2009:035 CIV
M A S T E R ' S T H E S I S
Potential Symmetries
Jaime Dols Duxans
Luleå University of Technology MSc Programmes in Engineering
Engineering Physics Department of Mathematics
2009:035 CIV - ISSN: 1402-1617 - ISRN: LTU-EX--09/035--SE
Abstract
We derive and analyze the so-called potential symmetries of certain partial differ- ential equations. At first we calculate adjoint symmetries to find integrating factors that provide the conserved currents and flux densities for the conservation law of the given partial differential equations. For some partial differential equations one is able to gain a potentialisation of the given equations by introducing a new depen- dent variable, the so-called potential variables. This process can in certain cases be repeated to obtain a chain of potentialisations for the same starting equation. The combined auxiliary systems associated to the chain of potentialisations give, in some cases, interesting higher degree potential symmetries for PDEs which are realized as Lie point symmetries of the auxiliary systems.
iii
Acknowledgements
I would like to thank specially professor Norbert Euler for his dedication and time for helping me during the study of this subject of Mathematics. I want to thank as well Marianna Euler and Staffan Lundberg for their advise in mathematical programming and the use of L
ATEX editor.
v
Contents
Abstract iii
Acknowledgments v
1 Introduction 1
2 Lie Point Symmetry 3
2.1 Prolongations . . . . 4
3 Vertical Lie Group Generator 9
4 Conservation laws 13
4.1 Integrating factor . . . 13 4.2 Adjoint symmetry . . . 14 4.3 Direct conservation law method . . . 15
5 Potential symmetries 17
5.1 Examples . . . 20 5.2 Equation u
t= u
3x+ 3u
2x+ 3uu
xx + 3u
2u
x. . . 22 5.3 Equation u
t= u
2u
xx. . . 23
6 Conclusion 37
vii
Chapter 1 Introduction
The field of ordinary differential equations (ODE) and partial differential equations (PDE) have been a source of study for mathematicians since long time ago. If these equations become non-linear, usual in almost all processes taking place in nature, researchers have to take advantage of approximations, truncations or special condi- tions, leading to linear models. However exact solutions, are difficult to reach.
At the end of nineteenth century, the Norwegian mathematician Sophus Lie intro- duced a method that transform a set of solutions to another without variation and create what is known as symmetry groups or Lie groups. By this modification, this method allow us to get information of the group, or even diminish the amount of independent variables needed to get a solution.
All the technics used before his theory to solve non-linear ODE or PDE turned out to be special cases of his theory and nowadays the Lie-groups have mainly importance in mathematics and physics and also in solving methods of engineering applications.
This thesis begins with a short introduction to the definitions and methods needed to calculate these desired symmetries in order to define potential symmetries of PDEs.
This last chapter is the main goal of the thesis in which we try to give a feeling of what kind of symmetries we can get from an initial PDE by enlarging our boundary of the study to higher degree potential auxiliary systems by introducing auxiliary dependent variables, also called potentials.
1
Chapter 2
Lie Point Symmetry
The Lie transformation groups that define point symmetry transformations for dif- ferential equations, provide one-to-one mappings of solutions in a space of a subset of R
N[2]. These are infinitesimal transformations for a small group parameter .
Suppose x = (x
1, x
2, ..., x
p) and u = (u
1, u
2, ..., u
q). Then the Taylor expansion of the Lie symmetry transformation is:
Γ
:
x
∗1= ϕ
1(x, u; ) = x
1+ ξ
1(x, u) + O(
2) ≡ x
1+ Zx
1+ O(
2) x
∗2= ϕ
2(x, u; ) = x
2+ ξ
2(x, u) + O(
2) ≡ x
2+ Zx
2+ O(
2)
·
·
·
x
∗p= ϕ
p(x, u; ) = x
p+ ξ
p(x, u) + O(
2) ≡ x
p+ Zx
p+ O(
2) u
∗1= ψ
1(x, u; ) = u
1+ η
1(x, u) + O(
2) ≡ u
1+ Zu
1+ O(
2) u
∗2= ψ
2(x, u; ) = u
2+ η
2(x, u) + O(
2) ≡ u
2+ Zu
2+ O(
2)
·
·
·
u
∗q= ψ
q(x, u; ) = u
q+ η
q(x, u) + O(
2) ≡ u
q+ Zu
q+ O(
2),
(2.1)
where the generator, Z, of this transformation acting on the (x, u)-space is of the form
3
Z =
p
X
i=1
ξ
i(x, u) ∂
∂x
i+
q
X
j=1
η
j(x, u) ∂
∂u
j. (2.2)
The symmetry of the differential equation is characterized by generators.
