂x
+ p(1)∂H
∂u
+
〈
∂H
∂p(1)
, p(2)
〉)
.
Combining now the first two relationships of (3.7) with the found above relations (3.10) we get
a final form for the characteristic vector field (3.5),
KH =
(
aH ;< p(1), aH >,< p(2), aH >,−µ(1|0),∗⊗
⊗
(
∂H
∂x
+ p(1)∂H
∂u
+
〈
∂H
∂p(1)
, p(2)
〉))ᵀ
, (3.11)
where aH = < µ(1|0), ∂H/∂p(2) > and µ(1|0) ∈ C1(SH ; Rn) is some smooth vector-valued
function on SH . Thereby, we can construct, as before, solutions to our partial second order
differential equation (3.1) by means of solving the equivalent Cauchy problem for the set of
ordinary differential equations (3.5) on the characteristic surface SH .
4. The characteristic vector field method: partial differential equations of higher orders.
Consider a general nonlinear partial differential equation of higher order m ∈ Z+,
H(x;u, ux, uxx, ..., umx) = 0, (4.1)
where it is assumed that H ∈ C2(Rn+1 × (Rn)⊗m(m+1)/2; R). Within the generalized Cartan –
Monge geometric characteristic method, we need to construct the related characteristic surface
SH as
SH :=
{
(x;u, p(1), p(2), ..., p(m)) ∈
∈ Rn+1 × (Rn)⊗m(m+1)/2 : H(x;u, p(1), p(2), ..., p(m)) = 0
}
, (4.2)
where we put p(1) := ux ∈ Rn, p(2) := uxx ∈ Rn ⊗ Rn, ..., p(m) ∈ (Rn)⊗m for x ∈ Rn. The
corresponding solution manifold ΣH ⊂ SH is defined naturally as the integral submanifold of
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
THE CARTAN – MONGE GEOMETRIC APPROACH TO THE CHARACTERISTIC METHOD . . . 33
the following set of one- and two-forms on SH :
α
(1)
1 := du− < p(1), dx > |ΣH
⇒ 0,
dα
(1)
1 := < dx,∧dp(1) > |ΣH
⇒ 0,
α
(1)
2 := dp(1)− < p(2), dx > |ΣH
⇒ 0,
dα
(1)
2 := < dx,∧dp(2) > |ΣH
⇒ 0, (4.3)
..................................................
α(1)
m := dp(m−1)− < p(m), dx > |ΣH
⇒ 0,
dα(1)
m := < dx,∧dp(m) > |ΣH
⇒ 0,
vanishing on ΣH .The set of differential forms (4.3) is augmented with the determining characteri-
stic surface SH invariance condition
α
(1)
m+1 := dH|SH
⇒ 0. (4.4)
Proceed now to constructe the characteristic vector fieldKH : SH → T (SH) on the hypersurface
SH within the developed above generalized characteristic method. Take the expressions
dx
dτ
= aH(x;u, p(1), p(2), ..., p(m))
du
dτ
= cH(x;u, p(1), p(2), ..., p(m)
dp(1)
dτ
= b
(1)
H (x;u, p(1), p(2), ..., p(m))
dp(2)
dτ
= b
(2)
H (x;u, p(1), p(2), ..., p(m))
..................................................
dp(m)
dτ
= b
(m)
H (x;u, p(1), p(2), ..., p(m))
:= KH(x;u, p(1), p(2)), (4.5)
for (x;u, p(1), p(2), ..., p(m)) ∈ SH and satisfy the corresponding Cartan compatibility conditions
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
34 D. L. BLACKMORE, N. K. PRYKARPATSKA, V. HR. SAMOYLENKO, E. WACHNICKI, M. PYTEL-KUDELA
in the following geometric form:
iKH
α
(1)
1 |ΣH
⇒ 0, iKH
dα
(1)
1 |ΣH
⇒ 0,
iKH
α
(1)
2 |ΣH
⇒ 0, iKH
dα
(1)
2 |ΣH
⇒ 0,
(4.6)
...........................................................
iKH
α(1)
m |ΣH
⇒ 0, iKH
dα(1)
m |ΣH
⇒ 0.
As a result of suitable calculations in (4.6) one gets the following expressions:
cH = < p(1), aH >, b
(1)
H = < p(2), aH >,
β
(1)
1 := < aH , dp
(1) > − < b
(1)
H , dx > |SH
⇒ 0,
β
(1)
2 := < aH , dp
(2) > − < b
(2)
H , dx > |SH
⇒ 0,
.............................................................. (4.7)
β(1)
m := < aH , dp
(m) > − < b
(m)
H , dx > |SH
⇒ 0,
being satisfied on SH identically.
