In this paper, we extend the classical Lie symmetry analysis from partial differential equations to integro-differential equations with functional derivatives. We continue the work of OBERLACK and WACŁAWCZYK (2006, Arch. Mech., 58, 597), (2013, J. Math. Phys., 54, 072901) where the extended Lie symmetry analysis is performed in the Fourier space.
Here, we introduce a method to perform the extended Lie symmetry analysis in the physical space where we have to deal with the transformation of the integration variable in the appearing integral terms. The method is based on the transformation of the product y(x)dx appearing in the integral terms and applied to the functional formulation of the viscous Burgers equation.
The extended Lie symmetry analysis furnishes all known symmetries of the viscous Burgers equation and is able to provide new symmetries associated with the Hopf formulation of the viscous Burgers equation. Hence, it can be employed as an important tool for applications in continuum mechanics.
Abstract
In this paper, we extend the classical Lie symmetry analysis from partial differential equations to integro-
differential equations with functional derivatives. We continue the work of O
BERLACK
and W
ACLAWCZYK
(2006, Arch. Mech., 58, 597), (2013, J. Math. Phys., 54, 072901) where the extended Lie symmetry
analysis is performed in the Fourier space. Here, we introduce a method to perform the extended Lie
symmetry analysis in the physical space where we have to deal with the transformation of the inte-
gration variable in the appearing integral terms. The method is based on the transformation of the
product y
(x)dx appearing in the integral terms and applied to the functional formulation of the viscous
Burgers equation. The extended Lie symmetry analysis furnishes all known symmetries of the viscous
Burgers equation and is able to provide new symmetries associated with the Hopf formulation of the
viscous Burgers equation. Hence, it can be employed as an important tool for applications in continuum
mechanics.
1
Contents
1. Introduction
3
1.1. Three complete descriptions of turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2. Hopf functional and multi-point correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.2.1. Viscous Hopf-Burgers functional integro-differential equation . . . . . . . . . . . . .
6
2. Extension of the Lie symmetry analysis towards functional integro-differential equations
8
2.1. One-parameter Lie point transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.2. Differential operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.3. Infinitesimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.4. Infinitesimal generator, determining system of equations for the infinitesimals . . . . . . . .
14
2.5. Global transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
3. Lie symmetry analysis of the viscous Hopf-Burgers functional integro-differential equation 16
3.1. Three different approaches to Lie symmetry analysis . . . . . . . . . . . . . . . . . . . . . . .
16
3.2. Local transformations of the viscous Hopf-Burgers functional integro-differential equation
17
3.2.1. Determining system of equations for the infinitesimals . . . . . . . . . . . . . . . . . .
17
3.2.2. Solution of the determining system of equations for the infinitesimals . . . . . . . .
21
3.3. Symmetry breaking restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
4. Conclusions
29
5. Acknowledgments and contributions
30
A. Infinitesimals of the dependent variables
31
2
1 Introduction
1.1 Three complete descriptions of turbulence
Turbulence research knows three complete descriptions of turbulence. In each of these descriptions,
the aim is to calculate the statistics of a turbulent flow by determining the so-called multi-point velocity
correlations or shortly multi-point correlations
U
t
i
1
(x
1
) . . . U
t
i
n
(x
n
) , i
1
, . . . , i
n
= 1, 2, 3.
In this term,
U
t
is a random vector describing the velocity field in time t and x
1
, . . . , x
n
are positions of
different fluid particles in the space occupied by the fluid.
U
t
is given by
U
t
(x
i
) = U
t
1
(x
i
), U
t
2
(x
i
), U
t
3
(x
i
) = U
1
(x
i
, t
), U
2
(x
i
, t
), U
3
(x
i
, t
) .
In the language of stochastics, the multi-point correlations are the covariances of the velocity compo-
nents.
We briefly introduce the three complete descriptions of turbulence research.
· In the multi-point correlation approach an infinite dimensional chain of linear but non-local differ-
ential equations have to be solved. On the n-th level, the unknown
(n + 1)-point correlation is
present. Solving the infinitely many equations provides directly all multi-point correlations. In [3],
the Lie symmetries of the infinite set of multi-point correlation equations are investigated.
· In the Lundgren-Monin-Novikov approach [5] it is assumed that the velocity field
U
t
admit proba-
bility density functions (PDF's)
f
t
n
:
= f
t
(vvv
1
, . . . , vvv
n
; x
1
, . . . , x
n
)
given in terms of the Dirac delta distribution and describing the correlation of the velocity com-
ponents at multiple points in space. To be more precise, f
t
(vvv
1
; x
1
) dvvv
1
expresses the proba-
bility that the velocity vector
U
t
(x
1
) = U(x
1
, t
) is contained within the infinitesimal interval
[vvv
1
, vvv
1
+ dvvv
1
]. The Lundgren-Monin-Novikov hierarchy is an infinite dimensional chain of non-
local differential equations for the PDF's where on the n-th level the unknown
(n + 1)-point PDF
f
t
n
+1
= f
t
(vvv
1
, . . . , vvv
(n+1)
; x
1
, . . . , x
(n+1)
) is present. Solving the infinitely many equations provides
all PDF's. The multi-point correlations can be calculated by integrating the PDF's via
U
t
i
1
(x
1
) . . . U
t
i
n
(x
n
) =
3
· · ·
3
v
i
1
. . . v
i
n
f
t
(vvv
1
, . . . , vvv
n
; x
1
, . . . , x
n
) dvvv
1
. . . dvvv
n
.
In [4], the Lie symmetries of the Lundgren-Monin-Novikov hierarchy are investigated.
· In the Hopf approach the characteristic functions
of the PDF's f
t
for n
are investigated, cf.
Ref. [6]. The n-point characteristic function is defined as
(y
1
, . . . , y
n
, t
) := e
i
(
t
,y
)
=
3
· · ·
3
e
i
(vvv ,y)
f
t
(vvv
1
, . . . , vvv
n
; x
1
, . . . , x
n
) dvvv
1
. . . dvvv
n
3
with
t
:
= U
t
(x
1
), . . . , U
t
(x
n
) . (·, ·) is defined as the Euclidean scalar product
(vvv , y) :=
n
k
=1
3
i
=1
v
k
i
y
k
i
for vvv
= (vvv
1
, . . . , vvv
n
), y = (y
1
, . . . , y
n
) with vvv
k
, y
k
3
, k
= 1, . . . , n.
In hydromechanics, we assume that the mean free path is negligible, hence we take the continuum limit
n
. Thus, instead of the n velocity vectors vvv
1
, . . . , vvv
n
, we consider a continuous set
[vvv (x)] such
that the velocity vvv is a continuous function depending on the spatial variable x . In the continuum limit,
the probability density function f
t
becomes a probability density functional and the function
becomes
the functional
([y(x)], t) = e
i
(U
t
,y
)
=
e
i
(vvv ,y)
f
t
([vvv (x)]) d[vvv (x)].
(1.1)
In this paper, we pursuit the Hopf approach defined by equation (1.1). A relation between the Hopf
approach and the Lundgren-Monin-Novikov approach is discussed in Ref. [7] where also equations in
the Lagrangian multi-particle framework are investigated.
1.2 Hopf functional and multi-point correlations
In equation (1.1) it is not directly apparent what is meant by
(·, ·) and how the integration domain should
be chosen. In order to define
= ([y(x)], t) properly, we introduce the L
2
space.
Definition 1.2.1 (L
2
space). For G
3
define
L
2
(G,
3
) := vvv : G -
3
G
v
v
v
(x) · vvv (x) dx =
G
3
i
=1
v
i
(x)v
i
(x) dx
and equip L
2
(G,
3
) with (·, ·) defined by
(vvv , y) :=
G
v
v
v
(x) · y(x) dx =
G
3
i
=1
v
i
(x)y
i
(x) dx
(1.2)
for vvv , y
L
2
(G,
3
).
If we equip L
2
(G,
3
) with (·, ·) defined by equation (1.2), it is a Hilbert space, cf. Ref. [11].
Definition 1.2.2 (Hopf functional, Hopf functional differential equation). Let
y
L
2
(G,
3
) and let (·, ·)
be the L
2
scalar product defined by equation (1.2). The characteristic functional or Hopf functional is
defined by
([y(x)], t) := e
i
(U
t
,y
)
=
L
2
(G,
3
)
e
i
(vvv ,y)
f
t
([vvv (x)]) d[vvv (x)].
(1.3)
Instead of dealing with an infinite dimensional chain of differential equations, the Hopf approach
works with one scalar functional differential equation (FDE).
A FDE is a generalization of a partial differential equation (PDE). A PDE becomes a FDE if the number
of independent variables tends to infinity. Precisely:
Definition 1.2.3 (Functional, functional derivative, functional differential equation).
4
· Let
be a function space. A functional is a mapping
:
- .
· Let
= ([y(x)]) be a functional. We define
1
the functional derivative of
as the limit
([y(x)])
y
(z)dz
:
= lim
h
0
([y(x) + he
(x - z)]) - ([y(x)])
h
,
= 1, 2, 3.
Here,
e
denotes the
th
unit vector and
denotes the Dirac delta distribution. We denote the
functional derivative by
, y
(z)
=
y
(z)dz
=
y
(z)
.
· Let
= ([y(x )], x, t) be a functional. A functional differential equation (FDE) of order q is an
equation F
= 0 where F is a functional relating and all its derivatives up to order q, which can
include partial derivatives with respect to t, partial derivatives with respect to x and functional
derivatives with respect to each y
(x ), = 1, 2, 3:
F
([y(x )], x, t, ,
1
,
2
, . . . ,
q
) = 0.
(1.4)
In a FDE, the finite set
(y
1
, . . . , y
n
) (
3
)
n
is replaced by an infinite set
[y(x )]
with a continuous
parameter
x representing the continuum analogon of the discrete counting parameter k
= 1, . . . , n.
Hence, the dependent variables are functionals as they depend on functions.
