On the structure of characteristic surfaces related with partial differential equations of first and higher orders. Pt 2
The generalized characteristics method is developed in the framework of the geometric Monge picture. The Hopf – Lax-type extremality solutions to a wide class of Cauchy problem for nonlinear partial differential equations of first and higher orders are derived. The special Hamilton – Jacobi-type ca...
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irk-123456789-1780212021-02-18T01:27:43Z On the structure of characteristic surfaces related with partial differential equations of first and higher orders. Pt 2 Prykarpatska, N.K. The generalized characteristics method is developed in the framework of the geometric Monge picture. The Hopf – Lax-type extremality solutions to a wide class of Cauchy problem for nonlinear partial differential equations of first and higher orders are derived. The special Hamilton – Jacobi-type case is analized separately. The exact extremality Hopf – Lax-type solution for Cauchy problem to the nonlinear Burgers equation is received, its linearization to the Hopf – Cole expression and to the related Airy-type linear partial differential equation is found and discussed. Розвинуто узагальнений метод характеристик у рамках геометричного пiдходу Монжа. Отримано екстремальнi розв’язки типу Хопфа – Лакса широкого класу задач Кошi для нелiнiйних диференцiальних рiвнянь з частинними похiдними першого та вищих порядкiв. Окремо дослiджено спецiальний випадок типу Гамiльтона – Якобi. Отримано точний екстремальний розв’язок типу Хопфа – Лакса задачi Кошi для нелiнiйного рiвняння Бюргерса. Знайдено та проаналiзовано його лiнеаризацiю у виглядi виразу Хопфа – Коула та пов’язаного з ним лiнiйного диференцiального рiвняння з частинними похiдними типу Ейрi. 2005 Article On the structure of characteristic surfaces related with partial differential equations of first and higher orders. Pt 2 / N.K. Prykarpatska // Нелінійні коливання. — 2005. — Т. 8, № 4. — С. 529-543. — Бібліогр.: 20 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/178021 517.9 en Нелінійні коливання Інститут математики НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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The generalized characteristics method is developed in the framework of the geometric Monge picture. The
Hopf – Lax-type extremality solutions to a wide class of Cauchy problem for nonlinear partial differential equations of first and higher orders are derived. The special Hamilton – Jacobi-type case is analized
separately. The exact extremality Hopf – Lax-type solution for Cauchy problem to the nonlinear Burgers
equation is received, its linearization to the Hopf – Cole expression and to the related Airy-type linear
partial differential equation is found and discussed. |
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Article |
| author |
Prykarpatska, N.K. |
| spellingShingle |
Prykarpatska, N.K. On the structure of characteristic surfaces related with partial differential equations of first and higher orders. Pt 2 Нелінійні коливання |
| author_facet |
Prykarpatska, N.K. |
| author_sort |
Prykarpatska, N.K. |
| title |
On the structure of characteristic surfaces related with partial differential equations of first and higher orders. Pt 2 |
| title_short |
On the structure of characteristic surfaces related with partial differential equations of first and higher orders. Pt 2 |
| title_full |
On the structure of characteristic surfaces related with partial differential equations of first and higher orders. Pt 2 |
| title_fullStr |
On the structure of characteristic surfaces related with partial differential equations of first and higher orders. Pt 2 |
| title_full_unstemmed |
On the structure of characteristic surfaces related with partial differential equations of first and higher orders. Pt 2 |
| title_sort |
on the structure of characteristic surfaces related with partial differential equations of first and higher orders. pt 2 |
| publisher |
Інститут математики НАН України |
| publishDate |
2005 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/178021 |
| citation_txt |
On the structure of characteristic surfaces related with partial differential equations of first and higher orders. Pt 2 / N.K. Prykarpatska // Нелінійні коливання. — 2005. — Т. 8, № 4. — С. 529-543. — Бібліогр.: 20 назв. — англ. |
| series |
Нелінійні коливання |
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2025-07-15T16:23:16Z |
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2025-07-15T16:23:16Z |
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1837730729863151616 |
| fulltext |
UDC 517 . 9
ON THE STRUCTURE OF CHARACTERISTIC SURFACES
RELATED WITH PARTIAL DIFFERENTIAL EQUATIONS
OF FIRST AND HIGHER ORDERS. Pt 2
ПРО СТРУКТУРУ ХАРАКТЕРИСТИЧНИХ ПОВЕРХОНЬ,
ПОВ’ЯЗАНИХ IЗ ДИФЕРЕНЦIАЛЬНИМИ РIВНЯННЯМИ
З ЧАСТИННИМИ ПОХIДНИМИ
ПЕРШОГО ТА ВИЩИХ ПОРЯДКIВ. Ч. 2
N. K. Prykarpatska
AGH Univ. Sci. and Technol.
Krakow, 30059, Poland
The generalized characteristics method is developed in the framework of the geometric Monge picture. The
Hopf – Lax-type extremality solutions to a wide class of Cauchy problem for nonlinear partial differenti-
al equations of first and higher orders are derived. The special Hamilton – Jacobi-type case is analized
separately. The exact extremality Hopf – Lax-type solution for Cauchy problem to the nonlinear Burgers
equation is received, its linearization to the Hopf – Cole expression and to the related Airy-type linear
partial differential equation is found and discussed.
