Galaktionov V.A., Svirshchevskii S.R. Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics
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7 Partially Invariant Subspaces and GSV 363
7.5.1 Scaling order, index and evolutionary invariant sets
Simple exact solutions presented here admit other treatments to be discussed later
on. Currently, we apply a version explaining connection with the most well-known
group-invariantsolutions that are admitted by many scaling invariant PDEs.
1. Group of scalings, order, and index. Consider a simple Lie group of scaling
transformations
x
∗
= e
ε
x, t
∗
= e
µε
t, (7.129)
where µ ∈ IR \{0} is a fixed constant, with the infinitesimal generator
X = x
∂
∂x
+ µt
∂
∂t
. (7.130)
Then, equation (7.128) is invariant under (7.129) if operator F satisfies the following
homogenuity condition: for any s > 0,
F
sx, u,
1
s
u
1
, ...,
1
s
m
u
m
≡
1
s
µ
F(x , u, u
1
, ..., u
m
). (7.131)
From the invariance equation Xu = 0, it follows that (7.128) admits self-similar
solutions, depending on a single variable (the invariant of the group):
u(x , t) = θ(ξ), ξ =
x
t
1/µ
. (7.132)
On the set of invariant solutions (7.132), the PDE (7.128) reduces to the following
ODE for θ(ξ):
F
ξ,θ,θ
, ..., θ
(m)
+
1
µ
θ
ξ = 0.
Then, µ = µ(F) is said to be the scaling order of the operator F. For the operator
with power-like nonlinearities in derivatives
F[u] = φ(u)x
β
*
m
j =1
(u
j
)
α
j
, (7.133)
where φ is a smooth function, the scaling order is µ(F) =
m
j =1
jα
j
− β.
Consider a more general evolution PDE
u
t
= F[u] ≡
I
i=1
F
i
[u], (7.134)
where each operator F
i
is of the form (7.128)with the scaling orders µ
i
= µ
j
for all
i = j. We call I = I (F) the scaling index of the operator F.
2. Evolution invariance and algebraic differentiation. If I > 1, (7.134) does not
admit any Lie group (7.129). In this case, we begin our analysis by introducing the
set of functions
M
0
=
v ∈ C
∞
(IR ) : v
x
=
1
x
H (v)
, (7.135)
where H is an unknown C
∞
-function. As a standard practice, M
0
is said to be evo-
lutionary invariant for PDE (7.134) if
u(·, 0) ∈ M
0
⇒ u(·, t) ∈ M
0
for t ∈ (0, 1]. (7.136)
The contact, first-order structure of the equality
u
x
=
1
x
H (u) (7.137)
© 2007 by Taylor & Francis Group, LLC
364 Exact Solutions and Invariant Subspaces
(a differential constraint) includes the scaling invariant (7.132). This is seen by in-
tegration. The simple differential structure of M
0
makes it possible to introduce the
following algebraic differentiation in the set, that reduces the PDE on M
0
to a system
of ODEs.
Lemma 7.32
Let
u ∈ M
0
. Then,
u
k
=
1
x
k
G
k
[H ], k = 1, 2, ... , (7.138)
where the ordinary differential operators
G
k
satisfy the recursion relation: for any
H (u) ∈ C
∞
,
G
k+1
[H ] = H
d
du
G
k
[H ] − kG
k
[H ]
for
k = 1, 2, ... , G
1
[H ] = H.
Proof. Differentiating (7.138) yields
u
k+1
=−
k
x
k+1
G
k
[H ] +
1
x
k
d
du
G
k
[H ]u
1
≡
1
x
k+1
G
k+1
[H ],
since u
1
=
1
x
H (u) by the definition (7.135).
We present below a few of the quasilinear polynomial differential operators G
k
,
which are sufficient to deal with PDEs up to the fifth order:
G
2
[H ] = H (H
− 1),
G
3
[H ] = HG
2
− 2G
2
= H [(HH
)
− 3H
+ 2],
G
4
[H ] = HG
3
− 3G
3
= H [(H (HH
)
)
− 6(HH
)
+ 11H
− 6],
G
5
[H ] = H [(H (H (HH
)
)
)
− 10(H (HH
)
)
+35(HH
)
− 50H
+ 24].
(7.139)
4. Evolutionary invariant sets. Using the rule (7.138) and the scaling properties
(7.131) with µ = µ
i
of operators F
i
,wefind that, on M
0
,
F[u] =
I
i=1
F
i
x, u,
1
x
G
1
, ...,
1
x
m
G
m
≡
I
i=1
x
−µ
i
F
i
(1, u, G
1
, ..., G
m
) ≡
I
i=1
x
−µ
i
HC
i
[H ],
(7.140)
where, for convenience, we introduce the operators
C
i
[H ] =
1
H
F
i
(1, u, G
1
[H ], ..., G
m
[H ]) for i = 1, ..., I.