It is important to note that we may obtain all multi-point correlations by differentiating
and evalu-
ating the derivatives at
y
= 0 via
U
t
1
(x
1
) . . . U
t
n
(x
n
) =
1
i
n
, y
n
(x
n
)...y
1
(x
1
)
y
=0
.
(1.5)
Consequently, the Hopf functional
provides the full statistical description of the velocity field U
t
since
all multi-point correlations can be expressed in terms of
and its functional derivatives.
In order to solve FDE's containing the Hopf functional
, so-called Hopf FDE's, we use an extension of
the classical Lie symmetry analysis. The extension is done in [1] where it is shown how the Lie symmetry
analysis is performed upon FDE's. The extended Lie symmetry analysis is applied to the Hopf formulation
of the viscous and inviscid Burgers equation in [2].
In order to circumvent the partial derivatives
/ x appearing both in the viscous and in the inviscid
Hopf-Burgers FDE, in [1] and [2] the extended Lie symmetry analysis is performed in the Fourier space by
considering the Fourier transform of the Hopf-Burgers FDE. In the Fourier space, derivatives with respect
to x become multipliers with ik. Hence, in Refs. [1] and [2] k was not transformed separately but was
treated as a continuous index. In [2] we also derived invariant solutions for the Hopf equations, however,
due to the restriction of the method (no transformation of k), the solutions contained unknown functions
of k governed by an infinite chain of equations. We note that the Hopf functional was also considered
in Refs. [8], [9] where, using numerical integration, scaling in k and decay of energy was calculated.
As far as symmetry methods are concerned, interesting aspects of energy scaling for turbulence were
discovered in [10] based on self-similar solutions of the Leith model.
1
In fact, this is a corollary of the definition of the Gâteaux derivative, cf. Ref. [12]. As the presented formula is more
suitable for calculations, we use it as definition.
5
In this paper, the extended Lie symmetry analysis is performed on the viscous Hopf-Burgers FDE in
the physical space. Hence, we have to deal with partial derivatives with respect to x. There are several
approaches how to do that. The approaches are introduced in section 3.1.
1.2.1 Viscous Hopf-Burgers functional integro-differential equation
Our primary long-term goal is to investigate the Hopf functional of turbulent velocity, i.e. the Hopf-
Navier-Stokes FDE. For the sake of convenience, we presently restrict ourselves to the one-dimensional
case, i.e. replace
3
by
. Instead of the incompressible Navier-Stokes equations, we use the viscous
Burgers equation to derive the so-called viscous Hopf-Burgers FDE. We assume that y
, U
t
L
2
(G, ) with
y
= y(x),
U
t
= U
t
(x) = U(x, t)
and that U
t
fulfills the viscous Burgers equation
U
t
t
+ U
t
U
t
x
=
2
U
t
x
2
which can equivalently by written as
U
t
t
+
1
2
(U
t
)
2
x
=
2
U
t
x
2
.
By simple rescaling we may eliminate the factor
1
/2. We call the following equation viscous Burgers
equation:
U
t
t
+
(U
t
)
2
x
=
2
U
t
x
2
.
(1.6)
We differentiate the Hopf functional (1.3) with respect to t to get
,t
([y(x)], t) =
t
e
i
(U
t
, y
)
= i
U
t
t
, y e
i
(U
t
, y
)
= i -
(U
t
)
2
x
+
2
U
t
x
2
e
i
(U
t
, y
)
(1.7)
where equation (1.6) was inserted in order to eliminate
U
t
/ t. Using the definition of the character-
istic functional (1.1), the following relations can be calculated:
-
(U
t
)
2
x
e
i
(U
t
, y
)
=
x
2
y(x) y(x)
=
,x y
(x)y(x)
,
i
2
U
t
x
2
e
i
(U
t
, y
)
=
x
2
y(x)
=
,x x y
(x)
.
After introducing the above formulas into equation (1.7), we finally obtain the viscous Hopf-Burgers FDE
,t
([y(x)], t) =
G
y
(x) i
,x y
(x)y(x)
+
,x x y
(x)
dx
(1.8)
with y
L
2
(G, ). Since is part of the integrand, the viscous Hopf-Burgers FDE is an integro-
differential equation.
In the following, we want to perform the extended Lie symmetry analysis on the viscous Hopf-Burgers
FDE. Before doing that, we review the extended Lie symmetry analysis developed in [1] and [2] and
advance it for our purposes.
6
This paper is structured as follows: In the second section, we present an extension of the classical Lie
symmetry analysis based on the two papers [1] and [2] which allows us to analyze functional equations
in the physical space. The third section presents the main part of the paper: We apply the extended Lie
symmetry analysis to the viscous Hopf-Burgers functional integro-differential equation. First, we present
three different approaches to perform the extended Lie symmetry analysis on the viscous Hopf-Burgers
FDE. Subsequently, we solve the system of equations for the infinitesimals. Then, we discuss symmetry
breaking restrictions and indicate physically relevant symmetries. In the end of the third section, we
compare the symmetries of the viscous Hopf-Burgers FDE with the symmetries of the viscous Burgers
equation and calculate the associated global transformations. Finally, a conclusion and perspectives are
given in the fourth section.
7
2 Extension of the Lie symmetry analysis
towards functional integro-differential
equations
2.1 One-parameter Lie point transformations
To start with, we introduce some basic notions, cf. Ref. [1]. As in this paper we consider the viscous
Hopf-Burgers FDE being of third order, we set q
= 3 in equation (1.4). Furthermore, as we use the viscous
Burgers equation as the underlying equation for fluid motion instead of the incompressible Navier-Stokes
equations, we replace
3
by
. The whole theory can easily be extended to higher dimensions and to
differential equations of arbitrary order q.
In Lie symmetry analysis, we only consider continuous symmetry transformations depending on a
continuous parameter
S where S is a Lie group. We restrict ourselves to so-called one-paramater Lie
point transformations, cf. Ref. [14]:
Definition 2.1.1 (One-parameter Lie point transformations of PDE's). Consider the PDE
F
(y
1
, . . . , y
n
, x
1
, . . . , x
n
, t,
,
1
,
2
, . . . ,
3
) = 0
and let
y
i
= y
i
(y
1
, . . . , y
n
, x
1
, . . . , x
n
, t,
, ),
i
= 1, . . . , n,
x
i
= x
i
(y
1
, . . . , y
n
, x
1
, . . . , x
n
, t,
, ),
i
= 1, . . . , n,
t
= t(y
1
, . . . , y
n
, x
1
, . . . , x
n
, t,
, ),
= (y
1
, . . . , y
n
, x
1
, . . . , x
n
, t,
, )
be the transformed variables. The transformation is called one-parameter Lie point transformation if and
only if the transformed variables are given by
y
i
= y
i
+
y
i
(y
1
, . . . , y
n
, x
1
, . . . , x
n
, t,
) ,
i
= 1, . . . , n,
x
i
= x
i
+
x
i
(y
1
, . . . , y
n
, x
1
, . . . , x
n
, t,
) ,
i
= 1, . . . , n,
t
= t +
t
(y
1
, . . . , y
n
, x
1
, . . . , x
n
, t,
) ,
= +
(y
1
, . . . , y
n
, x
1
, . . . , x
n
, t,
) .
In a one-parameter Lie point transformation, the derivatives of
are not transformed separately. The
transformations of
1
,
2
, . . . ,
q
are calculated as functions of
y
1
, . . . ,
y
n
,
x
1
, . . . ,
x
n
,
t
,
. We write
,t
=
,t
+
;t
(y
1
, . . . , y
n
, x
1
, . . . , x
n
, t,
,
1
) ,
, y
i
=
, y
i
+
; y
i
(y
1
, . . . , y
n
, x
1
, . . . , x
n
, t,
,
1
) ,
i
= 1, . . . , n,
, y
i
y
j
=
, y
i
y
j
+
; y
i
y
j
(y
1
, . . . , y
n
, x
1
, . . . , x
n
, t,
,
1
,
2
) ,
i
, j
= 1, . . . , n,
..
.
8
The functions
y
i
,
x
i
,
t
,
,
;...
are called infinitesimals. Notice that we separate indices of
by a
semicolon to distinguish them from derivatives. The transformations are expanded in a Taylor series
(T f )
0
( ) :=
k
=0
1
k
!
k
f
k
=
0
( -
0
)
k
(2.1)
about
0
= 0, hence the infinitesimals are defined by
y
i
:
=
y
i
=0
,
x
i
:
=
x
i
=0
,
t
:
=
t
=0
,
:
=
=0
,
;t
:
=
,t
=0
,
; y
i
:
=
, y
i
=0
,
. . . .
We extend this definition to FDE's.
Definition 2.1.2 (One-parameter Lie point transformations of FDE's). Consider the FDE
F
([y(x )], x, t, ,
1
,
2
,
3
) = 0
and let
y
(z)dz = y(z)dz([y(x )], z, x, t, , ),
x
, z
G,
x
= x([y(x )], x, t, , ),
x
G,
t
= t([y(x )], x, t, , ),
x
G,
= ([y(x )], x, t, , ),
x
G,
be the transformed variables where G
is an uncountable set. The transformation is called one-
parameter Lie point transformation if and only if the transformed variables are given by
y
(z)dz = y(z)dz +
y
(z)
dz
([y(x )], z, x, t, ) ,
x
, z
G,
x
= x +
x
([y(x )], x, t, ) ,
x
G,
t
= t +
t
([y(x )], x, t, ) ,
x
G,
= +
([y(x )], x, t, ) ,
x
G.
Again, the derivatives of
are not transformed separately. The transformations of
1
,
2
,
3
are calcu-
lated as functionals depending on
y
(z)
dz,
x
,
t
,
. We write
,t
=
,t
+
;t
([y(x )], x, t, ,
1
) ,
x
G,
, y
(z
1
)
=
, y
(z
1
)
+
; y
(z
1
)
([y(x )], z
1
, x, t,
,
1
) ,
x
, z
1
G,
, y
(z
2
) y(z
1
)
=
, y
(z
2
)y(z
1
)
+
; y
(z
2
)y(z
1
)
([y(x )], z
1
, z
2
, x, t,
,
1
,
2
) ,
x
, z
1
, z
2
G,
..