2.1 Prolongations
Now consider R
3. The Lie symmetry transformation is now:
Γ
:
ˆ
x = ϕ
1(x, t, u; ) = x + ξ
1(x, t, u) + O(
2) ˆ t = ϕ
2(x, t, u; ) = t + ξ
2(x, t, u) + O(
2) ˆ
u = ψ(x, t, u; ) = u + χ(x, t, u) + O(
2),
(2.3)
where
Z = ξ
1(x, t, u) ∂
∂x + ξ
2(x, t, u) ∂
∂t + η(x, t, u) ∂
∂u . (2.4)
The first prolongation of Γ will include two extra dependent variables in the space, turning to the so called jetspace, namely to the first order x-derivative and t- derivative of the dependent variable u:
Γ
(1):
ˆ
x = ϕ
1(x, t, u; ) = x + ξ
1(x, t, u) + O(
2) ˆ t = ϕ
2(x, t, u; ) = t + ξ
2(x, t, u) + O(
2) ˆ
u = ψ(x, t, u; ) = u + χ(x, t, u) + O(
2) ˆ
u
xˆ= ψ(x, t, u, u
x, u
t; ) = u
x+ γ
(x)(1)+ O(
2) ˆ
u
ˆt= ψ(x, t, u, u
x, u
t; ) = u
t+ γ
(t)(1)+ O(
2),
(2.5)
where
γ
(x)(1)= D
xχ − u
xD
xξ
1− u
tD
xξ
2γ
(t)(1)= D
xχ − u
xD
tξ
1− u
tD
tξ
2,
(2.6)
2.1. PROLONGATIONS 5 and where D
xand D
tdenotes the total x- and t-derivative operator, respectively,
D
x:= ∂
∂x + u
x∂
∂u + u
xt∂
∂u
t+ u
xxt∂
∂u
xt+ u
xtt∂
∂u
tt+ u
2x∂
∂u
x+ u
3x∂
∂u
2x+ · · · (2.7)
D
t:= ∂
∂t + u
t∂
∂u + u
tt∂
∂u
t+ u
xt∂
∂u
x+ u
xtt∂
∂u
xt+ u
xxt∂
∂u
2x+ u
3xt∂
∂u
3x+ · · ·. (2.8) The generator is prolonged due to the extra variables:
Z
(1)= ξ(x, t, u) ∂
∂x + η(x, t, u) ∂
∂u +γ
x(1)(x, t, u, u
x, u
t) ∂
∂u
x+ γ
t(1)(x, t, u, u
x, u
t) ∂
∂u
t. (2.9)
The second prolongation of Z takes the form,
Z
(2)= ξ(x, t, u) ∂
∂x + η(x, t, u) ∂
∂u + γ
x(1)(x, t, u, u
x, u
t) ∂
∂u
x+γ
t(1)(x, t, u, u
x, u
t) ∂
∂u
t+ γ
xx(2)(x, t, u, u
x, u
t, u
xx, u
xt, u
tt) ∂
∂u
xx+γ
xt(2)(x, t, u, u
x, u
t, u
xx, u
xt, u
tt) ∂
∂u
xt+γ
tt(2)(x, t, u, u
x, u
t, u
xx, u
xt, u
tt) ∂
∂u
tt,
(2.10)
where
γ
(xx)(2)= D
xγ
x(1)− u
xxD
xξ
1− u
xtD
xξ
2γ
(xt)(2)= D
xγ
x(1)− u
xxD
xξ
1− u
xtD
tξ
2γ
(tt)(2)= D
tγ
t(1)− u
xtD
tξ
1− u
ttD
tξ
2.
