Name: Lie symmetry analysis of the Hopf functional-differential equation
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Author: Daniel Janocha
ISBN: 978-3-668-05847-7

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.

Excerpt

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.

Contents

1. Introduction

1.1. Three complete descriptions of turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2. Hopf functional and multi-point correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2.1. Viscous Hopf-Burgers functional integro-differential equation . . . . . . . . . . . . .

2. Extension of the Lie symmetry analysis towards functional integro-differential equations

2.1. One-parameter Lie point transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2. Differential operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3. Infinitesimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4. Infinitesimal generator, determining system of equations for the infinitesimals . . . . . . . .

2.5. Global transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3. Lie symmetry analysis of the viscous Hopf-Burgers functional integro-differential equation 16

3.1. Three different approaches to Lie symmetry analysis . . . . . . . . . . . . . . . . . . . . . . .

3.2. Local transformations of the viscous Hopf-Burgers functional integro-differential equation

3.2.1. Determining system of equations for the infinitesimals . . . . . . . . . . . . . . . . . .

3.2.2. Solution of the determining system of equations for the infinitesimals . . . . . . . .

3.3. Symmetry breaking restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4. Conclusions

5. Acknowledgments and contributions

A. Infinitesimals of the dependent variables

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

) , i

, . . . , i

= 1, 2, 3.

In this term,

is a random vector describing the velocity field in time t and x

, . . . , x

are positions of

different fluid particles in the space occupied by the fluid.

is given by

) = U

), U

) = U

, t

), U

, t

), U

, 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

admit proba-

bility density functions (PDF's)

= f

(vvv

, . . . , vvv

; x

, . . . , x

)

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

(vvv

; x

) dvvv

expresses the proba-

bility that the velocity vector

) = U(x

, t

) is contained within the infinitesimal interval

[vvv

, vvv

+ dvvv

]. 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

(vvv

, . . . , vvv

(n+1)

; x

, . . . , 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

) =

· · ·

. . . v

(vvv

, . . . , vvv

; x

, . . . , x

) dvvv

. . . dvvv

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

for n

are investigated, cf.

Ref. [6]. The n-point characteristic function is defined as

, . . . , y

, t

) := e

(

)

· · ·

(vvv ,y)

(vvv

, . . . , vvv

; x

, . . . , x

) dvvv

. . . dvvv

with

= U

), . . . , U

) . (·, ·) is defined as the Euclidean scalar product

(vvv , y) :=

for vvv

= (vvv

, . . . , vvv

), y = (y

, . . . , y

) with vvv

, y

, k

= 1, . . . , n.

In hydromechanics, we assume that the mean free path is negligible, hence we take the continuum limit

. Thus, instead of the n velocity vectors vvv

, . . . , vvv

, 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

becomes a probability density functional and the function

becomes

the functional

([y(x)], t) = e

)

(vvv ,y)

([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

space.

Definition 1.2.1 (L

space). For G

define

(G,

) := vvv : G -

(x) · vvv (x) dx =

(x)v

(x) dx

and equip L

(G,

) with (·, ·) defined by

(vvv , y) :=

(x) · y(x) dx =

(x)y

(x) dx

(1.2)

for vvv , y

(G,

If we equip L

(G,

) 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

(G,

) and let (·, ·)

be the L

scalar product defined by equation (1.2). The characteristic functional or Hopf functional is

defined by

([y(x)], t) := e

)

(G,

)

(vvv ,y)

([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).

· Let

be a function space. A functional is a mapping

- .

· Let

= ([y(x)]) be a functional. We define

the functional derivative of

as the limit

([y(x)])

(z)dz

= lim

([y(x) + he

(x - z)]) - ([y(x)])

= 1, 2, 3.

Here,

denotes the

unit vector and

denotes the Dirac delta distribution. We denote the

functional derivative by

, y

(z)

(z)dz

(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:

([y(x )], x, t, ,

, . . . ,

) = 0.

(1.4)

In a FDE, the finite set

, . . . , y

) (

)

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

= 0 via

) . . . U

) =

, y

)...y

)

(1.5)

Consequently, the Hopf functional

provides the full statistical description of the velocity field U

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.

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.