Розвинуто узагальнений метод характеристик у рамках геометричного пiдходу Монжа. Отри-
мано екстремальнi розв’язки типу Хопфа – Лакса широкого класу задач Кошi для нелiнiйних ди-
ференцiальних рiвнянь з частинними похiдними першого та вищих порядкiв. Окремо дослiджено
спецiальний випадок типу Гамiльтона – Якобi. Отримано точний екстремальний розв’язок ти-
пу Хопфа – Лакса задачi Кошi для нелiнiйного рiвняння Бюргерса. Знайдено та проаналiзовано
його лiнеаризацiю у виглядi виразу Хопфа – Коула та пов’язаного з ним лiнiйного диференцiаль-
ного рiвняння з частинними похiдними типу Ейрi.
1. Introduction. It is well known [1, 2] that solutions to linear partial differential equations can
be studied effectively by making use of many different approaches, such as the Fourier method,
the spectral theory and the Green function method. Nevertheless, all of them, regrettably, can
not be applied for analysing solution manifolds of general nonlinear partial differential equati-
ons even of the first and second orders. Since the classical Cauchy works on the problem, by
now there exist [2 – 4] only a few approaches to treating such equations, among which the
famous characteristics method that appears to be the most effective and fruitful. During the
last century this method was further developed by many mathematicians, amongst whom are
P. Lax, H. Hopf, O. A. Oleinik, S. N. Kruzhkov, V. Maslov, P. Lions, L. Evans, Blackmore [2,
4 – 10] and others. Still long ago it was observed that there is a deep connection between
the characteristics method and the Hamiltonian analysis, reducing the problem to studying
some systems of ordinary differential equations. This aspect had become prevailing in works
of H. Hopf, P. Lax, and O. Oleinik (see [2, 9, 11]), who described, doing this way, a wide class
of so-called generalized solutions to first order nonlinear partial differential equations. The
most known result within this field is attributed to H. Hopf and P. Lax, who have found for
the first time a very interesting variational representation for solutions of first order nonlinear
partial differential equations called a Hopf – Lax-type representation. Since these results were
c© N. K. Prykarpatska, 2005
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 4 529
530 N. K. PRYKARPATSKA
strongly based on some geometrical notions, it was natural to analyze the Cauchy characteris-
tics method from the differential-geometrical point of view, initiated still in the classical works
of G. Monge and E. Cartan. Within the framework of the Monge geometric approach to study-
ing solutions of partial differential equations we proposed in Part 1 [12] a generalization of the
Cauchy characteristic method for equations of first and higher orders, making use of the speci-
al tensor fields intimately related with them. These tensor fields appear very naturally within
the developed Monge approach as some geometric objects, generalizing the classical Hamilton-
type equations for characteristic vector fields. Moreover, this geometric approach jointly with
some Cartan’s compatibility considerations is naturally extended to a wide class of nonlinear
partial diffrerential equations of the second and higher orders. Namely, if for instance a first
order differential equation is given as
H(x;u, ux) = 0, (1.1)
where x ∈ Rn, H ∈ C1(Rn × Rn+1; R), ‖Hux‖ 6= 0, the characteristics vector fields on the
related Monge hypersurface
SH :=
{
(x;u, p) ∈ Rn × Rn+1 : H(x;u, p) = 0
}
(1.2)
are represented [12] as follows:
dx
dτ
= µ(1|1)∂H
∂x
,
du
dτ
=
〈
p, µ(1|1)∂H/∂p
〉
,
dp
dτ
= −µ(1|1),∗
(
∂H
∂x
+ p
∂H
∂u
)
. (1.3)
Here µ(1|1) ∈ C1(Rn+1 × Rn; Rn ⊗ Rn) are some smooth tensor fields on SH and τ ∈ R is
an evolution parameter. Vector fields (1.3) ensure [12] the tangency to the hypersurface SH ⊂
⊂ Rn × Rn+1 and the projection compatibility condition with the dual Monge cone K∗ upon
the corresponding solution hypersurface S̄H ⊂ Rn+1 (see Fig. 1), generated by the characte-
ristic strips ΣH ⊂ SH through smoothly imbedded sets Σ ⊂ SH consisting of points carrying
the solutions to our problem (1.1). Similar results were obtained in [12] also for both partial
differential equations of higher orders and systems.
In general, the problem (1.1) is endowed with some boundary condition on a smooth hyper-
surface Γϕ ⊂ Rn like
u|Γϕ = u0, (1.4)
where u0 ∈ C1(Γϕ; R) is a given function. The hypersurface Γϕ ⊂ Rn can be, for simplicity,
defined as
Γϕ := {x ∈ Rn : ϕ(x) = 0} , (1.5)
where ϕ : Rn → R is a smooth mapping endowed with some local coordinates s(x) ∈ Rn−1 in
the corresponding open neighborhoods Oε(x) ⊂ Γϕ of all points x ∈ Γϕ for some ε > 0. Thus,
we are interested in constructing analytical solutions to the boundary problem (1.1), (1.4), and
(1.5) and studying their properties. This and related aspects of this problem will be discussed in
detail below.