(2.11)
Proposition 2.1 The n-prolongation of (2.4) is
Z
(n)= ξ
1(x, t, u) ∂
∂x + ξ
2(x, t, u) ∂
∂t + η(x, t, u) ∂
∂u + γ
(x)(1)(x, t, u, u
x) ∂
∂u
x+γ
(t)(1)(x, t, u, u
x, u
t) ∂
∂u
t+ γ
xx(2)(x, t, u, u
x, u
t, u
xx, u
xt, u
tt) ∂
∂u
xx+γ
xt(2)(x, t, u, u
x, u
t, u
xx, u
xt, u
tt) ∂
∂u
xt+ γ
tt(2)(x, t, u, u
x, u
t, u
xx, u
xt, u
tt) ∂
∂u
tt+ · · ·
· · · + γ
(nx(n)(x, t, u, u
x, u
t, · · ·, u
nx) + u
(x,t,··· un)) ∂
∂u
nx,
(2.12)
where
γ
nx(n)= D
xγ
(n−1)({n−1}x)− u
nxD
xξ
1− u
(n−1)xtD
xξ
2. (2.13)
The n-prolongation of Γ is:
Γ
(n):
ˆ
x = ϕ
1(x, t, u; ) = x + ξ
1(x, t, u) + O(
2) t = ϕ ˆ
2(x, t, u; ) = t + ξ
2(x, t, u) + O(
2) ˆ
u = ψ(x, t, u; ) = u + χ(x, t, u) + O(
2) ˆ
u
xˆ= ψ(x, t, u, u
x, u
t; ) = u
x+ γ
(x)(1)+ O(
2) ˆ
u
ˆt= ψ(x, t, u, u
x, u
t; ) = u
t+ γ
(t)(1)+ O(
2)
·
·
· ˆ
u
nˆx= ψ(x, t, u, u
x, u
t· · · u
nx; ) = u
nx+ γ
(nx)(n)+ O(
2).
(2.14)
2.1. PROLONGATIONS 7 The prolongations are needed for the invariance condition of differential equa- tions. The concept of vertical Lie symmetry generator is described in the following Chapter 3. This way to write prolongations have a more simple structure to work with and the Lie symmetry generators can easily be extended to higher order.
We remind that we are always working with a one-parameter Lie group of transfor-
mations in a space of independent and dependent variables. This space is enlarged
by all the derivatives of the dependent variables up to a finite order, where the corre-
sponding infinitesimals characterize the symmetry group transformation of a given
PDE. These symmetries are local at any point x, but they can belong to other classes
like the so-called contact, Lie-B¨ acklund or nonlocal symmetries. A special type of
nonlocal symmetries are studied in Chapter 5, these are the potential symmetries.
Chapter 3
Vertical Lie Group Generator
The invariant surface condition
Q(x, t, u
x, u
t) = ξ
1(x, t, u)u
t+ ξ
2(x, t, u)u
x− η(x, t, u) = 0. (3.1)
is the equation that defines all the invariants of the corresponding Lie transforma- tion group. We define now the general evolution equation E:
E := u
t− F (x, u, u
x, ...) = 0. (3.2)
Observe that in (3.2) F is not depending on the independent variable t. This is also known as the Cauchy-Kovalevskaya form [2], where the equation is expressed in a solved form in terms of one pure derivative.
Proposition 3.1 The PDE of n
th-order is invariant under a Lie Symmetry trans- formation group generated by Z if and only if the following condition is fulfilled:
Z
v(n)E
E=0= 0, (3.3)
where
Z
v(n)= Q ∂
∂u + D
xQ ∂
∂u
x+ D
tQ ∂
∂u
t+ D
x◦ D
tQ ∂
∂u
xt+D
2xQ ∂
∂u
xx+ · · · + D
xnQ ∂
∂u
nx.
(3.4)
The generator Z
v= Q
∂u∂is known as the vertical symmetry generator where Q is given in (3.1). Z
(n)given by (3.4) is the n
thprolongation of Z
v.
9
Condition (3.3) is equivalent to L
E[u]Q
E=0
= 0, (3.5)
where L is the linear operator, L
E[u] = ∂E
∂u + ∂E
∂u
tD
t+ ∂E
∂u
ttD
t2+ ∂E
∂u
xtD
x◦ D
t+ ∂E
∂u
xxD
x2+ ... + ∂E
∂u
kxD
xk.