.
The functionals
y
(z)
dz,
x
,
t
,
,
;t
, . . . are called infinitesimals. Again, we separate indices of
by
a semicolon to distinguish them from derivatives. The transformations are expanded in a Taylor series
(2.1) about
0
= 0, hence the infinitesimals are defined by
y
(z)
dz :
=
y(z)dz
=0
,
x
:
=
x
=0
,
t
:
=
t
=0
,
:
=
=0
,
;t
:
=
,t
=0
,
; y
(z)
:
=
, y
(z)
=0
,
. . . .
9
Comparing the notion of one-parameter Lie point transformations of PDE's and of FDE's, one sees the
considered transformed functions y
i
, x
i
, t,
are replaced by functionals y(z)dz, x, t, depending on
the infinite set
[y(x )]. Notice that we do not transform y(z) but y(z)dz as
G
y
(z) dz is the continuum analogon of
n
i
=1
y
i
.
Following Ref. [16], the variable y
(z)dz can be represented as a test series
y
(z)dz =
n
=1
y
n
h
n
(z)dz.
(2.2)
where
{h
n
(z)}
n
is a set of orthogonal functions. In this respect, the transformed variable y
(z)dz reads
y
(z)dz =
n
=1
y
n
h
n
(z)dz.
(2.3)
A possible approach in extending the Lie group analysis would be to account for the transformations
of y
n
and z separately. However, due to the presence of functional derivatives in equation (1.8), i.e.
derivatives with respect to y
(z)dz, we prefer to consider y(z)dz as a variable to be transformed. Still,
the decomposition (2.2) will be taken into account in the definitions of the infinitesimals in section 2.3.
An additional option would be to transform y
(z) instead of y(z)dz. Then, one has to take into consid-
eration the transformation of
dz. There are two ways how this could be done, cf. section 3.1; however,
we do not pursuit those methods here. In order to be consistent, the infinitesimal associated with the
transformation y
(z)dz is called
y
(z)
dz instead of
y
(z)
. The infinitesimal
y
(z)
dz has to depend ex-
plicitly on z as it defines a new variable for each z
G. Analogous considerations hold true for the
transformations of the functional derivatives of the dependent variable
,...
and their infinitesimals
;...
.
2.2 Differential operators
In the classical Lie symmetry analysis, the variables
(y
1
, . . . , y
n
, x
1
, . . . , x
n
, t,
,
1
,
2
,
3
) are treated
as independent variables.
Hence, in the extended Lie symmetry analysis, we treat the variables
([y(x )], x, t, ,
1
,
2
,
3
) as independent variables. The Hopf functional does not depend explicitly
on x, hence all derivatives of
with respect to x vanish except when is derived first with respect
to y
(x ) and then with respect to x. Neglecting the vanishing summands and considering the non-
commutativity
2
x y(x)dx
=
2
y(x)dx x
,
we can define:
Definition 2.2.1 (Differential operators for Hopf FDE's). For a Hopf FDE of third order
F
([y(x )], x, t, ,
1
,
2
,
3
) = 0
10
we introduce
t
=
t
+
,t
+
,t t
,t
+
G
,t y
(x )
dx
, y
(x )
dx
+
,t t t
,t t
+
G
,t t y
(x )
dx
,t y
(x )
dx
+
G
,t x y
(x )
dx
,x y
(x )
dx
+
G
G
,t y
(x )y(x )
dx dx
, y
(x )y(x )
dx dx
,
x
=
x
+
G
,x y
(x )
dx
, y
(x )
dx
+
G
,x t y
(x )
dx
,t y
(x )
dx
+
G
,x x y
(x )
dx
,x y
(x )
dx
+
G
G
,x y
(x )y(x )
dx dx
, y
(x )y(x )
dx dx
,
y
(z)dz
=
y(z)dz
+
, y
(z)
+
, y
(z)t
,t
+
G
, y
(z)y(x )
dx
, y
(x )
dx
+
, y
(z)t t
,t t
+
G
, y
(z)t y(x )
dx
,t y
(x )
dx
+
G
, y
(z)x y(x )
dx
,x y
(x )
dx
+
G
G
, y
(z)y(x )y(x )
dx dx
, y
(x )y(x )
dx dx
.
2.3 Infinitesimals
By means of the differential operators presented in definition 2.2.1, we can calculate the infinitesimals
;...
as functionals of the infinitesimals
y
(z)
dz,
x
,
t
,
. For the viscous Hopf-Burgers FDE, we need the
three infinitesimals
;t
,
;x y
(x)y(x)
and
;x x y
(x)
.
· In order to calculate
;t
, we differentiate the transformed Hopf functional
with respect to t
taking into account the decomposition (2.2):
t
=
t
t
t
+
x
x
t
+
G
y
(x )dx
n
=1
y
n
t
h
n
(x )dx .
(2.4)
Hence, in the definition above we account for the fact that t, x and the infinite set
{ y
n
}
n
are
the independent variables. Still, as argued in section 2.1, in the extended Lie group analysis we
transform the continuum variable y
(x)dx, hence we will express the last RHS term in equation
(2.4) in terms of y
(x)dx. This step is crucial for the further successful recovery of symmetries of
the Burgers equation. Using (2.2) again, the time differential operator applied to y
(x )dx can be
decomposed into
y
(x)dx
t
=
n
=1
y
n
t
h
n
(x)dx +
n
=1
y
n
h
n
(x)dx
t
.
(2.5)
The second RHS term can be rewritten as
n
=1
y
n
t
h
n
(x)dx =
y
(x)dx
t
-
y
(x)dx
x
x
t
(2.6)
11
and substituted into equation (2.4). Taking into account the fact that
does not depend explicitely
on x, from equation (2.4) we obtain
t
=
t
t
t
+
G
y
(x )dx
y
(x )dx
t
-
G
y
(x )dx
x
t
y
(x )dx
x
.
(2.7)
We integrate the last term by parts and assume that
and its derivatives are zero at the boundaries.
This step is necessary, as we want to introduce Lie point transformations, cf. definition 2.1.2, for
the quantities in the integral. We finally obtain
t
=
t
t
t
+
G
y
(x )dx
y
(x )dx
t
+
G
x
y
(x )dx
x
t
y
(x )dx .
(2.8)
With the one-parameter Lie point transformations, cf. definition 2.1.2, equation (2.8) reads
( +
)
t
= (
,t
+
;t
)
(t +
t
)
t
+
G
(
, y
(x )
+
; y
(x )
)
(y(x )dx +
y
(x )
dx
)
t
+
G
(
,x y
(x )
+
;x y
(x )
)
(x +
x
)
t
y
(x )dx +
y
(x )
dx .
(2.9)
Evaluating this equation in
(1), we get
t
=
,t
.
Using this, evaluating in
( ) leads to
t
=
,t
t
t
+
;t
+
G
, y
(x )
y
(x )
dx
t
+
G
,x y
(x )
x
t
y
(x )dx
which constitutes an equation for
;t
:
;t
=
t
-
,t
t
t
-
G
, y
(x )
y
(x )
dx
t
-
G
,x y
(x )
x
t
y
(x )dx .
Since the appearing infinitesimals
y
(x )
dx ,
t
,
do not depend on derivatives of
, we immedi-
ately get
;t
=
t
+
,t
-
t
t
- (
,t
)
2
t
-
G
, y
(x )
y
(x )
dx
t
-
G
, y
(x )
,t
y
(x )
dx
-
G
,x y
(x )
x
t
y
(x )dx -
G
,x y
(x )
,t
x
y
(x )dx .
(2.10)
· In order to calculate
; y
(x)
, we differentiate
with respect to y(x). An analog calculation leads to
; y
(x)
=
y
(x)dx
-
,t
t
y
(x)dx
-
G
, y
(x )
y
(x )
dx
y
(x)dx
-
,x y
(x )
x
y
(x)dx
y
(x )dx
=
y(x)dx
+
, y
(x)
-
,t
t
y(x)dx
-
,t
, y
(x)
t
-
G
, y
(x )
y
(x )
dx
y(x)dx
-
G
, y
(x )
, y
(x)
y
(x )
dx
-
,x y
(x )
x
y(x)dx
y
(x )dx -
,x y
(x )
, y
(x)
d
x
y
(x )dx .
12
In order to calculate
; y
(x)y(x)
, we differentiate
, y
(x)
with respect to y
(x). As
, y
(x)
does depend
on x, we have
, y
(x)
y
(x)dx
=
, y
(x)
t
t
y
(x)dx
+
G
, y
(x)
y
(x )dx
y
(x )dx
y
(x)dx
-
y
(x )dx
x
x
y
(x)dx
+
, y
(x)
x
x
y
(x)dx
=
,t y
(x)
t
y
(x)dx
+
G
, y
(x ) y(x)
y
(x )dx
y
(x)dx
+
G
,x y
(x ) y(x)
x
y
(x)dx
y
(x )dx
+
,x y
(x)
x
y
(x)dx
.
If we insert the one-parameter Lie point transformations, cf. definition 2.1.2, we get
; y
(x)y(x)
=
; y
(x)
y
(x)dx
-
,t y
(x)
t
y
(x)dx
-
G
, y
(x )y(x)
y
(x )
dx
y
(x)dx
-
,x y
(x)
x
y
(x)dx
-
G
,x y
(x )y(x)
x
y
(x)dx
y
(x )dx .
In order to get
;x y
(x)y(x)
, we differentiate
, y
(x) y(x)
with respect to x:
, y
(x) y(x)
x
=
, y
(x) y(x)
t
t
x
+
G
, y
(x) y(x)
y
(x )dx
y
(x )dx
x
+
, y
(x) y(x)
x
x
x
=
,t y
(x) y(x)
t
x
+
G
, y
(x ) y(x) y(x)
y
(x )dx
x
+
,x y
(x) y(x)
x
x
.