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

. 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

(G, ) with

= y(x),

= U

(x) = U(x, t)

and that U

fulfills the viscous Burgers equation

+ U

which can equivalently by written as

)

By simple rescaling we may eliminate the factor

/2. We call the following equation viscous Burgers

equation:

)

(1.6)

We differentiate the Hopf functional (1.3) with respect to t to get

([y(x)], t) =

, y

)

= i

, y e

, y

)

= i -

)

, y

)

(1.7)

where equation (1.6) was inserted in order to eliminate

/ t. Using the definition of the character-

istic functional (1.1), the following relations can be calculated:

)

, y

)

y(x) y(x)

,x y

(x)y(x)

, y

)

y(x)

,x x y

(x)

After introducing the above formulas into equation (1.7), we finally obtain the viscous Hopf-Burgers FDE

([y(x)], t) =

(x) i

,x y

(x)y(x)

,x x y

(x)

(1.8)

with y

(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.

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.

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

. 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

, . . . , y

, x

, . . . , x

, t,

, . . . ,

) = 0

and let

= y

, . . . , y

, x

, . . . , x

, t,

, ),

= 1, . . . , n,

= x

, . . . , y

, x

, . . . , x

, t,

, ),

= 1, . . . , n,

= t(y

, . . . , y

, x

, . . . , x

, t,

, ),

= (y

, . . . , y

, x

, . . . , x

, 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

, . . . , y

, x

, . . . , x

, t,

) ,

= 1, . . . , n,

= x

, . . . , y

, x

, . . . , x

, t,

) ,

= 1, . . . , n,

= t +

, . . . , y

, x

, . . . , x

, t,

) ,

= +

, . . . , y

, x

, . . . , x

, t,

) .

In a one-parameter Lie point transformation, the derivatives of

are not transformed separately. The

transformations of

, . . . ,

are calculated as functions of

, . . . ,

. We write

, . . . , y

, x

, . . . , x

, t,

) ,

, y

; y

, . . . , y

, x

, . . . , x

, t,

) ,

= 1, . . . , n,

, y

; y

, . . . , y

, x

, . . . , x

, t,

) ,

, j

= 1, . . . , n,

The functions

;...

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 )

( ) :=

( -

)

(2.1)

about

= 0, hence the infinitesimals are defined by

; y

, y

. . . .

We extend this definition to FDE's.

Definition 2.1.2 (One-parameter Lie point transformations of FDE's). Consider the FDE

([y(x )], x, t, ,

) = 0

and let

(z)dz = y(z)dz([y(x )], z, x, t, , ),

, z

= x([y(x )], x, t, , ),

= t([y(x )], x, t, , ),

= ([y(x )], x, t, , ),

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

(z)dz = y(z)dz +

(z)

([y(x )], z, x, t, ) ,

, z

= x +

([y(x )], x, t, ) ,

= t +

([y(x )], x, t, ) ,

= +

([y(x )], x, t, ) ,

Again, the derivatives of

are not transformed separately. The transformations of

are calcu-

lated as functionals depending on

(z)

dz,

. We write

([y(x )], x, t, ,

) ,

, y

)

, y

)

; y

)

([y(x )], z

, x, t,

) ,

, z

, y

) y(z

)

, y

)y(z

)

; y

)y(z

)

([y(x )], z

, z

, x, t,

) ,

, z

The functionals

(z)

dz,

, . . . are called infinitesimals. Again, we separate indices of

a semicolon to distinguish them from derivatives. The transformations are expanded in a Taylor series

(2.1) about

= 0, hence the infinitesimals are defined by

(z)

dz :

y(z)dz

; y

(z)

, y

(z)

. . . .

Comparing the notion of one-parameter Lie point transformations of PDE's and of FDE's, one sees the

considered transformed functions y

, x

, 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

(z) dz is the continuum analogon of

Following Ref. [16], the variable y

(z)dz can be represented as a test series

(z)dz =

(z)dz.

(2.2)

where

(z)}

is a set of orthogonal functions. In this respect, the transformed variable y

(z)dz reads

(z)dz =

(z)dz.

(2.3)

A possible approach in extending the Lie group analysis would be to account for the transformations

of y

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

(z)

dz instead of

(z)

. The infinitesimal

(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

, x

, . . . , x

, t,

) are treated

as independent variables.

Hence, in the extended Lie symmetry analysis, we treat the variables

([y(x )], x, t, ,

) 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

x y(x)dx

y(x)dx x

we can define:

Definition 2.2.1 (Differential operators for Hopf FDE's). For a Hopf FDE of third order

([y(x )], x, t, ,

) = 0

we introduce

,t t

,t y

(x )

, y

(x )

,t t t

,t t

,t t y

(x )

,t y

(x )

,t x y

(x )

,x y

(x )

,t y

(x )y(x )

dx dx

, y

(x )y(x )

dx dx

,x y

(x )

, y

(x )

,x t y

(x )

,t y

(x )

,x x y

(x )

,x y

(x )

,x y

(x )y(x )

dx dx

, y

(x )y(x )

dx dx

(z)dz

y(z)dz

, y

(z)

, y

(z)t

, y

(z)y(x )

, y

(x )

, y

(z)t t

,t t

, y

(z)t y(x )

,t y

(x )

, y

(z)x y(x )

,x y

(x )

, 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

(z)

dz,

. For the viscous Hopf-Burgers FDE, we need the

three infinitesimals

;x y

(x)y(x)

and

;x x y

(x)

· In order to calculate

, we differentiate the transformed Hopf functional

with respect to t

taking into account the decomposition (2.2):

(x )dx

(x )dx .