(3.6)
Equation (3.3) provides an over determined system of linear PDEs in ξ
1, ξ
2and η for its corresponding symmetry generators.
For a system of n equations and m dependent variables u
1, u
2, ..., u
mthe condition will appear in the following proposition.
Proposition 3.2 For n equations and m dependent variables, the invariance con- dition for E
i= 0 (i = 1, .., n) is
m
X
j=1
L
Ei[u
j]Q
jE=0
= 0,
i = 1, .., n. j = 1, ..., m.
(3.7)
where E
i= {E
1, E
2..., E
n} and E
i= 0.
For example, consider two k
th-order equations E
1= 0, E
2= 0 and two indepen- dent variables (j = 2). Following Proposition 3.2 the invariance condition follows:
In order to work with a simpler notation we rename here u
1= u and u
2= v.
(L
E1[u]Q
1+ L
E1[v]Q
2)
E1=0,E2=0
= 0
(L
E2[u]Q
1+ L
E2[v]Q
2)
E1=0,E2=0
= 0, (3.8)
where
11
L
E1[u]Q
1= Q
1∂E
1∂u + ∂E
1∂u
tD
tQ
1+ ∂E
1∂u
ttD
t2Q
1+ ∂E
1∂u
xt(D
x◦ D
t)(Q
1) + ∂E
1∂u
xxD
2xQ
1+ ... + ∂E
1∂u
kxD
xkQ
1L
E1[v]Q
2= Q
2∂E
1∂v + ∂E
1∂v
tD
tQ
2+ ∂E
1∂v
ttD
t2Q
2+ ∂E
1∂v
xt(D
x◦ D
t)(Q
2) + ∂E
1∂v
xxD
2xQ
2+ ... + ∂E
1∂v
kxD
xkQ
2L
E2[u]Q
1= Q
1∂E
2∂u + ∂E
2∂u
tD
tQ
1+ ∂E
2∂u
ttD
t2Q
1+ ∂E
2∂u
xt(D
x◦ D
t)(Q
1)
+ ∂E
2∂u
xxD
2xQ
1+ ... + ∂E
2∂u
kxD
xkQ
1L
E2[v]Q
2= Q
2∂E
2∂v + ∂E
2∂v
tD
tQ
2+ ∂E
2∂v
ttD
t2Q
2+ ∂E
2∂v
xt(D
x◦ D
t)(Q
2) + ∂E
2∂v
xxD
x2Q
2+ ... + ∂E
2∂v
kxD
kxQ
2.
These conditions will set the over determined linear system of equations that will
give us the symmetry generators.
Chapter 4
Conservation laws
The conservation laws in nature can appear in several formats like integral conser- vation law or differential conservation law. For example describing diffusion for a chemical component. For our case the differential form is the most appropriate to remark. The equation for this property in a chemical concentration looks like this:
∂u
∂t + ∂φ
∂x = 0 (4.1)
where
∂u∂tis the variation of the density with respect to time, and
∂φ∂xis the variation of flux with respect to position.
The standard conservation law used in our study of PDE will be the form
D
tΦ
t+ D
xΦ
x= 0. (4.2)
where, Φ
tis the current density and Φ
xis the current flux.
4.1 Integrating factor
The integrating factor, Λ, is the multiplying term to the given equation
E := u
t− F (x, u, u
x, ...) = 0, (4.3)
fulfilling the conservation law
Λ(x, t, u, u
x, ..., u
qx)E = D
tΦ
t+ D
xΦ
x= 0. q even (4.4) Every conservation law has at least one integrating factor.