If we insert the one-parameter Lie point transformations, cf. definition 2.1.2, we get
;x y
(x)y(x)
=
; y
(x)y(x)
x
-
,t y
(x)y(x)
t
x
-
G
, y
(x )y(x)y(x)
y
(x )
dx
x
-
,x y
(x)y(x)
x
x
. (2.11)
· In order to calculate
;x y
(x)
, we differentiate
, y
(x)
with respect to x. We have
, y
(x)
x
=
, y
(x)
t
t
x
+
G
, y
(x)
y
(x )dx
y
(x )dx
x
+
, y
(x)
x
x
x
=
,t y
(x)
t
x
+
G
, y
(x ) y(x)
y
(x )dx
x
+
,x y
(x)
x
x
.
If we insert the one-parameter Lie point transformations, cf. definition 2.1.2, we get
;x y
(x)
=
; y
(x)
x
-
,t y
(x)
t
x
-
G
, y
(x )y(x)
y
(x )
dx
x
-
,x y
(x)
x
x
.
In order to get
;x x y
(x)
, we differentiate
,x y
(x)
with respect to x:
,x y
(x)
x
=
,x y
(x)
t
t
x
+
G
,x y
(x)
y
(x )dx
y
(x )dx
x
+
,x y
(x)
x
x
x
=
,t x y
(x)
t
x
+
G
, y
(x ) x y(x)
y
(x )dx
x
+
,x x y
(x)
x
x
.
13
If we insert the one-parameter Lie point transformations, cf. definition 2.1.2, we get
;x x y
(x)
=
;x y
(x)
x
-
,t x y
(x)
t
x
-
G
, y
(x )x y(x)
y
(x )
dx
x
-
,x x y
(x)
x
x
.
(2.12)
Applying the differential operators introduced in definition 2.2.1, one can represent the infinitesimals
(2.11) and (2.12) as sums of partial and functional derivatives of
y
(z)
dz,
x
,
t
,
, cf. appendix A.
2.4 Infinitesimal generator, determining system of equations for the infinitesimals
Let F
([y(x )], x, t, ,
1
,
2
,
3
) = 0 be a transformed FDE. Consider F as a function depending on the
group parameter
and expand F in a Taylor series (2.1) about
0
= 0, i.e. consider the equation
F
=0
+
F
=0
+ (
2
) = 0.
(2.13)
We calculate
F
=0
=
G
y
(x )
dx
y(x )dx
+
x
x
+
t
t
+
+
;t
,t
+
G
; y
(x )
dx
, y
(x )
dx
+ . . . F
and define
Definition 2.4.1 (Infinitesimal generator and its prolongation).
· The differential operator
X
:
=
G
y
(x )
dx
y(x )dx
+
x
x
+
t
t
+
is called infinitesimal generator or simply generator.
· The differential operator
X
(3)
:
= X +
;t
,t
+
G
; y
(x )
dx
, y
(x )
+
;t t
,t t
+
G
;t y
(x )
dx
,t y
(x )
+
G
;x y
(x )
dx
,x y
(x )
+
G
G
; y
(x )y(x )
dx dx
, y
(x )y(x )
+
;t t t
,t t t
+
G
;t t y
(x )
dx
,t t y
(x )
+
G
G
;t y
(x )y(x )
dx dx
,t y
(x )y(x )
+
G
;t x y
(x )
dx
,t x y
(x )
+
G
G
G
; y
(x )y(x )y(x )
dx dx dx
, y
(x )y(x )y(x )
+
G
G
; y
(x )x y(x )
dx dx
, y
(x )x y(x )
+
G
G
;x y
(x )y(x )
dx dx
,x y
(x )y(x )
+
G
;x x y
(x )
dx
,x x y
(x )
is called prolongation of X .
14
Using this definition and employing F
=0
= F, equation (2.13) reads
F
+ X
(3)
F
+ (
2
) = 0.
This equation is fulfilled in
( ) if and only if
X
(3)
F
F
=0
= 0.
(2.14)
Equation (2.14) constitutes an overdetermined system of linear FDE's for the infinitesimals
y
(z)
dz,
x
,
t
,
. In order to formulate the system of equations, one has to insert the necessary infinitesimals,
calculated in section 2.3, into equation (2.14). Since
y
(z)
dz,
x
,
t
,
do not depend on derivatives
of
, all coefficients of all appearing derivatives of have to vanish which leads to a system of linear
FDE's.
2.5 Global transformations
Knowing
y
(z)
dz,
x
,
t
,
, one obtains the global transformations using Lie's first theorem, cf. Ref.
[14]:
Theorem 2.5.1 (Lie's first theorem). The global tansformation can be obtained by solving the following
initial value problems:
y(z)dz
=
y
(z)
dz,
x
=
x
,
t
=
t
,
=
with the initial values
y
(z)dz( = 0) = y(z)dz,
x
( = 0) = x,
t
( = 0) = t,
( = 0) = .
15
3 Lie symmetry analysis of the viscous
Hopf-Burgers functional integro-differential
equation
3.1 Three different approaches to Lie symmetry analysis
As already mentioned in the introduction, when working in the physical space, the main problem is to
deal with the partial derivatives
/ x and the transformation of the integration variable x. Different
Lie symmetry analysis approaches for integro-differential equations have been proposed in literature, cf.
Ref. [13]. In the following, we present three methods which could possibly be applied in our case of
functional equations with functional derivatives.
1. We transform y
(z) instead of y(z)dz and have to take into account the transformation of the
integral term appearing in Hopf FDE's. In [14] and [15], I
BRAGIMOV
suggests to use the fact that
X given by definition 2.4.1 is equivalent to a canonical Lie-Bäcklund operator ~
X which does not
contain the term
x
/ x. This implies ~X is very suitable for the symmetry analysis of integro-
differential equations. Hence, one might replace X by ~
X and perform the extended Lie symmetry
analysis on functional integro-differential equations.
2. We transform y
(z) instead of y(z)dz and consider the differential equation as an equation F = 0
where F depends on an integral term I and an integral-free term H, i.e. F
(H, I) = 0. In order
to get the correct determining system of equations for the infinitesimals, the transformation of
F is expanded in a two-dimensional Taylor series about H and I . This method is presented by
Z
AWISTOWSKI
in [17].
In [18] it is shown that using Z
AWISTOWSKI
's approach leads to results being equivalent to the
results presented in I
BRAGIMOV
's study and that the Lie algebra of symmetry group transforma-
tions spanned by the infinitesimal generators containing integral terms is solvable, cf. Ref. [17]
for the Vlasov-Maxwell integro-differential equation and [18] for the Benney integro-differential
equations. Z
AWISTOWSKI
's approach leads to the following determining system of equations for the
infinitesimals
y
(z)
dz,
x
,
t
,
:
(3)
F
-
G
x
x
f
dx
F
=0
= 0.
One has to pay attention that an analogous formula for the transformed integral should be used
during the calculation of
;t
,
;x y
(x)y(x)
and
;x x y
(x)
, cf. section 2.3, in order to determine the
generator
(3)
given by definition 2.4.1, which makes this approach more complicated in our
particular case of an equation with functional derivatives.
3. We transform y
(z)dz instead of y(z) and introduce a transformation of x. Then, we perform the
extended Lie symmetry analysis on the viscous Hopf-Burgers FDE (1.8). Presently, this approach is
succesfully applied to (1.8) and we rediscover all known symmetries of the usual viscous Burgers
equation (1.6). The results and a discussion comparing the symmetries of the viscous Hopf-Burgers
FDE and the symmetries of the viscous Burgers equation are presented in the following subsections.
16
3.2 Local transformations of the viscous Hopf-Burgers functional integro-differential equation
3.2.1 Determining system of equations for the infinitesimals
In this section, we perform the extended Lie symmetry analysis on the viscous Hopf-Burgers FDE
F
=
,t
-
G
y
(x) i
,x y
(x)y(x)
+
,x x y
(x)
dx
= 0.
We consider the one-parameter Lie point transformations given by definition 2.1.2.
In order to calculate the determining system of equations for the infinitesimals
y
(z)
dz,
x
,
t
,
, we
consider equation (2.14):
X
(3)
F
F
=0
= 0.
The generator X
(3)
is given by definition 2.4.1. The contributing summands of X
(3)
are given by
X
(3)
modif
=
G
y
(x )
dx
y(x )dx
+
;t
,t
+
G
G
;x y
(x )y(x )
dx dx
,x y
(x )y(x )
dx dx
+
G
;x x y
(x )
dx
,x x y
(x )
dx
.
Applying X
(3)
modif
to F , we get the equation
X
(3)
modif
F
=
;t
-
G
y
(x)
dx i
,x y
(x)y(x)
+
,x x y
(x)
-
G
y
(x) i
;x y
(x)y(x)
dx
+
;x x y
(x)
dx
= 0. (3.1)
We insert the infinitesimals
;t
(cf. equation (2.10)),
;x y
(x)y(x)
and
;x x y
(x)
(cf. appendix A) and employ
F
= 0 in order to eliminate
,t
.
In the extended Lie symmetry analysis we deal with different types of derivatives. Inside the integral
the functional derivatives
/ y(x) are present. It was argued that these derivatives correspond to
partial derivatives
/ y
k
in the discrete case, cf. section 2.1. Additionally, in (1.8), partial derivatives
with respect to x are present, consider
,x y
(x)y(x)
and
,x x y
(x)
, which do not have a correspondence in
the discrete case. Also, different types of mixed functional-partial derivatives are present in the final
formula (3.1). In the further procedure, in order to obtain proper symmetry transformations, for the
mixed derivatives we have to assume that
·
, y
(x)
is not independent of
,x y
(x)
,
·
,t y
(x)
is not independent of
,t x y
(x)
,
·
, y
(x)y(z)
is not independent of
,x y
(x)y(z)
etc. Hence, in the final step, we integrate by parts in order to remove the x-derivatives from the func-
tional derivatives of
. In order to do this, we have to eliminate the boundary integrals. One may choose
between the following two options:
· The first option is to demand that G
is bounded: G
= (a, b) with - a b . If this
holds true, one has to demand additionally that all appearing terms evaluated at x
= b equal to
the same terms evaluated at x
= a. Then all boundary integrals vanish. As this demands a huge
number of restrictions, we do not choose G to be bounded. Instead, we choose the second option.