(2.4)

Hence, in the definition above we account for the fact that t, x and the infinite set

{ y

}

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

(x)dx

(x)dx +

(x)dx

(2.5)

The second RHS term can be rewritten as

(x)dx =

(x)dx

(2.6)

and substituted into equation (2.4). Taking into account the fact that

does not depend explicitely

on x, from equation (2.4) we obtain

(x )dx

(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

(x )dx

(x )dx .

(2.8)

With the one-parameter Lie point transformations, cf. definition 2.1.2, equation (2.8) reads

( +

)

= (

)

(t +

)

(

, y

(x )

; y

(x )

)

(y(x )dx +

(x )

)

(

,x y

(x )

;x y

(x )

)

(x +

)

(x )dx +

(x )

dx .

(2.9)

Evaluating this equation in

(1), we get

Using this, evaluating in

( ) leads to

, y

(x )

,x y

(x )

(x )dx

which constitutes an equation for

, y

(x )

,x y

(x )

(x )dx .

Since the appearing infinitesimals

(x )

dx ,

do not depend on derivatives of

, we immedi-

ately get

- (

)

, y

(x )

, y

(x )

,x y

(x )

(x )dx -

,x y

(x )

(x )dx .

(2.10)

· In order to calculate

; y

(x)

, we differentiate

with respect to y(x). An analog calculation leads to

; y

(x)

(x)dx

, y

(x )

(x)dx

,x y

(x )

(x)dx

(x )dx

y(x)dx

, y

(x)

y(x)dx

, y

(x)

, y

(x )

y(x)dx

, y

(x )

, y

(x)

(x )

,x y

(x )

y(x)dx

(x )dx -

,x y

(x )

, y

(x)

(x )dx .

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)

(x)dx

, y

(x)

(x)dx

, y

(x)

(x )dx

(x)dx

(x )dx

(x)dx

, y

(x)

(x)dx

,t y

(x)

(x)dx

, y

(x ) y(x)

(x )dx

(x)dx

,x y

(x ) y(x)

(x)dx

(x )dx

,x y

(x)

(x)dx

If we insert the one-parameter Lie point transformations, cf. definition 2.1.2, we get

; y

(x)y(x)

; y

(x)

(x)dx

,t y

(x)

(x)dx

, y

(x )y(x)

(x )

(x)dx

,x y

(x)

(x)dx

,x y

(x )y(x)

(x)dx

(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)

, y

(x) y(x)

, y

(x) y(x)

(x )dx

, y

(x) y(x)

,t y

(x) y(x)

, y

(x ) y(x) y(x)

(x )dx

,x y

(x) y(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)

,t y

(x)y(x)

, y

(x )y(x)y(x)

(x )

,x y

(x)y(x)

. (2.11)

· In order to calculate

;x y

(x)

, we differentiate

, y

(x)

with respect to x. We have

, y

(x)

, y

(x)

, y

(x)

(x )dx

, y

(x)

,t y

(x)

, y

(x ) y(x)

(x )dx

,x y

(x)

If we insert the one-parameter Lie point transformations, cf. definition 2.1.2, we get

;x y

(x)

; y

(x)

,t y

(x)

, y

(x )y(x)

(x )

,x y

(x)

In order to get

;x x y

(x)

, we differentiate

,x y

(x)

with respect to x:

,x y

(x)

,x y

(x)

,x y

(x)

(x )dx

,x y

(x)

,t x y

(x)

, y

(x ) x y(x)

(x )dx

,x x y

(x)

If we insert the one-parameter Lie point transformations, cf. definition 2.1.2, we get

;x x y

(x)

;x y

(x)

,t x y

(x)

, y

(x )x y(x)

(x )

,x x y

(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

(z)

dz,

, cf. appendix A.

2.4 Infinitesimal generator, determining system of equations for the infinitesimals

Let F

([y(x )], x, t, ,

) = 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, i.e. consider the equation

+ (

) = 0.