13
4.2 Adjoint symmetry
The necessary and sufficient conditions for the integrating factor Λ, are L
∗E[u]Λ
E(j)=0
= 0 (4.5)
L
Λ[u]E = L
∗Λ[u]E = 0, (4.6)
where
L
∗E[u] = ∂E
∂u − D
x◦ ∂E
∂u
x− D
t◦ ∂E
∂u
t+ D
2x◦ ∂E
∂u
2x+ ...(−1)
nD
nx◦ ∂E
∂u
nx. (4.7) The expression (4.5) is the adjoint symmetry condition of (4.3). The operator L
∗is the adjoint of the linear operator L (see 3.6). If we apply this adjoint linear operator on the integrating factor, we will end up with an expression of the form:
D
tΛ
E=0= −L
∗F[u]Λ, (4.8)
where L
∗F[u] is L
∗F[u] = ∂F
∂u − D
x◦ ∂F
∂u
x+ D
2x◦ ∂F
∂u
2x+ ...(−1)
nD
xn◦ ∂F
∂u
nx. (4.9)
This equation provides all possible integrating factors that can be used to fulfill the conservation law. To check the solution of the adjoint symmetry condition we can apply the Euler operator, ˆ E
[u], for the condition
E ˆ
[u](ΛE) ≡ 0. (4.10)
Definition 4.1 ˆ E
[u]is the operator which annihilates any divergence expression, D
tΦ
t+ D
xΦ
xand is defined by
E ˆ
[u]= ∂
∂u − D
x◦ ∂
∂u
x− D
t◦ ∂
∂u
t+ D
2x◦ ∂
∂u
2x− D
x3◦ ∂
∂u
3x+ · · ·. (4.11) Thus,
E ˆ
[u](D
tΦ
t+ D
xΦ
x) ≡ 0. (4.12)
4.3. DIRECT CONSERVATION LAW METHOD 15
4.3 Direct conservation law method
In [1] a direct method to find all the integrating factors from a conservation law of PDE’s in Cauchy-Kovalevskaya form is shown.
For a given Λ, Φ
tis obtained by solving
Λ = ˆ E
[u]Φ
t. (4.13)
Consider an n
th-order evolution equation of the general form
u
t= F (x, u, u
x, u
xx, u
3x, . . . , u
nx). (4.14)
Proposition 4.1: [3]Let Λ be an integrating factor for the evolution equations (4.14) and assume that the corresponding conserved current, Φ
t, admits the dependence
Φ
t= Φ
t(x, u, u
x, u
xx, u
3x). (4.15)
Then the flux, Φ
x, for (4.14) is given by Φ
x= −D
x−1(ΛF ) − ∂Φ
t∂u
xF − ∂Φ
t∂u
xxD
xF − ∂Φ
t∂u
3xD
2xF
+F D
x∂Φ
t∂u
xx− F D
x2∂Φ
t∂u
3x+ (D
xF )D
x∂Φ
t∂u
3x.
(4.16)
Remark:
D
tΦ
t+ D
xΦ
x= 0
⇔ Φ
x= D
x−1(D
tΦ
t).
(4.17)
Integrating (4.17) by parts leads to (4.16).
Chapter 5
Potential symmetries
Assume now that the evolution equation
u
t= F (x, u, u
x, u
xx, u
3x, . . . , u
nx) (5.1)
admits a conserved current, Φ
t1, and flux, Φ
x1. By [5] a first potential variable v is then defined by the auxiliary system:
( v
x= Φ
t1(x, u, u
x, . . .) v
t= −Φ
x1(x, u, u
x, . . .).
(5.2)
We name system (5.2) the first auxiliary system of (5.1). Assume further that v
tcan be expressed in terms of the first potential variable v, i.e. v
tbecomes by v
xin (5.2) the first potential equation of the general form
v
t= G(x, v
x, v
xx, . . . , v
nx) = −Φ
x1u→v
(5.3)
which may again admit a conserved current, Φ
t2, and flux, Φ
x2. A further potential w is then introduced for (5.3), and named the second potential for (5.1), by the second auxiliary system
( w
x= Φ
t2(x, v, v
x, . . .) w
t= −Φ
x2(x, v, v
x, . . .).
(5.4)
The corresponding potential equation for (5.3) is then obtained from the two equa- tions of the system (5.4), which we assume to have the general form
w
t= H(x, w
x, w
xx, . . . , w
nx). (5.5)
17
We name (5.5) the second potential equation for (5.1). A step further bring us to the third auxiliary system. This is build again from the second potential equation (5.5) with the new potential variable q:
( q
x= Φ
t3(x, w, w
x, . . .) q
t= −Φ
x3(x, w, w
x, . . .)
(5.6)
with the third potential equation for (5.1)
q
t= V (x, q
x, q
xx, . . . , q
nx). (5.7)
Definition 5.1: The Lie point symmetry generators Z = ξ
1(x, t, u, v) ∂
∂x + ξ
2(x, t, u, v) ∂
∂t + η
1(x, t, u, v) ∂
∂u + η
2(x, t, u, v) ∂
∂v (5.8)
of the first auxiliary system (5.2) for (5.1), i.e.