17
· The second option is to demand that G
is not bounded. We restrict ourselves to the case
G
= (-, +). If this holds true, one may impose the condition
U
t
(x = ±) = 0.
(3.2)
As equation (1.5) states
, y
(x)
= i U
t
(x)e
i
(U
t
, y
)
,
we have
, y
(x)
x
=±
= 0.
Additionally, we impose that all functional derivatives of
vanish for x ±, i.e.
,t y
(x)
x
=±
= 0,
,x y
(x)
x
=±
= 0,
, y
(x)y(x)
x
=±
= 0,
. . .
Hence, all appearing boundary integrals vanish. For example,
G
y
(x)
dx
,x y
(x)y(x)
= -
G
y
(x)
dx
x
, y
(x)y(x)
+
y
(x)
, y
(x)y(x)
x
=+
x
=-
=0
.
The resulting equation (3.1) has the form
0
=
+
G
, y
(z)
dz
+
G
G
, y
(x)y(z)
dzdx
+
G
G
, y
(x)
, y
(z)
dzdx
+
G
G
G
, y
(x)y(a)
, y
(z)
dadzdx
+
G
G
(
, y
(x)
)
2
, y
(z)
dzdx
+
G
,t y
(x)
dx
+
G
G
(
, y
(x)
)
2
, y
(z)y(z)
dzdx
+
G
, y
(x)
,t y
(x)
dx
+
G
,t y
(x)y(x)
dx
+
G
G
, y
(z)y(x)y(x)
dzdx
+
G
G
, y
(x)y(x)
, y
(z)y(z)
dzdx.
Since the infinitesimals
y
(z)
dz,
x
,
t
,
do not depend on derivatives of
, all coefficients of all
appearing derivatives of
have to vanish:
=
=
= = =
=
=
=
=
=
=
= 0.
This leads to the following system of linear FDE's for the infinitesimals
y
(z)
dz,
x
,
t
,
:
·
= 0 reads
t
-
G
y
(x) i
3
x (y(x)dx)
2
+
3
x
2
y(x)dx
dx
= 0,
(3.3)
18
·
= 0 reads
0
=
2
z
2
y
(z)
-
2
z
2
y
(z)
t
t
-
y
(z)
t
-
2
y
(z)
z
2
- 2i y(z)
3
z y(z)dz
+ i
G
2
z
2
y
(x)y(z)
3
t
x (y(x)dx)
2
dx
+ i
G
y
(x)
3
y
(z)
x (y(x)dx)
2
dx
- i
z
y
(z)
2
z
z y(z)dz
+ 2i
z
y
(z)
2
y(z)dz
- i
G
x
y
(x)
2
y
(z)
(y(x)dx)
2
dx
+ i
2
z
2
y
(z)
z
y(z)dz
+
2
G
2
z
2
y
(x)y(z)
3
t
x
2
y(x)dx
dx
+
G
y
(x)
3
y
(z)
x
2
y(x)dx
dx
- 2
G
x
y
(x)
2
y
(z)
x y(x)dx
dx
-
z
y
(z)
2
z
z
2
-
2
z
2
y
(z)
+
G
2
x
2
y
(x)
y
(z)
y(x)dx
dx
+ 2
2
z
2
y
(z)
z
z
+ y (z)
z
t
+ y(z)
2
z
x t
+ i
2
z
2
y
(z)
z
y(z)dz
- i
z
y
(z)
2
z
z y(z)dz
+ i
G
2
z x
y
(z)y(x)
2
z
y(x)dx y(x)dx
dx
- i
G
y
(x)
z
y
(z)
3
z
x( y(x)dx)
2
dx
-
G
y
(z)y(x)
3
z
x
2
y(x)dx
dx
-
G
y
(z)y(x)
4
z
z x
2
y(x)dx
dx
-
G
3
x
2
z
y
(z)y(x)
z
y(x)
dx
+ 2
G
2
x z
y
(z)y(x)
z
x y(x)dx
dx,
(3.4)
·
= 0 reads
0
= -i
x
y
(x)
(x - z) + i
x
y
(x)
t
t
(x - z) + i
y
(x)
x
(x - z)
+
G
z
y
(a)y(z)
3
t
a (y(a)da)
2
(x - z) da + 2i y(x)
2
y
(z)
x y(x)dx
- 2i
x
y
(x)
y
(z)
y(x)dx
+ i
x
y
(x)
(x - z) - i
x
y
(x)
x
x
(x - z)
- i
G
z
y
(a)y(z)
3
t
a
2
y(a)da
(x - z) da + y(x)
3
y
(z)
x
2
+ y(x)
2
y
(z)
x
2
-
x
y
(x)
y
(z)
x
-
x
y
(x)
y
(z)
x
- 2i y(x)
z
y
(z)
2
z
x y(x)dx
+ 2i
2
x z
y
(x)y(z)
z
y(x)dx
,
(3.5)
19
·
= 0 reads
0
= -
2
4
z
2
x
2
y
(x)y(z)
t
-
2
x
2
y
(x)
y
(z)
+ 2i
2
z
2
y
(x)y(z)
3
t
x y(x)dx
+ 2i y(x)
3
y
(z)
x y(x)dx
- i
x
y
(x)
2
x
x
(x - z) - 2i
3
z
2
x
y
(x)y(z)
2
t
y(x)dx
- 2i
x
y
(x)
2
y
(z)
y(x)dx
(x - z) - 2i
x
y
(x)
2
y
(z)
y(x)dx
+ 2i
x
y
(x)
2
2
(x - z) + i
2
x
2
y
(x)
x
(x - z) + i
2
x
2
y
(x)
x
(x - z)
-
x
y
(x)
2
y
(z)
x
- 2
x
y
(x)
2
y
(z)
x
-
x
y
(x)
2
y
(z)
x
+ 2
2
x
2
y
(x)
y
(z)
+
2
4
z
2
x
2
y
(x)y(z)
t
+
2
x
2
y
(x)
y
(z)
+
2
x
2
y
(x)
y
(z)
- 2
2
3
z
2
x
y
(x)y(z)
2
t
x
+
2
2
z
2
y
(x)y(z)
3
t
x
2
+ i
2
x
2
y
(x)
x
(x - z) + i
2
x z
y
(x)
x
(x - z) - i
x
y
(x)
2
x
x
(x - z)
- i
z
y
(x)y(z)
3
z
x y(x)
+ 2i
2
z x
y
(x)y(z)
2
z
y(x)
(3.6)
·
= 0 reads
0
= 2i
3
z
2
x
y
(x)y(z)
t
(x - a) + i
x
y
(x)
y
(z)
(x - a)
+ 2
x
y
(x)y(z)
3
t
z y(z)dz
(x - a) + 2i y(z)
2
y
(x)
z
(a - z) + i y(x)
2
y
(z)
x
(x - a)
- 2
2
z x
y
(x)y(z)
2
t
y(z)dz
(x - a) - 2i
z
y
(z)
y
(x)
(a - z)
- i
x
y
(x)
y
(z)
(x - a) - 2i
z
y
(z)
y
(x)
(a - z)
- i
3
z
2
x
y
(x)y(z)
t
(x - a) - i
x
y
(x)
y
(z)
(x - a)
- i
3
z
2
x
y
(x)y(z)
t
(x - a) + i
2
z
2
y
(x)y(z)
2
t
x
(x - a)
+ 2i
2
z x
y
(x)y(z)
2
t
z
(x - a) - 2i
x
y
(x)y(z)
3
t
z
2
(x - a)
+ i y (z)y (x)(a - z)
z
,
(3.7)
·
= 0 reads
0
= i y(x)
3
y
(z)
x
2
- 2i
3
z
2
x
y
(x)y(z)
2
t
2
- i
x
y
(x)
2
y
(z)
2
- 2i
x
y
(x)
2
y
(z)
2
+ i
2
z
2
y
(x)y(z)
3
t
x
2
,
(3.8)
20
·
= 0 reads
0
= 2i y(x)
2
t
x y(x)dx
- 2i
x
y
(x)
t
y(x)dx
-
x
y
(x)
t
x
+ y(x)
2
t
x
2
-
x
y
(x)
t
x
,
(3.9)
·
= 0 reads
0
= -2
2
z x
y
(x)y(z)
2
t
2
+
z
y
(x)y(z)
3
t
x
2
,
(3.10)
·
= 0 reads
0
= -2i
x
y
(x)
t
- 2i
x
y
(x)
t
+ 2i y(x)
2
t
x
,
(3.11)
·
= 0 reads
0
= i y(x)
t
x
,
(3.12)
·
= 0 reads
0
= i y(x)
y
(z)
x
,
(3.13)
·
= 0 reads
0
=
2
z x
y
(x)y(z)
t
-
2
z x
y
(x)y(z)
t
-
z
y
(x)y(z)
2
t
x
.
(3.14)
3.2.2 Solution of the determining system of equations for the infinitesimals
· First of all, consider equation (3.13). Since this equation has to hold for all choices of y
L
2
(G, ),
the coefficient of y has to vanish and we get
y
(z)
x
= 0.
(3.15)
· Now, consider equation (3.12). Similarly, we get
t
x
= 0.
(3.16)
· Then, consider equation (3.11). Similarly, we get
t
= 0.
(3.17)
· Due to equation (3.17), equations (3.10) and (3.14) are fulfilled identically.
21
· Next, we consider equation (3.9). We apply the product rule and make use of equation (3.16). We
get
t
y(x)dx
= 0.