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 4
ON THE STRUCTURE OF CHARACTERISTIC SURFACES . . . 531
Fig. 1. The boundary Γn−1
ϕ = {x0 ∈ Rn : ϕ(x0) = 0} , x0(s) ∈ Γn−1
ϕ , s ∈ Rn−1
are local coordinates.
2. Boundary problem analysis. Consider the set of characteristic equations (1.3) on the
hypersurface SH ⊂ R × Rn+1, which start at points (x0;u0, p0) ∈ Γϕ under the additional
condition that the corresponding projection Σ → Σ upon the subspace Rn+1 (see Fig. 2) coi-
ncides with the boundary set (Γϕ;u0) ⊂ Rn+1, that is,
Σ := (Γϕ;u0) , (2.1)
where u0 ∈ C1(Γϕ; R) is our boundary condition. The condition (2.1) assumes evidently that
the set Σ ⊂ SH can be defined as follows:
Σ =
(
Σ; p0
)
(2.2)
with p0 ∈ C1(Γϕ; Rn) being yet unknown smooth mappings. For it to be determined we need to
ensure for all points Σ ⊂ SH the mentioned above compatibility condition, that is the condition
du|Σ = 〈p, dx〉 |Σ, (2.3)
where Σ ⊂ SH is given by (2.2). As a result of (2.3) one finds easily that
∂u0(s)
∂s
−
〈
p0(s),
∂x0(s)
∂s
〉
= 0,
(2.4)
H(x0(s);u0(s), p0(s)) = 0
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 4
532 N. K. PRYKARPATSKA
Fig. 2. Geometric Monge method. The boundary conditions: Σ̄ =
�
Γn−1
ϕ , u0
�
⊂
⊂ S̄H , u0 ∈ C1(Γn−1
ϕ ; R), S̄H :=
�
(x, u) ∈ Rn+1 : u = ψ(x)
is the
boundary problem solution hypersurface.
for all points x0 = x0(s) ∈ Γϕ, s ∈ Rn−1. Here we took into account that any point x ∈ Γϕ
is parametrized by means of the corresponding local coordinates s = s(x0) ∈ Rn−1, defined in
the corresponding ε-vicinities Oε(x) ⊂ Γϕ, ε > 0.
The system of relationships (2.4) must be solvable for a mapping p0 : Γϕ → Rn at all points
x0 ∈ Γϕ, what gives rise to the determinant condition
det
(
∂x0(s)
∂s
;
∂H
∂p0
(x0(s), u0(x0(s)), p0(x0(s)))
)
6= 0, (2.5)
owing to the implicit function theorem [13]. If the condition (2.5) is satisfied at points(
x0;u0, p
(j)
0
)
∈ SH , where j = 1, N for some N ∈ Z+ and all points (x0;u0) ∈ Σ, the system
of equations (2.4) possesses exactly N ∈ Z+ different smooth solution p
(j)
0 ∈ C1(Γϕ; Rn),
j = 1, N, thereby determining the corresponding Cauchy data (2.2) for the characteristic
vector fields (1.3). It is clear enough that our boundary problem (1.1), (1.4), and (1.5) possesses,
in general, many solutions of different functional classes, depending on the kind of chosen
boundary conditions. For instance, as it was studied and analyzed in [2, 14] this boundary
problem can possess also so-called generalized solutions, which allow at some additional condi-
tions the so-called Hopf – Lax inf-type extremality form, being often very useful for studying
their asymptotic and other properties.
Concerning this Hopf – Lax-type extremality solution problem for our equation (1.1) under
the boundary conditions (1.4) and (1.5) the meaning of the involved before in [12] tensor fields
µ(1|1) ∈ C1(Rn × Rn+1; Rn ⊗ Rn) becomes more understandable. Namely, it consists in fin-
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 4
ON THE STRUCTURE OF CHARACTERISTIC SURFACES . . . 533
ding solutions to (1.1), which will satisfy both the imposed boundary conditions and the related
Hopf – Lax inf-type extremality representation [2, 11, 14].
3. The Hopf – Lax inf-type extremality representation. Assume now that p0 ∈
∈ C1(Γϕ; Rn) is a smooth solution to the system (2.4), thereby defining completely the sought
Cauchy data Σ ⊂ SH for the characteristic vector fields (1.3). Thus, making use of suitable if
any methods for solving these ordinary differential equations in the space Rn × Rn+1 at some
appropriately chosen tensor field µ(1|1) ∈ C1(Rn × Rn+1; Rn ⊗ Rn) one can find, in particular,
the function u ∈ C2(Rn; R) for each reachable point x = x(t) ∈ Rn in the form
u(x(t)) = u(x(0)) +
t∫
0
〈
p(τ), µ(1|1)∂H
∂p
(τ)
〉
dτ (3.1)
at any moment of ”time” t ∈ R. As, by definition, x(0) := x0(s) ∈ Γϕ and u(x(0)) :=
:= u0(x0(s)), s ∈ Rn−1, the solution (3.1) can be rewritten
u(x(t)) = u0(x0(s)) +
t∫
0
〈
p(τ), s(1|1)
∂H
∂p
(τ)
〉
dτ (3.2)
for any t ∈ R, where the integrand function in (3.2) is assumed to be known.