( v
x= Φ
t1(x, u, u
x, . . .) v
t= −Φ
x1(x, u, u
x, . . .),
are defined as the first-degree potential symmetries of (5.1) if the infinitesimals ξ
1, ξ
2and η
1depend essentially on the first potential variable v, that is
∂ξ
1∂v
2+ ∂ξ
2∂v
2+ ∂η
1∂v
26≡ 0. (5.9)
The second-degree potential symmetries of (5.1) are defined by the Lie point symme- try generators of the combined first- and second-auxiliary systems (5.2) and (5.4), that is the Lie point symmetry generators of the form
Z = ξ
1(x, t, u, v, w) ∂
∂x + ξ
2(x, t, u, v, w) ∂
∂t + η
1(x, t, u, v, w) ∂
∂u +η
2(x, t, u, v, w) ∂
∂v + η
3(x, t, u, v, w) ∂
∂w (5.10)
19 for the system
v
x= Φ
t1(x, u, u
x, . . .) v
t= −Φ
x1(x, u, u
x, . . .) w
x= Φ
t2(x, v, v
x, . . .) w
t= −Φ
x2(x, v, v
x, . . .),
where the infinitesimals ξ
1, ξ
2, η
1and η
2depend essentially on the second poten- tial variable w, that is
∂ξ
1∂w
2+ ∂ξ
2∂w
2+ ∂η
1∂w
2+ ∂η
2∂w
26≡ 0. (5.11)
The third-degree potential symmetries of (5.1) are defined by the Lie point sym- metry generators of the combined first-, second- and third auxiliary systems (5.2), (5.4) and (5.6), that is the Lie point symmetry generators of the form
Z = ξ
1(x, t, u, v, w, q) ∂
∂x + ξ
2(x, t, u, v, w, q) ∂
∂t + η
1(x, t, u, v, w, q) ∂
∂u +η
2(x, t, u, v, w, q) ∂
∂v + η
3(x, t, u, v, w, q) ∂
∂w + η
4(x, t, u, v, w, q) ∂
∂q
(5.12) for the system
v
x= Φ
t1(x, u, u
x, . . .) v
t= −Φ
x1(x, u, u
x, . . .) w
x= Φ
t2(x, v, v
x, . . .) w
t= −Φ
x2(x, v, v
x, . . .) q
x= Φ
t3(x, q, q
x, . . .) q
t= −Φ
t3(x, q, q
x, . . .),
where the infinitesimals ξ
1, ξ
2, η
1, η
2and η
3depend essentially on the third poten- tial variable q, that is
∂ξ
1∂q
2+ ∂ξ
2∂q
2+ ∂η
1∂q
2+ ∂η
2∂q
2+ ∂η
3∂q
26≡ 0. (5.13)
It should be clear that Definition 5.1 can easily be extended to m-degree potential symmetries.
5.1 Examples
Consider the Burgers’ equation [3] of second order
u
t= u
xx+ 2uu
x. (5.14)
It is well known that (5.14) admits only one local integrating factor and one local conservation law (see e.g. [6]), where
Λ = 1, Φ
t1= u, Φ
x1= − u
x+ u
2. (5.15) The first auxiliary system is
( v
x= u v
t= u
x+ u
2(5.16)
and the first potential equation has the form
v
t= v
xx+ v
x2. (5.17)
By Definition 5.1 the first-degree potential symmetries of (5.14) are the Lie point symmetries of (5.16). We obtain
Z
1= ∂
∂t , Z
2= ∂
∂x , Z
3= ∂
∂v (5.18a)
Z
4= x ∂
∂x + 2t ∂
∂t − u ∂
∂u , Z
5= 2t ∂
∂x − ∂
∂u − x ∂
∂v (5.18b)
Z
6= 4xt ∂
∂x + 4t
2∂
∂t − 2(x + 2tu) ∂
∂u − (2t + x
2) ∂
∂v (5.18c)
Z
∞= e
−v∂f
∂x − uf (x, t) ∂
∂u + f (x, t)e
−v∂
∂v , where f
t− f
xx= 0.