(3.18)
Considering equations (3.16) and (3.17), we have
t
=
t
(t).
· If we apply equations (3.17) and (3.15) to equation (3.7), we obtain
-4i
z
y
(z)
y
(x)
(a-z)-i
x
y
(x)
y
(z)
(x -a)+i y (z)y (x)(a-z)
z
= 0. (3.19)
Considering the case a
= z and taking into account that y L
2
(G, ) is an arbitrary function, we
get
y
(z)
= 0.
(3.20)
With the above relation, equation (3.19) for a
= x leads to
z
= 0.
(3.21)
which holds for each z
G. If we substitute this result back into equation (3.19) and assume x = z,
we find again the formula (3.20) which has to hold for each z
G. As y L
2
(G, ) is an arbitrary
function, equation (3.19) is fulfilled for x
= z as well.
· In virtue of equations (3.17) and (3.20), equation (3.8) is fulfilled identically.
· Now, we take a look at the remaining four equations (3.3) - (3.6). We start with equation (3.6).
Considering equations (3.17), (3.20) and (3.21), equation (3.6) reads
2i
x
y
(x)
2
2
(x - z) = 0,
hence
2
2
= 0.
(3.22)
Equation (3.22) means there are functionals f
, g such that
= f ([y(x )], t) + g([y(x )], t).
(3.23)
For f we choose the ansatz
f
([y(x )], t) =
G
f
1
(x , t)y(x )dx + f
2
(t).
(3.24)
22
· The next equation we solve is equation (3.5). If we use equations (3.18) and (3.15) and apply the
product rule, equation (3.5) reads
0
= i
x
y
(x)
t
t
(x - z) + i
y
(x)
x
(x - z)
- i
x
y
(x)
x
x
(x - z) - 2i y (x)
y
(z)
y(x)dx
+ 2i y (z)y (x)
z
y(x)dx
.
(3.25)
Considering the case x
= z we get
y
(z)
z
y(x)dx
-
y
(z)
y(x)dx
= 0.
(3.26)
Although equation (3.26) allows a broader range of solutions, we restrict our considerations to the
case
z
y(x)dx
= 0,
y
(z)
y(x)dx
= 0
(3.27)
and use the following ansätze for
y
(z)
dz and
z
:
y
(z)
dz
= c(z, t)dz + c
0
(z, t)y(z)dz,
z
= c
1
(z, t) + c
2
(z, t)y(z).
(3.28)
Next, we want to consider equation (3.25) without the restriction x
= z, hence we integrate
equation (3.25) with respect to z
G. This leads to
x
y
(x)
t
t
+
y
(x)
x
-
x
y
(x)
x
x
-2
G
y
(x)
y
(z)
y(x)dx
dz
+2y (x)
G
y
(z)
z
y(x)dx
dz
= 0.
Now, we put in ansatz (3.28), make use of
t
=
t
(t) and take into consideration that this equation
has to hold for all choices of y
L
2
(G, ), hence the coefficients of 1, y, y , y , y y , y y have to
vanish:
c(x, t)
x
= 0 = c = c(t),
(3.29)
c
0
(x, t)
x
-
2
c
1
x
2
= 0,
(3.30)
t
(t) -
c
1
x
- c
0
(x, t) = 0,
(3.31)
c
2
(x, t) = 0.
(3.32)
After differentiating equation (3.31) with respect to x we find
c
0
x
+
2
c
1
x
2
= 0
which, together with equation (3.30), gives
2
c
1
x
2
= 0, c
0
= c
0
(t).
(3.33)
Considering this and equations (3.29) and (3.31), ansatz (3.28) reads
z
= c
3
(t)z + c
4
(t),
(3.34)
y
(z)
dz
= c(t)dz +
t
(t) - c
3
(t) y(z)dz.
(3.35)
23
· Now, we are ready to deal with equation (3.4). If we use equations (3.35), (3.34), (3.24), (3.18)
and (3.33), equation (3.4) reads
0
= -y (z)
t
(t) - c (t) -
t
(t) -
2
x
t x
(t) y(z) -
t
(t) -
x
x
(t) y (z)
- 2i y(z)
f
1
(z, t)
z
+ 2i
z
(y(z)f
1
(z, t)) +
G
y
(x)
t
(t) -
z
z
(t)
2
(x - z)
x
2
dx
- 2
G
x
y
(x)
t
(t) -
x
x
(t)
(x - z)
x
dx
-
z
y
(z)
2
z
z
2
+
G
2
x
2
y
(x)
t
(t) -
x
x
(t) (x - z) dx + 2
2
z
2
y
(z)
z
z
(t) + y (z)
z
t
+ y(z)
2
z
t x
.
(3.36)
In this equation, the last two integrals involving the Dirac delta distribution vanish if we assume
that
y
(x), y (x) - 0 for x ±.
As equation (3.36) has to hold for all choices of y
L
2
(G, ), the coefficients of 1, y, y and y
have to vanish. We evaluate the coefficient of y in x
= z and get
-c (t) = 0 = c(t) = const. =: c ,
(3.37)
-
t
(t) + 2
2
z
t z
(t) = 0,
(3.38)
2i f
1
(z, t) +
z
t
= 0,
(3.39)
2
z
z
(t) -
t
(t) = 0.
(3.40)
From the above system we first use the relations (3.40) and (3.23), take into account (3.33) and
substitute this into equation (3.3) to get
1
2
i
2
z
t
2
y
(x)dx +
g
t
-
G
y
(x) i
3
g
x (y(x)dx)
2
+
3
g
x
2
y(x)dx
dx
= 0.
(3.41)
This equation has to hold for every
, hence the coefficient of has to vanish. Together with
(3.33) this furnishes
2
z
t
2
= 0,
(3.42)
hence equation (3.41) reads
g
t
-
G
y
(x) i
3
g
x (y(x)dx)
2
+
3
g
x
2
y(x)dx
dx
= 0,
(3.43)
i.e. g has to fulfill the viscous Hopf-Burgers FDE. This is expected by the classical Lie symmetry
analysis as the considered differential equation is linear; we see that this result is furnished by the
extended Lie symmetry analysis as well. With (3.42), the system (3.38) - (3.40) has the solution
t
= 2a
1
t
+ a
2
+ a
4
t
2
,
(3.44)
x
= a
1
x
+ a
3
+ t xa
4
+ a
5
t
,
(3.45)
f
1
=
1
2
i x a
1
+
1
2
ia
3
+ a
6
,
(3.46)
24
where a
1
, a
2
, a
3
, a
4
, a
5
, a
6
.
We insert equations (3.45), (3.44) and (3.37) into equation (3.35) to get
y
(z)
dz
= (a
1
+ ta
4
)y(z)dz + a
7
dz,
with a
7
.
Theorem 3.2.1 (Local transformations of the viscous Hopf-Burgers FDE). The infinitesimals of the vis-
cous Hopf-Burgers FDE are given by
t
= 2a
1
t
+ a
2
+ a
4
t
2
,
x
= a
1
x
+ a
3
+ t xa
4
+ a
5
t
,
y
(z)
dz
= (a
1
+ ta
4
)y(z)dz + a
6
dz,
=
1
2
i
(a
4
x
+ a
5
)y(x)dx + a
7
+ g([y(x )], t),
where a
1
, a
2
, a
3
, a
4
, a
5
, a
6
, a
7
are arbitrary constants and g is an arbitrary functional which has to
fulfill the viscous Hopf-Burgers FDE.
The associated generators read
X
1
= x
x
+ 2t
t
+
G
y
(x)dx
y(x)dx
,
X
2
=
t
,
X
3
=
x
,
X
4
= 2t x
x
+ 2t
2
x
+ 2t
G
y
(x)dx
y(x)dx
+ i
x y
(x)dx
,
X
5
= 2t
x
+ i
y
(x)dx
X
6
=
G
dx
y(x)dx
,
X
7
=
,
X
g
= g([y(x )], t)
.
3.3 Symmetry breaking restrictions
The Lie symmetry analysis furnishes symmetries of the viscous Hopf-Burgers FDE without respecting
physical restrictions. If we incorporate such physical restrictions, we lose some of the calculated sym-
metries which the viscous Hopf-Burgers FDE exhibits considered as a mathematical equation detached
from any physical conditions. The loss of symmetries by incorporating physical restrictions is called sym-
metry breaking. This section is devoted to restrictions on
breaking some of the calculated symmetries
X
1
, . . . , X
g
.
25
In [6], H
OPF
states conditions which have to be fulfilled by Hopf functionals
([y(x)], t) = e
i
(U
t
,y
)
=
L
2
(G,
3
)
e
i
(vvv ,y)
f
t
([vvv (x)]) d[vvv (x)],
cf. equation (1.3). These conditions may be derived from conditions which are imposed on the associated
probability density functional f
t
. The definition of a probability density functional requires f
t
to fulfill
the following two conditions:
Definition 3.3.1 (Probability density functional). f
t
is called a probability density functional if and only
if
1. f
t
is real-valued and non-negative, i.e.
f
t
([vvv (x)])
+
0
.
(3.47)
2. The integral of f
t
over the whole domain of integration equals 1, i.e.
L
2
(G,
3
)
f
t
([vvv (x)]) d[vvv (x)] = 1.
(3.48)
As in this paper we restrict ourselves to the one-dimensional case and make use of the viscous Burgers
equation instead of the incompressible Navier-Stokes equations, solutions of the viscous Hopf-Burgers
FDE do not have to fulfill any conditions related to incompressibility. There remain three conditions
which a solution
of the viscous Hopf-Burgers FDE has to fulfill. We define:
Definition 3.3.2 (Physically relevant solution of the viscous Hopf-Burgers FDE). Let
be a solution of
the viscous Hopf-Burgers FDE.
is a physically relevant solution of the viscous Hopf-Burgers FDE if and
only if
1.
([y(x)], t) = ([-y(x)], t) where
denotes the complex conjugate of
,
2.