(2.13)

We calculate

(x )

y(x )dx

; y

(x )

, y

(x )

+ . . . F

and define

Definition 2.4.1 (Infinitesimal generator and its prolongation).

· The differential operator

(x )

y(x )dx

is called infinitesimal generator or simply generator.

· The differential operator

(3)

= X +

; y

(x )

, y

(x )

;t t

,t t

;t y

(x )

,t y

(x )

;x y

(x )

,x y

(x )

; y

(x )y(x )

dx dx

, y

(x )y(x )

;t t t

,t t t

;t t y

(x )

,t t y

(x )

;t y

(x )y(x )

dx dx

,t y

(x )y(x )

;t x y

(x )

,t x y

(x )

; y

(x )y(x )y(x )

dx dx dx

, y

(x )y(x )y(x )

; y

(x )x y(x )

dx dx

, y

(x )x y(x )

;x y

(x )y(x )

dx dx

,x y

(x )y(x )

;x x y

(x )

,x x y

(x )

is called prolongation of X .

Using this definition and employing F

= F, equation (2.13) reads

+ X

(3)

+ (

) = 0.

This equation is fulfilled in

( ) if and only if

(3)

= 0.

(2.14)

Equation (2.14) constitutes an overdetermined system of linear FDE's for the infinitesimals

(z)

dz,

. 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

(z)

dz,

do not depend on derivatives

, all coefficients of all appearing derivatives of have to vanish which leads to a system of linear

FDE's.

2.5 Global transformations

Knowing

(z)

dz,

, 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

(z)

dz,

with the initial values

(z)dz( = 0) = y(z)dz,

( = 0) = x,

( = 0) = t,

( = 0) = .

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. 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

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

(z)

dz,

(3)

= 0.

One has to pay attention that an analogous formula for the transformed integral should be used

during the calculation of

;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.

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

(x) i

,x y

(x)y(x)

,x x y

(x)

= 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

(z)

dz,

, we

consider equation (2.14):

(3)

= 0.

The generator X

(3)

is given by definition 2.4.1. The contributing summands of X

(3)

are given by

(3)

modif

(x )

y(x )dx

;x y

(x )y(x )

dx dx

,x y

(x )y(x )

dx dx

;x x y

(x )

,x x y

(x )

Applying X

(3)

modif

to F , we get the equation

(3)

modif

(x)

dx i

,x y

(x)y(x)

,x x y

(x)

(x) i

;x y

(x)y(x)

;x x y

(x)

= 0. (3.1)

We insert the infinitesimals

(cf. equation (2.10)),

;x y

(x)y(x)

and

;x x y

(x)

(cf. appendix A) and employ

= 0 in order to eliminate

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

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.

· The second option is to demand that G

is not bounded. We restrict ourselves to the case

= (-, +). If this holds true, one may impose the condition

(x = ±) = 0.

(3.2)

As equation (1.5) states

, y

(x)

= i U

(x)e

, y

)

we have

, y

(x)

=±

= 0.

Additionally, we impose that all functional derivatives of

vanish for x ±, i.e.

,t y

(x)

=±

= 0,

,x y

(x)

=±

= 0,

, y

(x)y(x)

=±

= 0,

. . .

Hence, all appearing boundary integrals vanish. For example,

(x)

,x y

(x)y(x)

= -

(x)

, y

(x)y(x)

(x)

, y

(x)y(x)

The resulting equation (3.1) has the form

, y

(z)

, y

(x)y(z)

dzdx

, y

(x)

, y

(z)

dzdx

, y

(x)y(a)

, y

(z)

dadzdx

(

, y

(x)

)

, y

(z)

dzdx

,t y

(x)

(

, y

(x)

)

, y

(z)y(z)

dzdx

, y

(x)

,t y

(x)

,t y

(x)y(x)

, y

(z)y(x)y(x)

dzdx

, y

(x)y(x)

, y

(z)y(z)

dzdx.