Pose now the following ”inverse” vector field problem for the equation
dx
dτ
= µ(1|1)∂H
∂p
(3.3)
with the following ”inverse” Cauchy data
x|τ=t(x) = x ∈ Rn, x|τ=0 = x0(s(x)) ∈ Γϕ (3.4)
for some s(x) ∈ Rn−1 at the moment of ”time” t(x) ∈ R corresponding to an arbitrary
reachable point x ∈ Rn as it is shown on Fig. 3. Respectively, for each found above point
x0(s(x)) ∈ Γϕ, x ∈ Rn, one can suitably determine the unique point p0(s(x)) ∈ Rn, x ∈ Rn,
making use of the system (3.2). As a result, one can write down owing to the conditions (3.4)
the following expression:
u(x) = u0(x0(s(x))) +
t(x)∫
0
L(τ |x0(s(x));x)dτ, (3.5)
where L is the so-called ”Lagrangian” function:
L(τ |x0(s(x));x) :=
〈
p(τ), µ(1|1)∂H
∂p
(τ)
〉
, (3.6)
being defined by solutions to the equations (3.3), (3.4) and to the equation
dp
dτ
= −µ(1|1)∗
(
∂H
∂x
+
∂H
∂u
)
(3.7)
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 4
534 N. K. PRYKARPATSKA
Fig.3. The geometric Monge method. The characteristic surface: SH = {(x;u, p) ∈
∈ R2n+1 : H(x;u, p) = 0} and intial conditions for the vector field KH : SH →
→ T (SH), satisfying the Cartan’s compatibility conditions: du−< p, dx>|KH ,Γn−1
ϕ
=
= 0 iff S̄H ||K∗ and there exist data Σ = (Σ̄, p0) defining the characteristic strip ΣH .
under the corresponding ”inverse” Cauchy data
p|τ=0 = p0(s(x)) ∈ Rn (3.8)
for any reachable point x ∈ Rn.
By integrating the expressing (3.5) one finds the following solution to the boundary problem
(1.1), (1.4), and (1.5):
u(x) = u0 (x0(s(x))) + P (x0(s(x));x) , (3.9)
where x0(s(x)) ∈ Γϕ, x ∈ Rn, is, as above, any reachable by the vector field (3.3) point, and,
by definition, the ”kernel”
P(x0(s(x));x) :=
t(x)∫
0
L(τ |x0(s(x));x)dτ. (3.10)
The obtained solution (3.9) allows an additional interpretation strongly motivated by the previ-
ous Hopf – Lax-type results [2, 11, 14]. Namely, consider the expression (3.5) with the Lagrangi-
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 4
ON THE STRUCTURE OF CHARACTERISTIC SURFACES . . . 535
an function given by the expression (3.6) and the following extremality problem:
δu = δ
(u0(x0(x))) +
t∫
0
L0 (x(τ);u(τ), ẋ(τ)) dτ
= 0 (3.11)
under a fixed ending point x ∈ Rn and varying both a moment of ”time” t ∈ R+ and a point
x0(s) ∈ Γϕ. Here we put, by definition, ẋ := dx/dτ and
L0 (τ |x0(s(x));x) |ẋ=µ(1|1) ∂H
∂p
:=
〈
p(τ), µ(1|1)∂H
∂p
〉
(3.12)
for all (x, p) ∈ Rn × Rn and τ ∈ R.
By means of standard variational analysis calculations one gets easily that the condition
(3.11) is realized exactly upon solutions to the vector field (1.3) under the Cauchy data at points
in Σ ⊂ SH , defined by (2.2) and (2.3). Thus, one can formulate at some natural conditions the
following important theorem.
Theorem 3.1. The expression (3.9) at any reachable point x ∈ Rn, for suitable points
x0(s) ∈ Γϕ defined by the ”inverse” Cauchy problem (3.3) and (3.4), solves the boundary
problem (1.4), (1.5) for the partial differential equation (1.1) and allows, under some natural
conditions on the tensor field µ(1|1) ∈ C1(Rn × Rn+1; Rn ⊗ Rn), the following Hopf – Lax-type
extremality representation:
u(x) = inf
y∈Γϕ
{u0(y) + P(y;x)} , (3.13)
where the ”kernel” P : Rn × Rn → R is given by the analytical expression (3.10).
Proof. For the proof we need only to consider the extremum conditions (3.11), (3.12) and
ensure that the function [u0(·) + P(·;x)] : Γϕ → R attains its finite infimum at some point
y = x0(s(x)) ∈ Γϕ for all reachable points x ∈ Rn. The latter depends, in particular, on the
functional properties of the boundary conditions (Γϕ;u0) := Σ̄ ⊂ Rn+1 and on a choice of
the tensor field µ(1|1) ∈ C1(Rn × Rn+1; Rn ⊗ Rn), defining our vector field (1.3), describing
correspondingly the set of reachable points x ∈ Rn. Having assumed these natural conditions,
we find right away that the infimum (3.13) is attained exactly at the point x0(s(x)) ∈ Γϕ and
at the moment of ”time” t(x) ∈ R+, satisfying the conditions (3.3) and (3.4), giving rise to the
found before solution (3.9) of the boundary problem (1.1), (1.4), and (1.5), that ends the proof.