(0, t) = 1,
3.
|([ y(x)], t)| 1.
These three conditions are implied by restriction (3.47) and equation (3.48).
In subsection 3.2.2 we showed that the extended Lie symmetry analysis furnishes eight generators X
1
,
X
2
, X
3
, X
4
, X
5
, X
6
, X
7
, X
g
as the local transformations depend on seven parameters a
1
, a
2
, a
3
, a
4
, a
5
, a
6
, a
7
and on a functional g. Especially, the generators X
7
and X
g
associated with symmetries of
are inde-
pendent. As the conditions given by definition 3.3.2 do not influence any symmetries corresponding to
transformations of the independent variables
([y(x )], x, t), the generators associated with transforma-
tions of the independent variables are not changed if we are looking for physically relevant solutions:
We have
X
phys
i
= X
i
,
i
{1, 2, 3, 4, 5, 6}.
Thus, it suffices to have a look at X
7
and X
g
. If
shall be a physically relevant solution, X
7
and X
g
are
not independent. In order to see this, decompose g in
X
g
= g([y(x )], t)
26
into a constant part g
1
and a non-constant part g
2
= g
2
([y(x )], t), i.e.
g
([y(x )], t) = g
1
+ g
2
([y(x )], t).
Here, g
2
is a solution of the viscous Hopf-Burgers FDE, however, it is not a characteristic functional, i.e.
the conditions given by definition 3.3.2 must be fulfilled for the transformed functional
but not for g
2
separately. We get the decomposition X
g
= X
g
1
+ X
g
2
with
X
g
1
:
= g
1
,
g
1
,
X
g
2
:
= g
2
([y(x )], t)
.
We replace X
7
and X
g
by the two generators
X
7
+ X
g
1
= ( + g
1
)
,
f
, g
1
,
X
g
2
= g
2
([y(x )], t)
and calculate the associated global transformations by solving the Lie initial value problems
= + g
1
,
= g
2
([y(x )], t),
( = 0) = ,
( = 0) = .
The solutions are given by
([y(x)], t) = ([y(x)], t)e + (e - 1)g
1
,
,
g
1
,
(3.49)
([y(x)], t) = ([y(x)], t) + g
2
([y(x)], t) ,
.
(3.50)
In the following, we investigate the consequences of the conditions given by definition 3.3.2 for
and g
1
if
is given by equation (3.49).
1. As
has to fulfill
([y(x)], t) = ([-y(x)], t), we have
g
1
.
2. As
has to fulfill
([y(x)], t) = ([-y(x)], t) and (0, t) = 1 and since g
1
, we get
g
1
= -1.
This shows that X
7
and X
g
are not independent.
3. As
has to fulfill |([y(x)], t)| 1, we have
(-, 0]
and the generator X
7
+ X
g
1
associated with the physically relevant symmetry reads
X
phys
7
:
= (X
7
+ X
g
1
)
phys
= ( - 1)
.
27
Next, we investigate the consequences of the conditions given by definition 3.3.2 for
and g
2
=
g
2
([y(x)], t) if is given by equation (3.50).
1. As
has to fulfill condition
([y(x)], t) = ([-y(x)], t), we have
g
2
([y(x)], t) = g
2
([-y(x)], t).
(3.51)
2. As
has to fulfill
([y(x)], t) = ([-y(x)], t) and (0, t) = 1, using equation (3.51) we get
g
2
(0, t) = 0.
(3.52)
3. As
has to fulfill condition |([y(x)], t)| 1, we have
Re
()Re(g
2
) + Im()Im(g
2
) 0.
(3.53)
Altogether, the generator X
g
2
associated with the physically relevant symmetry reads
X
phys
g
:
= X
phys
g
2
= g
2
([y(x )], t)
where g
2
fulfills conditions (3.51) - (3.53).
In the end of this section, we want to compare the calculated physically relevant symmetries with the
symmetries of the viscous Burgers equation, cf. Ref. [14].
Symmetries of the viscous Hopf-Burgers FDE
Symmetries of the viscous Burgers equation
X
phys
1
= 2t
t
+ x
x
+
G
y
(x)dx
y(x)dx
,
1
= 2t
t
+ x
x
- U
U
,
X
phys
2
=
t
,
2
=
t
,
X
phys
3
=
x
,
3
=
x
,
X
phys
4
= 2t x
x
+ 2t
2
x
4
= 2t
2
t
+ 2t x
x
+ (x - 2tU)
U
,
+2t
G
y
(x)dx
y(x)dx
+ i x y(x)dx
,
X
phys
5
= 2t
x
+ i y(x)dx
,
5
= 2t
x
+
U
.
X
phys
6
=
G
dx
y(x)dx
,
X
phys
7
= ( - 1)
,
X
phys
g
= g
2
([y(x )], t)
.
Table 3.1.: Comparison between the symmetries of the viscous Hopf-Burgers FDE and the viscous Burgers
equation.
Here, g
2
is a solution of the viscous Hopf-Burgers FDE satisfying the three conditions (3.51) - (3.53).
For X
phys
7
, the group parameter
is restricted to be non-positive, i.e.
(-, 0]. For all the other
generators we have
.
We see that we rediscover the analog forms of the symmetries of the viscous Burgers equation gener-
ated by
1
(scaling),
2
(time translation),
3
(space translation),
4
and
5
(Galilei invariance).
28
Generator
Global transformations associated with the generator
X
phys
1
t
= te
2
,
x
= xe ,
y
(x)dx = y(x)dxe ,
= ,
X
phys
2
t
= t + , x = x,
y
(x)dx = y(x)dx,
= ,
X
phys
3
t
= t,
x
= x + ,
y
(x)dx = y(x)dx,
= ,
X
phys
4
t
=
t
1
-2t
,
x
=
x
(1-2t )
y
(x)dx =
y
(x)dx
1
-2t
,
= exp i
1
-2t
G
x y
(x)dx ,
X
phys
5
t
= t,
x
= x + 2t , y(x)dx = y(x)dx
= exp i
G
y
(x)dx ,
X
phys
6
t
= t,
x
= x,
y
(x)dx = y(x)dx + dx, = ,
X
phys
7
t
= t,
x
= x,
y
(x)dx = y(x)dx,
= e + (1 - e ),
X
phys
g
t
= t,
x
= x,
y
(x)dx = y(x)dx,
= + g([y(x )], t) .
Table 3.2.: Global transformations of the viscous Hopf-Burgers FDE.
4 Conclusions
This paper continues the work of O
BERLACK
and W
ACLAWCZYK
, cf. Ref. [1] and [2], where the classical Lie
symmetry analysis is extended from partial differential equations to equations with functional derivatives
and performed in the Fourier space. Here, we introduce the procedure of applying the extended Lie sym-
metry analysis in the physical space. This corresponds to the case when both functional derivatives and
spatial derivates with respect to the integration variable are present in the functional integro-differential
equation. The method is based on the transformation of the product y
(x)dx appearing in the integral
term.
As an example, we consider the viscous Hopf-Burgers functional integro-differential equation, i.e. the
functional formulation of the viscous Burgers equation. We perform the extended Lie symmetry analysis
on the viscous Hopf-Burgers FDE to find the eight symmetries given in table 3.1. Furthermore, we
take a brief look at symmetry breaking restrictions and indicate physically relevant symmetries, i.e.
symmetries such that
fulfills the conditions for the characteristic functional. We see that only statistical
symmetries, i.e. symmetries associated with transformations of the dependent variable
, are influenced
if we demand
to be a characteristic functional. The construction of physically relevant invariant
solutions remains a task for future work. Finally, we compare the symmetries of the viscous Hopf-
Burgers FDE with the symmetries of the viscous Burgers equation: We are able to rediscover all the five
symmetries.
The most significant result of this paper consists in demonstrating that the extended Lie symmetry
analysis works for the considered functional equation and that it is not only able to rediscover symme-
tries of the considered equation but also to furnish new, unknown symmetries associated with the Hopf
formulation of the viscous Burgers equation having purely statistical origin. The presented extension of
the Lie symmetry analysis can be a useful tool for the analysis of FDE's containing functional derivatives.
Presently, the underlying equation is the viscous Burgers equation. For future work one might con-
sider the Hopf-Navier-Stokes FDE and perform the extended Lie symmetry analysis on this equation to
29
determine the moments of the solutions of the Navier-Stokes equations (see also [19]). Furthermore,
one might choose more sophisticated ansätze during the solution procedure of the determining system
of equations for the infinitesimals. Hopefully, less restrictive ansätze will lead to further new symmetries.
We believe that the presented machinery is highly relevant to a variety of important functional dif-
ferential equations and functional integro-differential equations in physics, especially in continuum
mechanics. As the numerical treatment of FDE's is difficult because of the high dimensionality and
since very little is known on how to treat and solve FDE's analytically, the presented methods may give
a chance to treat equations which so far have been put aside because of the missing analytical meth-
ods. Additionally, unknown symmetries may be discovered which would be pleasant since symmetries
illuminate the properties of the physical model equations.
5 Acknowledgments and contributions
The authors are thankful to Wolfgang Kollmann for his useful comments and discussions concerning the
paper.
M. Oberlack and M. Waclawczyk contributed analysis methods; D. D. Janocha and M. Waclawczyk
solved the determining system of equations for the infinitesimals; D. D. Janocha wrote the paper within
the scope of his master's thesis.