Since the infinitesimals

(z)

dz,

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

(z)

dz,

= 0 reads

(x) i

x (y(x)dx)

y(x)dx

= 0,

(3.3)

= 0 reads

(z)

- 2i y(z)

z y(z)dz

+ i

(x)y(z)

x (y(x)dx)

+ i

(x)

(z)

x (y(x)dx)

- i

(z)

z y(z)dz

+ 2i

(z)

y(z)dz

- i

(x)

(z)

(y(x)dx)

+ i

(z)

y(z)dz

(x)y(z)

y(x)dx

(x)

(z)

y(x)dx

- 2

(x)

(z)

x y(x)dx

(z)

(x)

(z)

y(x)dx

+ 2

(z)

+ y (z)

+ y(z)

x t

+ i

(z)

y(z)dz

- i

(z)

z y(z)dz

+ i

z x

(z)y(x)

y(x)dx y(x)dx

- i

(x)

(z)

x( y(x)dx)

(z)y(x)

y(x)dx

(z)y(x)

z x

y(x)dx

(z)y(x)

y(x)

+ 2

x z

(z)y(x)

x y(x)dx

dx,

(3.4)

= 0 reads

= -i

(x)

(x - z) + i

(x)

(x - z) + i

(x)

(x - z)

(a)y(z)

a (y(a)da)

(x - z) da + 2i y(x)

(z)

x y(x)dx

- 2i

(x)

(z)

y(x)dx

+ i

(x)

(x - z) - i

(x)

(x - z)

- i

(a)y(z)

y(a)da

(x - z) da + y(x)

(z)

+ y(x)

(z)

(x)

(z)

(x)

(z)

- 2i y(x)

(z)

x y(x)dx

+ 2i

x z

(x)y(z)

y(x)dx

(3.5)

= 0 reads

= -

(x)y(z)

(x)

(z)

+ 2i

(x)y(z)

x y(x)dx

+ 2i y(x)

(z)

x y(x)dx

- i

(x)

(x - z) - 2i

(x)y(z)

y(x)dx

- 2i

(x)

(z)

y(x)dx

(x - z) - 2i

(x)

(z)

y(x)dx

+ 2i

(x)

(x - z) + i

(x)

(x - z) + i

(x)

(x - z)

(x)

(z)

- 2

(x)

(z)

(x)

(z)

+ 2

(x)

(z)

(x)y(z)

(x)

(z)

(x)

(z)

- 2

(x)y(z)

+ i

(x)

(x - z) + i

x z

(x)

(x - z) - i

(x)

(x - z)

- i

(x)y(z)

x y(x)

+ 2i

z x

(x)y(z)

y(x)

(3.6)

= 0 reads

= 2i

(x)y(z)

(x - a) + i

(x)

(z)

(x - a)

+ 2

(x)y(z)

z y(z)dz

(x - a) + 2i y(z)

(x)

(a - z) + i y(x)

(z)

(x - a)

- 2

z x

(x)y(z)

y(z)dz

(x - a) - 2i

(z)

(x)

(a - z)

- i

(x)

(z)

(x - a) - 2i

(z)

(x)

(a - z)

- i

(x)y(z)

(x - a) - i

(x)

(z)

(x - a)

- i

(x)y(z)

(x - a) + i

(x)y(z)

(x - a)

+ 2i

z x

(x)y(z)

(x - a) - 2i

(x)y(z)

(x - a)

+ i y (z)y (x)(a - z)

(3.7)

= 0 reads

= i y(x)

(z)

- 2i

(x)y(z)

- i

(x)

(z)

- 2i

(x)

(z)

+ i

(x)y(z)

(3.8)

= 0 reads

= 2i y(x)

x y(x)dx

- 2i

(x)

y(x)dx

(x)

+ y(x)

(x)

(3.9)

= 0 reads

= -2

z x

(x)y(z)

(3.10)

= 0 reads

= -2i

(x)

- 2i

(x)

+ 2i y(x)

(3.11)

= 0 reads

= i y(x)

(3.12)

= 0 reads

= i y(x)

(z)

(3.13)

= 0 reads

z x

(x)y(z)

z x

(x)y(z)

(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

(G, ),

the coefficient of y has to vanish and we get

(z)

= 0.

(3.15)

· Now, consider equation (3.12). Similarly, we get

= 0.

(3.16)

· Then, consider equation (3.11). Similarly, we get

= 0.

(3.17)

· Due to equation (3.17), equations (3.10) and (3.14) are fulfilled identically.

· Next, we consider equation (3.9). We apply the product rule and make use of equation (3.16). We

get

y(x)dx

= 0.

(3.18)

Considering equations (3.16) and (3.17), we have

(t).

· If we apply equations (3.17) and (3.15) to equation (3.7), we obtain

-4i

(z)

(x)

(a-z)-i

(x)

(z)

(x -a)+i y (z)y (x)(a-z)

= 0. (3.19)

Considering the case a

= z and taking into account that y L

(G, ) is an arbitrary function, we

get

(z)

= 0.

(3.20)

With the above relation, equation (3.19) for a

= x leads to

= 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

(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

(x)

(x - z) = 0,

hence

= 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

([y(x )], t) =

(x , t)y(x )dx + f

(t).