The Hopf – Lax-type extremality property of the solution to boundary problem (1.1), (1.4),
and (1.5) appears to have a very interesting and important for applications form in the case of
Cauchy problems for generalized Hamilton – Jacobi-type equations, which will be discussed in
the section below.
4. The Hopf – Lax-type extremality solutions to generalized Hamilton – Jacobi equations.
Assume we are given the following generalized scalar Hamilton – Jacobi equation
ut +H(x, t;u, ux) = 0 (4.1)
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 4
536 N. K. PRYKARPATSKA
with a Hamiltonian function H ∈ C2(Rn+1 × Rn+1; R) and the related Cauchy data
u|t=t0 = u0, (4.2)
where u0 ∈ C1(Rn; R) and t0 ∈ R.
Having applied the results obtained above, one gets easily that the corresponding characteri-
stic vector fields on SH are defined as
dx
dτ
= µ(1|1)∂H
∂p
,
dp
dτ
= −µ(1|1)
(
∂H
∂x
+ p
∂H
∂u
)
,
(4.3)
du
dτ
=
〈
p, µ(1|1)∂H
∂p
〉
−H,
dt
dτ
= 1,
where µ(1|1) ∈ C1(Rn+1 × Rn+1; Rn ⊗ Rn) is some suitable tensor field, allowing to solve the
following ”inverse” Cauchy problem:
dx
dτ
= µ(1|1)∂H
∂p
, t = τ, x|τ=t = x, x|τ=t0 = x0(x, t), (4.4)
where τ ∈ [t0, t] ⊂ R, x ∈ Rn, is any reachable point of the vector field (4.4) in Rn and
x0(x, t0) ∈ Rn is the corresponding initial point at which our orbit x : [t0, t] → Rn starts.
Thus, we can now write down a solution to the Cauchy problem (4.1), (4.2), making use of the
previous results (3.9) and (3.10),
u(x, t) = u0(x0(x, t)) + P(x0(x, t), t0;x, t), (4.5)
where, by definition, the ”kernel”
P(x0(x, t), t0;x, t) :=
t∫
t0
[〈
p(τ), µ(1|1)∂H
∂x
(τ)
〉
−H(x(τ), t;u(τ), p(τ)
]
dτ (4.6)
is defined for any reachable point x ∈ Rn and t0 ≤ t ∈ R. As the expression (4.5) solves
the Cauchy problem (4.1), (4.2) for some mapping x0 : Rn+1 → Rn defined by the ”inverse”
Cauchy data, we obtain the following theorem.
Theorem 4.1. The Hopf – Lax-type extremality expression
u(x, t) = inf
y∈Rn
{u0(y) + P(y, t0;x, t)} (4.7)
solves the Cauchy problem (4.1), (4.2), where the ”kernel” P : Rn+1 × Rn+1 → R is suitably
defined by (4.6) for any reachable point x ∈ Rn and t ≥ t0 ∈ R in such a way that the infimum
for the mapping [u0(·) + P(·, t0;x, t)] : Rn → R is attainable and finite.
It is to be mentioned here that the ”resolving kernel” P : Rn+1×Rn+1 → R, defined by the
expression (4.6), depends strongly on the choice of a tensor field µ(1|1) ∈ C1(Rn+1×Rn+1; Rn⊗
⊗Rn), ensuring both the effective solvability of the ordinary differential equations (4.3) and the
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ON THE STRUCTURE OF CHARACTERISTIC SURFACES . . . 537
corresponding existence of the infimum of the expression (4.7). Concerning these aspects, the
related Cauchy data for (4.3) at τ = t0 ∈ R are given, owing to (4.4) and (2.4), as follows:
x|τ=t0 = x0(x, t), p|τ=t0 = p0(x, t), (4.8)
where the point p0(x, t) ∈ Rn satisfies the compatibility condition
∂u0(x0, t)
∂x
−
〈
p0(x, t),
∂x0(x, t)
∂x
〉
= 0 (4.9)
for all reachable points x ∈ Rn and moments of time t ∈ R. For the system (4.9) to be solvable,
the natural condition
det
(
∂x0(x, t)
∂x
)
6= 0 (4.10)
must be satisfied at all points (x, t) ∈ Rn × R.
As the existence of the infimum (4.7) depends implicitly also on the Cauchy data u0 : Rn →
→ R, it looks very suggestive to represent a wide class of Cauchy problems for the Hamilton –
Jacobi equation (4.1) in the Hopf – Lax-type form by choosing suitable tensor
fields µ(1|1) ∈ C1(Rn+1 × Rn+1; Rn × Rn). Concerning this aspect a very important yet too
complicated problem of finding the related relationships between the Cauchy data and suitable
tensor fields µ(1|1) ∈ C1(Rn+1×Rn+1; Rn⊗Rn) at which the problem (4.7) is reasonably posed,
remains up to now to be unsolved. We can mention here also that results similar to those obtai-
ned above hold also for boundary problems posed for nonlinear partial differential equations
of higher orders and suitable systems.