Bibliography
[1] Oberlack, M.; Waclawczyk, M.: On the extension of Lie group analysis to functional differential
equations. Arch. Mech. 2006, 58, 597-618
[2] Waclawczyk, M.; Oberlack, M.: Application of the extended Lie group analysis to the Hopf func-
tional formulation of the Burgers equation. J. Math. Phys. 2013, 54, 072901
[3] Oberlack, M., Rosteck, A.: New statistical symmetries of the multi-point equations and its impor-
tance for turbulent scaling laws. Disc. and Cont. Dyn. Sys., Ser. S 2010, 3, 451-471
[4] Waclawczyk, M.; Staffolani, N.; Oberlack, M.; Rosteck, A.; Wilczek, M.; Friedrich, R.: Statistical
Symmetries of the Lundgren-Monin-Novikov Hierarchy. Phys. Rev. E 2014, 90, 013022
[5] Lundgren, T. S.: Distribution functions in the statistical theory of turbulence. Phys. Fluids, 1967,
10, 969-975
[6] Hopf, E.: Statistical Hydromechanics and Functional Calculus. J. Rational Mech. Anal. 1952, 1,
87-123
[7] Hosokawa, I.: Monin-Lundgren hierarchy versus the Hopf equation in the statistical theory of
turbulence. Phys. Rev. E, 2006, 73, 067301
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Characteristic Functional Formalism. Phys. Fluids 1970, 13, 1683-1692
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[9] Hosokawa, I.; Yamamoto K.: Energy decay of Burgers' model of turbulence. Phys. Fluids 1976, 19,
1423-1424
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solution in the Leith model of turbulence: anomalous power law and asymptotic analysis. J. Phys.
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[12] Gelfand, I. M.; Fomin, S. W.: Calculus of variations. Prentice Hall 1963, New Jersey
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survey of methods illustrated by the Benny equations. Nonl. Dyn. 2002, 28, 135-153
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tries, Exact Solutions and Conservation Laws. CRC Press 1994
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Trends in Theoretical Developments and Computational Methods. CRC Press 1996
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269-275
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tions, Mech. Res. Commun. 2005 , 32, 241-254
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statistical turbulence theory, Mechanical Engineering Reviews, 2015, 2, 15-00157
A Infinitesimals of the dependent variables
For the viscous Hopf-Burgers FDE we need the three infinitesimals
;t
,
;x y
(x)y(x)
and
;x x y
(x)
.
;t
is given
by equation (2.10):
;t
=
t
+
,t
-
t
t
- (
,t
)
2
t
-
G
, y
(x )
y
(x )
dx
t
-
G
, y
(x )
,t
y
(x )
dx
-
G
,x y
(x )
x
t
y
(x )dx -
G
,x y
(x )
,t
x
y
(x )dx .
;x y
(x)y(x)
and
;x x y
(x)
are given below:
;x y
(x)y(x)
=
3
x (y(x)dx)
2
+ 2
, y
(x)
3
x y(x)dx
-
,t
3
t
x (y(x)dx)
2
- 2
,t
, y
(x)
3
t
x y(x)dx
-
G
, y
(z)
3
y
(z)
dz
x (y(x)dx)
2
- 2
G
, y
(z)
, y
(x)
3
y
(z)
dz
x y(x)dx
+ (
, y
(x)
)
2
3
x
2
-
,t
(
, y
(x)
)
2
3
t
x
2
-
G
, y
(z)
(
, y
(x)
)
2
3
y
(z)
dz
x
2
- 2
,t y
(x)
2
t
x y(x)dx
- 2
, y
(x)
,t y
(x)
2
t
x
31
- 2
G
, y
(z)y(x)
2
y
(z)
dz
x y(x)dx
- 2
G
, y
(x)
, y
(z)y(x)
2
y
(z)
dz
x
-
,x y
(x)
2
x
x y(x)dx
-
, y
(x)
,x y
(x)
2
x
x
+
, y
(x)y(x)
2
x
-
,t
, y
(x)y(x)
2
t
x
-
G
, y
(z)
, y
(x)y(x)
2
y
(z)
dz
x
+ 2
,x y
(x)
2
y(x)dx
- 2
,t
,x y
(x)
2
t
y(x)dx
-
G
,x y
(z)
2
y
(z)
dz
(y(x)dx)
2
- 2
G
,x y
(z)
, y
(x)
2
y
(z)
dz
y(x)dx
- 2
G
, y
(z)
,x y
(x)
2
y
(z)
dz
y(x)dx
+ 2
, y
(x)
,x y
(x)
2
2
- 2
,t
, y
(x)
,x y
(x)
2
t
2
-
G
,x y
(z)
(
, y
(x)
)
2
2
y
(z)
dz
2
- 2
G
, y
(z)
, y
(x)
,x y
(x)
2
y
(z)
dz
2
- 2
,x y
(x)
,t y
(x)
t
- 2
G
,x y
(x)
, y
(z)y(x)
y
(z)
dz
- (
,x y
(x)
)
2
x
-
G
,x y
(z)
, y
(x)y(x)
y
(z)
dz
- 2
,x t y
(x)
t
y(x)dx
- 2
, y
(x)
,x t y
(x)
t
-
,x x y
(x)
x
y(x)dx
-
, y
(x)
,x x y
(x)
x
- 2
G
,x y
(z)y(x)
y
(z)
dz
y(x)dx
- 2
G
, y
(x)
,x y
(z)y(x)
y
(z)
dz
+
,x y
(x)y(x)
-
,t
,x y
(x)y(x)
t
-
G
, y
(z)
,x y
(x)y(x)
y
(z)
dz
-
,t y
(x)y(x)
t
x
-
G
, y
(z)y(x)y(x)
y
(z)
dz
x
-
,x y
(x)y(x)
x
x
-
,x x y
(x)
x
y(x)dx
-
,x y
(x)
2
x
x y(x)dx
-
,x x y
(x)
, y
(x)
x
-
,x y
(x)
,x y
(x)
x
-
,x y
(x)
, y
(x)
2
x
x
-
G
, y
(x)
,z y
(z)
y
(z)
3
z
x y(x)dx
dz
- 2
G
,z y
(x)y(z)
y
(z)
2
z
x y(x)dx
dz
-
G
,z x y
(z)
y
(z)
2
z
( y(x)dx)
2
dz
-
G
,z y
(z)
y
(z)
3
z
x( y(x)dx)
2
dz
-
G
,x y
(x)
,z y
(z)
y
(z)
2
z
y(x)dx
dz
-
G
, y
(x)
,z x y
(z)
y
(z)
2
z
y(x)dx
dz
- 2
G
,z x y
(x)y(z)
y
(z)
z
y(x)dx
dz
- 2
G
,z x y
(x)y(z)
, y
(x)
y
(z)
z
dz
- 2
G
,z y
(x)y(z)
,x y
(x)
y
(z)
z
dz
-
G
,x y
(x)y(x)
,z y
(z)
y
(z)
z
dz
32
Frequently Asked Questions: Abstract on Lie Symmetry Analysis for Integro-Differential Equations
What is the main topic of this abstract?
The abstract discusses an extension of the classical Lie symmetry analysis from partial differential equations (PDEs) to integro-differential equations with functional derivatives.
What previous work does this paper build upon?
This work continues the research of OBERLACK and WACLAWCZYK (2006, 2013), where the extended Lie symmetry analysis is performed in the Fourier space.
What new method does the paper introduce?
The paper introduces a method to perform the extended Lie symmetry analysis in the physical space, addressing the transformation of the integration variable in integral terms.
What is the method based on?
The method is based on the transformation of the product y(x)dx appearing in the integral terms.
What equation is used as an application example?
The functional formulation of the viscous Burgers equation.
What are the capabilities of the extended Lie symmetry analysis described?
The extended Lie symmetry analysis furnishes all known symmetries of the viscous Burgers equation and can provide new symmetries associated with the Hopf formulation of the viscous Burgers equation.
What is the potential application of this method?
It can be employed as an important tool for applications in continuum mechanics.
What topics are covered in the table of contents?
The table of contents lists the following main topics: Introduction, Extension of Lie Symmetry Analysis, Lie Symmetry Analysis of the Viscous Hopf-Burgers Equation, Conclusions, Acknowledgements, and an Appendix on Infinitesimals of Dependent Variables.
What are the three complete descriptions of turbulence mentioned in the introduction?
The three complete descriptions of turbulence are: the multi-point correlation approach, the Lundgren-Monin-Novikov approach, and the Hopf approach.
What is a Hopf functional and how is it relevant to the context?
A Hopf functional is a characteristic functional used to describe the statistics of a turbulent flow. The document discusses using Lie symmetry analysis to study the viscous Hopf-Burgers functional integro-differential equation.
What is a functional differential equation (FDE)?
A functional differential equation is a generalization of a partial differential equation (PDE) where the number of independent variables tends to infinity.
What is a functional derivative?
A functional derivative measures the sensitivity of a functional to a small change in its input function.
What is meant by 'one-parameter Lie point transformations'?
One-parameter Lie point transformations are continuous symmetry transformations depending on a continuous parameter from a Lie group, which are used to find solutions to differential equations.
What are infinitesimals in the context of Lie symmetry analysis?
Infinitesimals are functions that define the local transformations of variables under a Lie group. They determine how the variables change infinitesimally under the symmetry transformation.
What is the infinitesimal generator in Lie symmetry analysis?
The infinitesimal generator is a differential operator that defines the infinitesimal transformation of the variables. It is used to derive the determining equations for the infinitesimals.
What is a determining system of equations for the infinitesimals?
The determining system of equations is an overdetermined set of linear partial differential equations that must be satisfied by the infinitesimals in order for the transformation to be a symmetry of the differential equation.
What is a symmetry-breaking restriction?
A symmetry-breaking restriction is a condition imposed on the system (such as a physical constraint) that reduces the number of symmetries that the system admits.
How are energy scaling and symmetry methods related to turbulence in this context?
Energy scaling behavior in turbulence can sometimes be related to self-similar solutions obtained using symmetry methods, providing insights into how energy is distributed across different scales in the turbulent flow.
How can a functional integro-differential equation be a viscous Hopf-Burgers equation?
The viscous Hopf-Burgers equation can be cast as a functional integro-differential equation by expressing it in terms of the Hopf functional, which involves integrals over the velocity field, thus combining functional and integro-differential aspects into a single equation.
Can the Lie Symmetry be applied to turbulence research?
Lie Symmetry Analysis is used in turbulence research to find symmetries, analyze invariant solutions, and study statistical properties in turbulent flow scenarios.
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