(5.18d) The first-degree potential symmetries (5.18a)–(5.18d) were firstly obtained by Vinogradov and Krasil’shchik [7]. The second auxiliary system is
( w
x= e
vw
t= v
xe
v(5.19)
5.1. EXAMPLES 21
and the second potential equation has the form
w
t= w
xx. (5.20)
The corresponding nonlocal Φ
t, Φ
x: ( Φ
t= e
R u dxΦ
x= −D
xe
R u dx= −ue
R u dx(5.21)
and the linearizing transformation w
x= e
R u dx.
Following Definition 5.1 the second-degree potential symmetries of the Burgers equation of second order are the Lie point symmetries of the combined auxiliary systems (5.16) and (5.19), i.e. the Lie point symmetries of the system
v
x= u v
t= u
x+ u
2w
x= e
vw
t= v
xe
v.
(5.22)
We obtain the following second-degree potential symmetries:
Z
1= ∂
∂t , Z
2= ∂
∂x (5.23a)
Z
3= x ∂
∂x + 2t ∂
∂t − u ∂
∂u + w ∂
∂w , Z
4= w ∂
∂w + ∂
∂v (5.23b)
Z
5= 2t ∂
∂x − 2 − uwe
−v∂
∂u − x + we
−v∂
∂v − xw ∂
∂w (5.23c)
Z
6= 2xt ∂
∂x + 2t
2∂
∂t − 2x + 2tu + we
−v− xuwe
−v∂
∂u
−
3t + 1
2 x
2+ xwe
−v∂
∂v −
tw + 1 2 x
2w
∂
∂w
(5.23d)
Z
∞= e
−vu ∂f
∂x − ∂
2f
∂x
2∂
∂u − e
−v∂f
∂x
∂
∂v − f (x, t) ∂
∂w , (5.23e)
where f
t− f
xx= 0.
5.2 Equation u t = u 3x + 3u 2 x + 3uu x x + 3u 2 u x
Consider the Burgers’equation [3] of third order
u
t= u
3x+ 3u
2x+ 3uu
xx + 3u
2u
x. (5.24)
The first auxiliary system is:
( v
x= u
v
t= u
xx+ 3uu
x+ u
3(5.25)
and the first potential equation has the form
v
t= v
3x+ 3v
xv
xx+ v
3x. (5.26)
The first-degree potential symmetries of (5.14) are then Z
1= ∂
∂t , Z
2= ∂
∂x , Z
3= ∂
∂v (5.27a)
Z
4= x ∂
∂x + 3t ∂
∂t − u ∂
∂u (5.27b)
Z
∞= e
−v∂f
∂x − uf (x, t) ∂
∂u + f (x, t)e
−v∂
∂v , (5.27c)
where f
t− f
3x= 0.
The second auxiliary system is ( w
x= e
vw
t= v
xxe
v+ v
x2e
v(5.28)
and the second potential equation has the form
w
t= w
3x. (5.29)
5.3. EQUATION U
T= U
2U
XX23 The second-degree potential symmetries of the Burgers’ of third order would follow from the Lie point symmetries of the combined auxiliary systems (5.25) and (5.28), i.e. the Lie point symmetries of the system
v
x= u
v
t= u
xx+ 3uu
x+ u
3w
x= e
vw
t= v
xxe
v+ v
x2e
v.
(5.30)
We obtain the following Lie point symmetries of system (5.30):
Z
1= ∂
∂t , Z
2= ∂
∂x (5.31a)
Z
3= x ∂
∂x + 3t ∂
∂t − u ∂
∂u + w ∂
∂w , Z
4= w ∂
∂w + ∂
∂v (5.31b)
Z
∞= e
−vu ∂f
∂x − ∂
2f
∂x
2∂
∂u − e
−v∂f
∂x
∂
∂v − f (x, t) ∂
∂w , (5.31c)
where f
t− f
3x= 0.
It is clear that the above Lie point symmetry generators (5.31a) are not potential symmetries of second degree for the third-order Burgers’ equation (5.26) since its condition in Definition 5.1
∂ξ
1∂w
2+ ∂ξ
2∂w
2+ ∂η
1∂w
2+ ∂η
2∂w
26≡ 0
is not satisfied. The same happens for the Burgers’ equation of fourth order, i.e.
second-degree potential symmetries for the Burgers’ hierarchy appear only for the Burgers’ equation of second order, namely the Burgers’ equation (5.14).