5. An example: the Burgers nonlinear differential equation of the second order. Consider
the following Cauchy problem for the well known [15] Burgers differential equation
ut + uxx + uux = 0, u|t=0+ = u0, (5.1)
where (x, t) ∈ R2 and it is assumed that u ∈ C(3,2)(R × R; R), u0 ∈ C2(R; R). The correspon-
ding Monge surface SH is defined as
SH := (x, t;u, p) ∈
(
R× R× R6
)
: H(x, t;u, p) = 0, (5.2)
where
p := (p(1,0), p(0,1), p(1,0), p(1,1), p(2,0), p(0,2)) ∈ R6,
H(x, t;u, p) := p(0,1) + p(2,0) + up(1,0). (5.3)
The projection upon the surface SH ⊂ (R × R+) × R of solutions to (5.1) is given [12] by the
following Cartan compatibility conditions:
du = p(1,0)dx+ p(0,1)dt, dp(1,0) = p(2,0)dx+ p(1,1)dt, dp(0,1) = p(1,1)dx+ p(0,2)dt, (5.4)
for all points (x, t) ∈ (R× R+). Put now
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538 N. K. PRYKARPATSKA
dx
dτ
= aH(x, t : u, p),
dt
dτ
= aH(x, t;u, p),
(5.5)
dp(j)
dτ
= b(j)(x, t;u, p),
du
dτ
= cH(x, t;u, p),
where τ ∈ R, (j) ∈ J := (j1, j2) ∈ {0, 2× 0, 2}\(0, 0), being the related characteristic fields on
the Monge hypersurface SH , subject to which it remains to be invariant. This means that
dH
dτ
=
∂H
∂x
aH +
∂H
∂t
aH +
∂H
∂u
cH +
∑
(j)
∂H
∂p(j)
b(j) = 0 (5.6)
for all τ ∈ R.Owing to the Cartan compatibility conditions (5.3) one gets that, along the vector
field (5.5), the equalities
cH = p(1,0)aH + p(0,1), b(0,1) = p(1,1)aH + p(0,2)aH , b(1,0) = p(2,0) + aH + p(1,1)aH (5.7)
hold on SH . Having substituted (5.7) into (5.6) one finds that the relationship
(
∂H
∂x
+
∂H
∂u
p(1,0) +
∂H
∂p(1,0)
p(2,0) +
∂H
∂p(1,1)
p(1,1)
)
aH+
+
(
∂H
∂t
+
∂H
∂u
p(0,1) +
∂H
∂p(1,0)
p(1,1) +
∂H
∂p(0,1)
p(0,2)
)
aH+
+
∂H
∂p(1,1)
b(1,1) +
∂H
∂p(2,0)
b(2,0) +
∂H
∂p(0,2)
b(0,2) = 0 (5.8)
is satisfied on SH . This means [12] that
aH = µ
(1|1)
(1,1)
∂H
∂p(1,1)
+ µ
(1|1)
(2,0)
∂H
∂p(2,0)
+ µ
(1|1)
(0,2)
∂H
∂p(0,2)
,
āH = µ̄
(1,1)
(1,1)
∂H
∂p(1,1)
+ µ̄
(1,1)
(2,0)
∂H
∂p(2,0)
+ µ̄
(1,1)
(0,2)
∂H
∂p(0,2)
,
(5.9)
b(j) = −µ(1|1)
(j)
(
∂H
∂x
+
∂H
∂u
p(1,0) +
∂H
∂p(1,0)
p(2,0) +
∂H
∂p(1,1)
p(1,1)
)
−
−µ̄(1|1)
(j)
(
∂H
∂t
+
∂H
∂u
p(0,1) +
∂H
∂p(1,0)
p(1,1) +
∂H
∂p(0,1)
p(0,2)
)
,
where j ∈ J, |j| = j1 + j2 = 2. Taking into account now the expression (5.2), on gets that
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ON THE STRUCTURE OF CHARACTERISTIC SURFACES . . . 539
aH = µ
(1|1)
(2,0), āH = µ̄
(1|1)
(2,0),
b(0,1) = p(1,1) µ
(1|1)
(2,0) + p(0,2) µ̄
(1|1)
(2,0),
b(1,0) = p(2,0) µ
(1|1)
(2,0) + p(1,1) µ̄
(1|1)
(2,0),
cH = p(1,0) µ
(1|1)
(2,0) + p(0,1) µ̄
(1|1)
(2,0), (5.10)
b(2,0) = −µ(1|1)
(2,0)
(
+p2
(1,0) + up(2,0) + p(1,1)
)
− µ
(1|1)
(2,0)
(
+p(1,0) p(0,1) + up(1,1) + p(0,2)
)
,
b(1,1) = −µ(1|1)
(1,1)
(
+p2
(1,0) + up(2,0) + p(1,1)
)
,
b(0,2) = −µ(1|1)
(0,2)
(
+p2
(1,0) + up(1,0) + p(1,1)
)
− µ̄
(1|1)
(0,2)
(
+p(1,0)p(0,1) + up(1,1) + p(0,2)
)
.