(3.24)

· 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

= i

(x)

(x - z) + i

(x)

(x - z)

- i

(x)

(x - z) - 2i y (x)

(z)

y(x)dx

+ 2i y (z)y (x)

y(x)dx

(3.25)

Considering the case x

= z we get

(z)

y(x)dx

(z)

y(x)dx

= 0.

(3.26)

Although equation (3.26) allows a broader range of solutions, we restrict our considerations to the

case

y(x)dx

= 0,

(z)

y(x)dx

= 0

(3.27)

and use the following ansätze for

(z)

dz and

(z)

= c(z, t)dz + c

(z, t)y(z)dz,

= c

(z, t) + c

(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)

-2

(x)

(z)

y(x)dx

+2y (x)

(z)

y(x)dx

= 0.

Now, we put in ansatz (3.28), make use of

(t) and take into consideration that this equation

has to hold for all choices of y

(G, ), hence the coefficients of 1, y, y , y , y y , y y have to

vanish:

c(x, t)

= 0 = c = c(t),

(3.29)

(x, t)

= 0,

(3.30)

(t) -

- c

(x, t) = 0,

(3.31)

(x, t) = 0.

(3.32)

After differentiating equation (3.31) with respect to x we find

= 0

which, together with equation (3.30), gives

= 0, c

= c

(t).

(3.33)

Considering this and equations (3.29) and (3.31), ansatz (3.28) reads

= c

(t)z + c

(t),

(3.34)

(z)

= c(t)dz +

(t) - c

(t) y(z)dz.

(3.35)

· 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

= -y (z)

(t) - c (t) -

(t) -

t x

(t) y(z) -

(t) -

(t) y (z)

- 2i y(z)

(z, t)

+ 2i

(y(z)f

(z, t)) +

(x)

(t) -

(t)

(x - z)

- 2

(x)

(t) -

(t)

(x - z)

(z)

(x)

(t) -

(t) (x - z) dx + 2

(z)

(t) + y (z)

+ y(z)

t x

(3.36)

In this equation, the last two integrals involving the Dirac delta distribution vanish if we assume

that

(x), y (x) - 0 for x ±.

As equation (3.36) has to hold for all choices of y

(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) + 2

t z

(t) = 0,

(3.38)

2i f

(z, t) +

= 0,

(3.39)

(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

(x)dx +

(x) i

x (y(x)dx)

y(x)dx

= 0.

(3.41)

This equation has to hold for every

, hence the coefficient of has to vanish. Together with

(3.33) this furnishes

= 0,

(3.42)

hence equation (3.41) reads

(x) i

x (y(x)dx)

y(x)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

= 2a

+ a

(3.44)

= a

+ a

+ t xa

+ a

(3.45)

i x a

+ a

(3.46)

where a

, a

We insert equations (3.45), (3.44) and (3.37) into equation (3.35) to get

(z)

= (a

+ ta

)y(z)dz + a

dz,

with a

Theorem 3.2.1 (Local transformations of the viscous Hopf-Burgers FDE). The infinitesimals of the vis-

cous Hopf-Burgers FDE are given by

= 2a

+ a

= a

+ a

+ t xa

+ a

(z)

= (a

+ ta

)y(z)dz + a

dz,

+ a

)y(x)dx + a

+ g([y(x )], t),

where a

, a

are arbitrary constants and g is an arbitrary functional which has to

fulfill the viscous Hopf-Burgers FDE.

The associated generators read

= x

+ 2t

(x)dx

y(x)dx

= 2t x

+ 2t

(x)dx

y(x)dx

+ i

x y

(x)dx

= 2t

+ i

(x)dx

y(x)dx

= 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

In [6], H

OPF

states conditions which have to be fulfilled by Hopf functionals

([y(x)], t) = e

)

(G,

)

(vvv ,y)

([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

. The definition of a probability density functional requires f

to fulfill

the following two conditions:

Definition 3.3.1 (Probability density functional). f

is called a probability density functional if and only

1. f

is real-valued and non-negative, i.e.

([vvv (x)])

(3.47)

2. The integral of f

over the whole domain of integration equals 1, i.e.

(G,

)

([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

([y(x)], t) = ([-y(x)], t) where

denotes the complex conjugate of

(0, t) = 1,

|([ 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

, X

as the local transformations depend on seven parameters a

, a

and on a functional g. Especially, the generators X

and X

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

phys

= X

{1, 2, 3, 4, 5, 6}.

Thus, it suffices to have a look at X

and X

. If

shall be a physically relevant solution, X

and X

are

not independent. In order to see this, decompose g in

= g([y(x )], t)

into a constant part g

and a non-constant part g

= g

([y(x )], t), i.e.