It is now easy to see that all of 1-forms β(1)
j ∈ Λ1(SH) ⊗ (Rn)⊗j , j = 1,m− 1, are vani-
shing identically on SH owing to the relationships (4.3). Thus, as a result, we obtain the only
relationship
β(1)
m := < aH , dp
(m) > − < b
(m)
H , dx > |SH
⇒ 0, (4.8)
which should be compatibly combined with that of (4.4). To do this suitably with the tensor
structure of the 1-forms (4.8), we take a smooth tensor function µ̄(m−1|0) ∈ C1(SH ; (Rn)⊗(m−1))
on SH and construct the parametrized set of scalar 1-forms
β(1)
m [µ] := < µ̄(m−1|0) ⊗ aH , dp
(m) > − < b
(m)
H , µ̄(m−1|0) ⊗ dx > |SH
⇒ 0, (4.9)
which can be now identified with the 1-form (4.4). This gives rise right away to the relationships
µ̄(m−1|0) ⊗ aH =
∂H
∂p(m)
,
(4.10)
< µ̄(m−1|0), b
(m)
H > = −
(
∂H
∂x
+ p(1)∂H
∂u
+
〈
∂H
∂p(1)
, p(2)
〉
+ . . .+
〈
∂H
∂p(m−1)
, p(m)
〉)
,
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
THE CARTAN – MONGE GEOMETRIC APPROACH TO THE CHARACTERISTIC METHOD . . . 35
holding on SH .Now we can take a dual tensor-valued function µ(m−1|0) ∈ C1(SH ; (Rn)⊗(m−1))
on SH such that < µ(m−1|0), µ̄(m−1|0) > = 1 for all points of SH . Then from (4.10) we easily get
the sought unkown expressions
aH =
〈
µ(m−1|0),
∂H
∂p(m)
〉
,
(4.11)
b
(m)
H = −µ(1|0),∗ ⊗
(
∂H
∂x
+ p(1)∂H
∂u
+
〈
∂H
∂p(1)
, p(2)
〉
+ . . .+
〈
∂H
∂p(m−1)
, p(m)
〉)
.
The obtained above result (4.11), combined with suitable expressions from (4.7), gives rise to
the following final form for the characteristic vector field (4.5):
KH =
(
aH ;< p(1), aH >,< p(2), aH >, . . . , < p(m), aH >,
− µ(m−1|0),∗ ⊗
(
∂H
∂x
+ p(1)∂H
∂u
+
+
〈
∂H
∂p(1)
, p(2)
〉
+ . . .+
〈
∂H
∂p(m−1)
, p(m)
〉))ᵀ
,
where aH = < µ(m−1|0), ∂H/∂p(m) > and µ(m−1|0) ∈ C1(SH ; (Rn)⊗(m−1)) is some smooth
tensor-valued function on SH . The resulting set (4.5) of ordinary differential equations on
SH allows to construct exact solutions to our partial differential equation (4.1) in a suitable
functional-analytic form, being often very useful for analyzing its properties important for
applications. On these and related questions we plan to stop in detail elsewhere later.
5. Acknowledgements. The authors are very grateful to their friends and collegues from
Dept. of Mathematics at AGH, Krakow, for helpful and constructive discussions of the problems
treated in the article.
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36 D. L. BLACKMORE, N. K. PRYKARPATSKA, V. HR. SAMOYLENKO, E. WACHNICKI, M. PYTEL-KUDELA
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Received 25.09.2006
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
|
| id | nasplib_isofts_kiev_ua-123456789-7240 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-3076 |
| language | English |
| last_indexed | 2025-12-01T09:06:31Z |
| publishDate | 2007 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Blackmore, D.L. Prykarpatska, N.K. Samoylenko, V.Hr. Wachnicki, E. Pytel-Kudela, M. 2010-03-26T09:59:25Z 2010-03-26T09:59:25Z 2007 The Cartan - Monge geometric approach to the characteristic method for nonlinear partial differential equations of the first and higher orders / D.L. Blackmore, N.К. Prykarpatska, V.Hr. Samoylenko, E. Wachnicki, M. Pytel-Kudela // Нелінійні коливання. — 2007. — Т. 10, № 1. — С. 26-36. — Бібліогр.: 13 назв. — англ. 1562-3076 https://nasplib.isofts.kiev.ua/handle/123456789/7240 517.9 We develop the Cartan – Monge geometric approach to the characteristic method for nonlinear part ial differential equations of the first and higher orders. The Hamiltonian structure of characteristic vector fields related with nonlinear partial differential equations of the first order is analyzed, the tensor fields of special structure are constructed for defining characteristic vector fields naturally related with nonlinear partial differential equations of higher orders. Розвинуто геометричний підхід Картана - Монжа до методу характеристик для нелінійних диференціальних рівнянь (НДР) з частинними похідними першого та вищих порядків. Досліджено гамільтонову структуру характеристичних векторних полів, пов'язаних із НДР з частинними похідними першого порядку, та побудовано тензорні поля зі спеціальною структурою для визначення характеристичних полів, природно пов'язаних із НДР з частинними похідними вищих порядків. en Інститут математики НАН України The Cartan – Monge geometric approach to the characteristic method for nonlinear partial differential equations of the first and higher orders Геометричний підхід Картана - Монжа до методу характеристик для нелінійних диференціальних рівнянь з частинними похідними першого та вищих порядків Article published earlier |
| spellingShingle | The Cartan – Monge geometric approach to the characteristic method for nonlinear partial differential equations of the first and higher orders Blackmore, D.L. Prykarpatska, N.K. Samoylenko, V.Hr. Wachnicki, E. Pytel-Kudela, M. |
| title | The Cartan – Monge geometric approach to the characteristic method for nonlinear partial differential equations of the first and higher orders |
| title_alt | Геометричний підхід Картана - Монжа до методу характеристик для нелінійних диференціальних рівнянь з частинними похідними першого та вищих порядків |
| title_full | The Cartan – Monge geometric approach to the characteristic method for nonlinear partial differential equations of the first and higher orders |
| title_fullStr | The Cartan – Monge geometric approach to the characteristic method for nonlinear partial differential equations of the first and higher orders |
| title_full_unstemmed | The Cartan – Monge geometric approach to the characteristic method for nonlinear partial differential equations of the first and higher orders |
| title_short | The Cartan – Monge geometric approach to the characteristic method for nonlinear partial differential equations of the first and higher orders |
| title_sort | cartan – monge geometric approach to the characteristic method for nonlinear partial differential equations of the first and higher orders |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/7240 |
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