For proceeding further put, for convenience,
µ
(1|1)
(0,2) = α, µ̄
(1|1)
(2,0) = 1,
∣∣∣µ(1|1)
(j)
∣∣∣
j 6=(0,2)
= 0 =
∣∣∣µ(1|1)
(j)
∣∣∣
j 6=(2,0)
. (5.11)
Then from (5.10), (5.11), and (5.6) one gets the system
dx
dτ
= α,
dt
dτ
= 1,
du
dτ
= αp(1,0) + p(0,1),
dp(1,0)
dτ
= αp(2,0) + p(1,1),
dp(0,1)
dτ
= αp(1,1) + p̄(0,2),
(5.12)
dp(2,0)
dτ
= −up(1,0) − p(0,1) + p̄(2,0),
dp(1,0)
dτ
= 0 =
dp(0,2)
dτ
,
dp(1,0)
dτ
= −αup(1,0) − αp(0,1) + α p̄(2,0) + p̄(1,1),
which reduces to the following three equations:
dp(1,0)
dτ
= −αup(0,1) − αp(0,1) + αp(2,0) + p̄(1,1),
(5.13)
dp(0,1)
dτ
= ap(1,1) + p̄(2,0),
du
dτ
= ap(1,0) + p(0,1),
where α ∈ C1(R×R+ ×R6; R), the quantities p(1,1) = p̄(1,1), p(0,2) = p̄(0,2) and p̄(2,0) ∈ R are
real constants not depending on the evolution parameter τ ∈ R. For solving the system (5.13)
we will make use of the ambiguous choice of the function α ∈ C1(R× R+ × R6; R). Namely, it
is convenient to put here
α =
c−
dp(1,0)
dτ
up(1,0) + p(0,1)
(5.14)
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540 N. K. PRYKARPATSKA
for all τ ∈ R and some c ∈ R. Then we easily deduce from (5.13) and (5.14) that simultaneously
one gets two relationships,
α =
c− p̄(1,1)
p̄(2,0)
:= −k (5.15)
where k ∈ R is some constant and
dp(1,0)
dτ
=
(
up(1,0) + p(0,1)
)
k + c,
du
dτ
= −kp(1,0) + p(0,1),
dp(0,1)
dτ
= −kp̄(1,1) + p̄(0,2).
(5.16)
From (5.16) we find that
p(0,1) =
(
p̄(0,2) − kp̄(1,1)
)
τ + p̄(0,1) := q (τ) ,
du
dτ
= −kp(1,0) + q(τ),
(5.17)
dp(1,0)
dτ
= (up(1,0) + p(0,1))k + c,
where q ∈ C1(R; R) is some smooth function. The latter two equations can be reduced to the
following one:
d
dτ
(
uτ −
ku2
2
)
= qτ − k2q − kuq + kc, (5.18)
which at the additional constraint q(τ) = 0 for all τ ∈ R gives rise to
uτ −
ku2
2
= kcτ + c0, (5.19)
where c0 ∈ R is constant, and owing to (5.15), (5.17),
k =
p̄(0,2)
p̄(1,1)
=
p̄(1,1) − c
p̄(2,0)
, p̄(0,1) = 0. (5.20)
Put now, by definition,
u = −2
k
d
dτ
lnψ, (5.21)
and substitute it into (5.19). We then find easily the second order linear ordinary differential
equation
ψττ +
2
k
(kcτ + c0)ψ = 0, (5.22)
being the standard Airy differential equation, whose solutions allow the following [16] integral
representation:
ψ = ψ̄0A i± (τ |c0, c, k) :=
ψ̄0
2π
∫
R
exp
[
± i
( c0
kc
+ τ
)
λ− 2λ3
3k2c
]
dλ, (5.23)
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ON THE STRUCTURE OF CHARACTERISTIC SURFACES . . . 541
with the norming constant parameter ψ̄0 ∈ C, and satisfying the boundary condition
lim
τ→+∞
ψ(τ) = 0 if ∓ c > 0.