5.3 Equation u t = u 2 u xx
In this example we illustrate the above procedure to obtain potential symmetries for
this equation. The steps are pretty standard and it is possible to do the calculations
using either the vertical symmetry generators, Z
v, or the regular Z.
We have the following starting equation
u
t= u
2u
xx+ λ(x
2u
x− 3xu) for λ = 0 : (5.32)
u
t= u
2u
xx. (5.33)
We define now
E = u
t− F (x, u, u
x) = 0 (5.34)
where
F = u
2u
xx. (5.35)
First of all we need to calculate the integrating factor, Λ, by using the necessary condition (4.5) seen in the preceding chapter,
L
∗E[u]Λ
E=0= 0 (5.36)
is equivalent to the following expression:
D
tΛ
E=0= −L
∗F[u]Λ (5.37)
or
D
tΛ
ut=F= − ∂F
∂u Λ − D
x∂F
∂u
xΛ
+ D
x2∂F
∂u
xxΛ
. (5.38)
Taking the independent coefficients of every variable, we can build up a set of determining equations that will profile our Λ:
Λ(x, u) = 1
u
2(C1x + C2) (5.39)
Now we check that the adjoint symmetry is an integrating factor:
E ˆ
[u](Λ · (u
t− u
2u
xx)) = 0
5.3. EQUATION U
T= U
2U
XX25 E ˆ
[u](Λ · (u
t− u
2u
xx)) = ∂
∂u − D
x◦ ∂
∂u
x+ D
2x◦ ∂
∂u
2x− D
x3◦ ∂
∂u
3x· 1
u
2(C1x + C2)(u
t− u
2u
xx)
= 0.
Case Λ = u
−2: Taking, for instance, C1 = 0 and C2 = 1, we get Λ = u
−2. Using the direct method by [1], shown in Chapter 4, we can find Φ
tby integration, i.e.
(4.13)
Λ = ˆ E
[u]Φ
t. That is,
u
−2= ∂
∂u − D
x◦ ∂
∂u
x+ D
2x◦ ∂
∂u
2x− D
x3◦ ∂
∂u
3xΦ
t+ · · · (5.40)
Assume the following dependencies for Φ
t: Φ
t(x, u).
Then
u
−2= ∂Φ
t∂u , (5.41)
so that Φ
t=
Z
u
−2du = −u
−1. (5.42)
To get Φ
xwe use equation (4.16):
Φ
x= −D
−1x{ΛF } − ∂Φ
t∂u
xF − ∂Φ
t∂u
xx(D
xF ) + · · · = −D
−1x{ΛF }
Φ
x= −D
−1xu
2u u
xx= −D
−1x(u
xx) = −u
x.
The auxiliary potential system is ( v
x= Φ
t= −u
−1v
t= −Φ
x= u
x.
(5.43)
We now calculate the first degree potential symmetries for (5.33). To find the de- termining equations the condition in Proposition 3.2 (see 3.7) will be applied having two equations E
1and E
2and two invariant surfaces Q
uand Q
v.
Consider now the system (5.43):
E
1= v
x+ u
−1E
2= v
t− u
x.
(5.44)
The invariant surfaces are
Q
u= ξ
1u
x+ ξ
2u
t− η Q
v= ξ
1v
x+ ξ
2v
t− χ.
(5.45)
The invariance condition (3.7) give the determining equations namely:
L
E1[u]Q
u+ L
E1[v]Q
vE1=0,E2=0
= 0 (5.46a)
L
E2[u]Q
u+ L
E2[v]Q
vE1=0,E2=0
= 0. (5.46b)
Now we need to find the following terms:
L
E1[u]Q
uE1=0,E2=0
=
∂E∂u1Q
u= −u
−2Q
uL
E1[v]Q
v|
E1=0,E2=0
=
∂E∂v1x
(D
xQ
v) = D
xQ
vL
E2[u]Q
uE1=0,E2=0
=
∂E∂u2x
(D
xQ
u) = −D
xQ
uL
E2[v]Q
v|
E1=0,E2=0
=
∂E∂v2t