([y(x )], t) = g

+ g

([y(x )], t).

Here, g

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

separately. We get the decomposition X

= X

+ X

with

= g

([y(x )], t)

We replace X

and X

by the two generators

+ X

= ( + g

)

, g

= g

([y(x )], t)

and calculate the associated global transformations by solving the Lie initial value problems

= + g

= g

([y(x )], t),

( = 0) = ,

( = 0) = .

The solutions are given by

([y(x)], t) = ([y(x)], t)e + (e - 1)g

(3.49)

([y(x)], t) = ([y(x)], t) + g

([y(x)], t) ,

(3.50)

In the following, we investigate the consequences of the conditions given by definition 3.3.2 for

and g

is given by equation (3.49).

1. As

has to fulfill

([y(x)], t) = ([-y(x)], t), we have

2. As

has to fulfill

([y(x)], t) = ([-y(x)], t) and (0, t) = 1 and since g

, we get

= -1.

This shows that X

and X

are not independent.

3. As

has to fulfill |([y(x)], t)| 1, we have

(-, 0]

and the generator X

+ X

associated with the physically relevant symmetry reads

phys

= (X

+ X

)

phys

= ( - 1)

Next, we investigate the consequences of the conditions given by definition 3.3.2 for

and g

([y(x)], t) if is given by equation (3.50).

1. As

has to fulfill condition

([y(x)], t) = ([-y(x)], t), we have

([y(x)], t) = g

([-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

(0, t) = 0.

(3.52)

3. As

has to fulfill condition |([y(x)], t)| 1, we have

()Re(g

) + Im()Im(g

) 0.

(3.53)

Altogether, the generator X

associated with the physically relevant symmetry reads

phys

= X

phys

= g

([y(x )], t)

where g

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

phys

= 2t

+ x

(x)dx

y(x)dx

= 2t

+ x

- U

phys

= 2t x

+ 2t

= 2t

+ 2t x

+ (x - 2tU)

+2t

(x)dx

y(x)dx

+ i x y(x)dx

phys

= 2t

+ i y(x)dx

= 2t

phys

y(x)dx

phys

= ( - 1)

phys

= g

([y(x )], t)

Table 3.1.: Comparison between the symmetries of the viscous Hopf-Burgers FDE and the viscous Burgers

equation.

Here, g

is a solution of the viscous Hopf-Burgers FDE satisfying the three conditions (3.51) - (3.53).

For X

phys

, 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

(scaling),

(time translation),

(space translation),

and

(Galilei invariance).

Generator

Global transformations associated with the generator

phys

= te

= xe ,

(x)dx = y(x)dxe ,

= ,

phys

= t + , x = x,

(x)dx = y(x)dx,

= ,

phys

= t,

= x + ,

(x)dx = y(x)dx,

= ,

phys

-2t

(1-2t )

(x)dx =

(x)dx

-2t

= exp i

-2t

x y

(x)dx ,

phys

= t,

= x + 2t , y(x)dx = y(x)dx

= exp i

(x)dx ,

phys

= t,

= x,

(x)dx = y(x)dx + dx, = ,

phys

= t,

= x,

(x)dx = y(x)dx,

= e + (1 - e ),

phys

= t,

= x,

(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

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

[8] Hosokawa, I.; Yamamoto K.: Numerical Study of the Burgers' Model of Turbulence Based on the

Characteristic Functional Formalism. Phys. Fluids 1970, 13, 1683-1692

[9] Hosokawa, I.; Yamamoto K.: Energy decay of Burgers' model of turbulence. Phys. Fluids 1976, 19,

1423-1424

[10] Grebenev, N. N.; Nazarenko, S. V.; Medvedev S. B.; Schwab I. V.; Chirkunov, Yu A.: Self-similar

solution in the Leith model of turbulence: anomalous power law and asymptotic analysis. J. Phys.