Return now back to our vector fields (5.12), taking into account the results obtained above,
dx
dτ
= k, x|τ=0 = x0(x, t), x|τ=t = x, k = [x− x0(x, t)]/t,
(5.24)
u(x, t) = u0(x0(x, t), 0) +
t∫
0
du
dτ
dτ = u0(x0(x, t), 0) +
2
k
t∫
0
u2(τ)dτ +
kct2
2
+ c0t,
where the function u0 ∈ C1(R; R) must satisfy the compatibility condition like (4.9),
∂u0(x0(x, t)
∂x
−
〈
p̄(1,0)(x0(x, t)),
∂x0(x, t)
∂x
〉
= 0 (5.25)
for all points (x, t) ∈ R× R+. Meanwhile, from (5.23) we get the general solution to (5.22),
ψ(τ) = A i+ (τ |c0, c, k) + ψ̄0A i−(τ |c0, c, k), (5.26)
whence from (5.21),
u(τ) = −
2
(
A i′+(τ |c0, c, k) + ψ̄0A i
′
−(τ |c0, c, k)
)
k
(
Ai+(τ |c0, c, k) + ψ̄0A i−(τ |c0, c, k)
) (5.27)
for all τ ∈ R. As at τ = 0 one has u|τ=0 = u0(x0(x, t)), from (5.27) we derive the following
three relationships:
u0(x0(x, t)) =
2t
x0(x, t)− x
Ai′+
(
0|c0, c,
x− x0(x, t)
t
)
+ Ψ̄0Ai
′
−
(
0|c0, c,
x− x0(x, t)
t
)
Ai+
(
0|c0, c,
x− x0(x, t)
t
)
+ Ψ̄0Ai−
(
0|c0, c,
x− x0(x, t)
t
) ,
c0(x, t|x0) =
(x0(x, t)− x)u2
0(x0(x, t))
2t
− 2t
(x− x0(x, t))
d2
dτ2
lnΨ(τ)|τ=0, (5.28)
c(x, t|x0) =
2t2
(x0(x, t)− x)2
d3
dt3
lnΨ(τ)|τ=0 +
2u0(x0(x, t))
(x− x0(x, t))
d2
dτ2
lnΨ(τ)|τ=0,
supplying us with before undetermined three functional parameters c0 = c0(x, t|x0), c =
= c(x, t|x0) and ψ̄0 = ψ̄0(x, t|x0) at x0 = x0(x, t) ∈ R for all (x, t) ∈ R× R+. Thereby, we can
substitute these functional parameters into the expression (5.27) having put τ = t ∈ R+ and
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542 N. K. PRYKARPATSKA
obtain the following expression:
u(x, t) =
2t
x0 − x
Ai′+
(
t|c0(x, t|x0), c(x, t|x0),
x− x0
t
)
+
+ψ̄0(x, t|x0)Ai′−
(
t|c0(x, t|x0), c(x, t|x0),
x− x0
t
)
(
Ai+
(
t|c0(x, t|x0), c(x, t|x0),
x− x0
t
)
+
+ ψ̄0(x, t|x0)Ai−
(
t|c0(x, t|x0), c(x, t|x0),
x− x0
t
))
, (5.29)
which is a solution of the Cauchy problem to the Burgers nonlinear differential equation (5.1)
completely defined by the functional parameter x0 = x0(x, t) ∈ R for all (x, t) ∈ R× R+,
solving the before discussed ”inverse” Cauchy problem for the vector field (5.1). As this problem
is, evidently, very hard and cumbersome, we can make use of the previously obtained results
and state that the solution (5.29) owing to Theorem 4.1 allows the following Hopf – Lax-type
extremality representation:
u(x, t) = inf
y∈R
2t
y − x
(
Ai′+
(
t|c0(x, t|y), c(x, t|y),
x− y
t
)
+
+ ψ̄0(x, t|y)A i′−
(
t|c0(x, t|y), c(x, t|y),
x− y
t
))
(
Ai+
(
t|c0(x, t|y), c(x, t|y),
x− y
t
)
+
+ ψ̄0(x, t|y)A i−
(
t|c0(x, t|y), c(x, t|y),
x− y
t
))
. (5.30)
Thus, we can formulate the obtained result as the next final theorem.
Theorem 5.1. The inf-type expression (5.30) is the Hopf – Lax-type extremality solution to
the Cauchy problem (5.1) for the nonlinear Burgers equation.
The method used for finding the extremality solution (5.30) to the nonlinear Burgers equati-
on can be naturally applied to other nonlinear natural differential equations, including Korte-
weg – de Vries, nonlinear Schrödinger and other equations for which the problem of solving the
Cauchy problem represents serious difficulties.
Remark 5.1. It is useful here to make a remark concerning the linearization result (5.22) for
the solution (5.21) at τ = t ∈ R+. Namely, this result means that by means of the mapping
(5.21), written in the invariant form u =
d
dx
lnΨ, our nonlinear Burgers partial differential
equation (5.1) transforms into the standard linear partial differential equation Ψxx + Ψt = 0,
(x, t) ∈ R× R+, what is the classical Hopf – Cole result [15].
6. Conclusion. The results of the previous [12, 14] and this work convince us firmly that
the geometrical Monge approach to studying solution of a wide class of nonlinear partial di-
fferential equations of first and higher order, based on our generalized characteristic method, is
effective enough for many possible applications. The Hopf – Lax-type extremality representati-
on of the corresponding solutions of both boundary and Cauchy problems gives rise to fin-
ding many new, in some sense, generalized solutions for a wide class of boundary and Cauchy
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ON THE STRUCTURE OF CHARACTERISTIC SURFACES . . . 543
data. Another still weakly investigated aspect of this approach is related with its application to
analyzing suitable multidimensional symplectic reductions of boundary and Cauchy problems,
giving rise [17 – 20] to new types of associated purely Hamiltonian nonlinear dynamical systems
on functional manifolds of smaller spatial dimension. We plan to discuss this topic elsewhere.
Concerning the Burgers equation example (5.1), discussed in Section 5, we could see that
the developed generalized characteristic method works well also for nonlinear partial differenti-
al equations of higher order. We proved the classical Hopf – Cole result [15] about the lineari-
zation of the Burgers equation. The exact Hopf – Lax-type extremality solution (5.30) was rep-
resented here through the classical Airy function. In particular, we obtained as a by product a
little generalized linearization (5.21) and (5.22) of the Burgers equation (5.1), which can have
some additional applications. Similar results are also valid for another nonlinear partial di-
fferential equations of first and higher orders like Korteweg — de Vries, nonlinear Schrödinger
and other important nonlinear equations.
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Received 04.07.2005
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