A: Math. Theoret. 2014, 47, 025501

[11] Alt, H. W.: Lineare Funktionalanalysis. Springer-Verlag, 2012

[12] Gelfand, I. M.; Fomin, S. W.: Calculus of variations. Prentice Hall 1963, New Jersey

[13] Ibragimov, N. H.; Kovalev, V. F.; Pustovalov V. V.: Symmetries of integro-differential equations: a

survey of methods illustrated by the Benny equations. Nonl. Dyn. 2002, 28, 135-153

[14] Ibragimov, N. H.: CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 1: Symme-

tries, Exact Solutions and Conservation Laws. CRC Press 1994

[15] Ibragimov, N. H.: CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 3: New

Trends in Theoretical Developments and Computational Methods. CRC Press 1996

[16] Klauder, J. R.: A Modern Approach to Functional Integration, Birkhäuser 2011

[17] Zawistowski, Z. J.: Symmetries of Integro-Differential Equations. Rep. Math. Phys. 2001, 48,

269-275

[18] Özer, T.: Symmetry group analysis of Benney system and an application for shallow-water equa-

tions, Mech. Res. Commun. 2005 , 32, 241-254

[19] Oberlack, M.; Waclawczyk, M.; Rosteck, A.; Avsarkisov, V.: Symmetries and their importance for

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

;x y

(x)y(x)

and

;x x y

(x)

is given

by equation (2.10):

- (

)

, y

(x )

, y

(x )

,x y

(x )

(x )dx -

,x y

(x )

(x )dx .

;x y

(x)y(x)

and

;x x y

(x)

are given below:

;x y

(x)y(x)

x (y(x)dx)

+ 2

, y

(x)

x y(x)dx

x (y(x)dx)

- 2

, y

(x)

x y(x)dx

, y

(z)

x (y(x)dx)

- 2

, y

(z)

, y

(x)

(z)

x y(x)dx

+ (

, y

(x)

)

(

, y

(x)

)

, y

(z)

(

, y

(x)

)

(z)

- 2

,t y

(x)

x y(x)dx

- 2

, y

(x)

,t y

(x)

- 2

, y

(z)y(x)

(z)

x y(x)dx

- 2

, y

(x)

, y

(z)y(x)

(z)

,x y

(x)

x y(x)dx

, y

(x)

,x y

(x)

, y

(x)y(x)

, y

(x)y(x)

, y

(z)

, y

(x)y(x)

(z)

+ 2

,x y

(x)

y(x)dx

- 2

,x y

(x)

y(x)dx

,x y

(z)

(y(x)dx)

- 2

,x y

(z)

, y

(x)

(z)

y(x)dx

- 2

, y

(z)

,x y

(x)

(z)

y(x)dx

+ 2

, y

(x)

,x y

(x)

- 2

, y

(x)

,x y

(x)

,x y

(z)

(

, y

(x)

)

(z)

- 2

, y

(z)

, y

(x)

,x y

(x)

(z)

- 2

,x y

(x)

,t y

(x)

- 2

,x y

(x)

, y

(z)y(x)

(z)

- (

,x y

(x)

)

,x y

(z)

, y

(x)y(x)

(z)

- 2

,x t y

(x)

y(x)dx

- 2

, y

(x)

,x t y

(x)

,x x y

(x)

y(x)dx

, y

(x)

,x x y

(x)

- 2

,x y

(z)y(x)

(z)

y(x)dx

- 2

, y

(x)

,x y

(z)y(x)

(z)

,x y

(x)y(x)

,x y

(x)y(x)

, y

(z)

,x y

(x)y(x)

(z)

,t y

(x)y(x)

, y

(z)y(x)y(x)

(z)

,x y

(x)y(x)

,x x y

(x)

y(x)dx

,x y

(x)

x y(x)dx

,x x y

(x)

, y

(x)

,x y

(x)

,x y

(x)

,x y

(x)

, y

(x)

, y

(x)

,z y

(z)

x y(x)dx

- 2

,z y

(x)y(z)

(z)

x y(x)dx

,z x y

(z)

( y(x)dx)

,z y

(z)

x( y(x)dx)

,x y

(x)

,z y

(z)

y(x)dx

, y

(x)

,z x y

(z)

y(x)dx

- 2

,z x y

(x)y(z)

(z)

y(x)dx

- 2

,z x y

(x)y(z)

, y

(x)

(z)

- 2

,z y

(x)y(z)

,x y

(x)

(z)

,x y

(x)y(x)

,z y

(z)

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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Title

Lie symmetry analysis of the Hopf functional-differential equation

Subtitle

Lie-Symmetrieanalyse der Hopf-Funktionaldifferentialgleichung

College

Technical University of Darmstadt (Fachbereich Maschinenbau, Fachgebiet für Strömungsdynamik, AG Turbulence theory and modelling)

Grade

1,0

Author

Daniel Janocha (Author)

Publication Year

2015

Pages

33

Catalog Number

V307089

ISBN (eBook)

9783668058460

ISBN (Book)

9783668058477

Language

German

Tags

Lie symmetries Hopf equation Burgers equation functional differential equations turbulence integro-differential equations Lie Symmetrieanalyse Hopf-Funktionaldifferentialgleichung

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