The main result is as follows:
Theorem 7.33
The set
M
0
is evolutionary invariant iff the function
H (u)
satisfies
the ODE system
H
d
du
C
i
[H ] = µ
i
C
i
[H ], i = 1, ..., I, (7.141)
and then solutions
u(x , t)
on
M
0
take the form
v ≡
u
1
dz
H(z)
= ln x + D(t). (7.142)
Here,
D(t)
solves the ODE
D
=
I
i=1
d
i
e
µ
i
D
,
where
d
i
= C
i
[H ]
u=1
. (7.143)
© 2007 by Taylor & Francis Group, LLC
7 Partially Invariant Subspaces and GSV 365
The ODE (7.143) is always integrated in quadratures and is the PDE (7.134) re-
stricted to the set M
0
.
Proof. In view of (7.142),
u
t
= H (u)D
(t). (7.144)
Substituting (7.144) into (7.134) and using (7.140) yields
D
(t) =
I
i=1
x
−µ
i
C
i
[H (u)] ≡ R(x, t). (7.145)
Therefore, R(x, t) does not depend on x , which results in
R
x
=
I
i=1
H
d
du
C
i
[H ] − µ
i
C
i
[H ]
x
−(µ
i
+1)
≡ 0.
Since functions {x
−(µ
i
+1)
} are linearly independent in IR
I
, we obtain the system
(7.141), meaning that
d
dv
C
i
[H ] ≡ H
d
du
C
i
[H ] = µ
i
C
i
[H ], so C
i
[H ] = d
i
e
µ
i
v
. (7.146)
Pluggingthe functions(7.146) with v = ln x +D givenby (7.142) into (7.145)yields
the ODE (7.143).
5. Group of scalings for I = 1. In the case I (F) = 1, for which the group of
scalings exists, one can solve (7.143) explicitly. Set µ
1
= µ and d
1
= d,so
D
= de
µD
⇒ D(t) =−
1
µ
lnt + constant.
Substituting this into (7.142) yields
u(x ,t)
1
dz
H(z)
= ln
x
t
1/µ
+ constant,
i.e., u(x, t) depends on the single scaling group invariant as in (7.132).
Remark 7.34 (On extensions) As an extension of the evolutionary invariant set
(7.137), it makes sense to consider
u
x
=
1
x
H (u) + G(u), (7.147)
with two unknown functions H and G. The ODE conditions on all the coefficients
(slightly harder than (7.138))can be obtained.The main difficultyis that (7.147) can-
not be integrated explicitly in general. A convenient integrable case (see Remarks)
corresponds to the Bernoulli equation v
x
=
1
x
v + v
n
for v = H (u). Then again a
single function H appears in (7.147). Are there other cases (an
OPEN PROBLEM)?
7.5.2 Back to invariant subspaces: equivalent formulation
We now comment on other equivalent (and sometimes simpler) forms of such exact
u(x , t) = U(ξ ), ξ = ln x + D(t), (7.148)
for solutions of (7.134). Differentiating (7.148) yields, as in the previous case,
u
1
= x
−1
U
(ξ) ≡ x
−1
P
1
[U],
u
2
= x
−2
{U
(ξ) − U
(ξ)}≡x
−1
P
2
[U],
... ... ...
u
k
= x
−k
{(P
k−1
[U])
− (k − 1)P
k−1
[U]}≡x
−k
P
k
[U],
© 2007 by Taylor & Francis Group, LLC
solutions. The first form is based on the representation (cf. (7.142))
366 Exact Solutions and Invariant Subspaces
where P
k
[U] = G
k
[U
] given in (7.139). Substituting into the PDE (7.134) yields
U
D
=
I
i=1
F
i
(x , U, x
−1
P
1
[U], ..., x
−m
P
m
[U])
≡
x
−µ
i
F
i
(1, U, P
1
[U], ..., P
m
[U]), or
D
=
I
i=i
x
−µ
i
1
U
˜
F
i
[U], where
˜
F
i
[U] = F
i
(1, U, P
1
[U], ..., P
m
[U]). (7.149)
Differentiating the last equality in x leads to
0 =
x
−µ
i
−1
1
U
˜
F[U]
− µ
i
1
U
˜
F[U]
.
Therefore, U satisfies I conditions
1
U
˜
F[U]
= µ
i
1
U
˜
F[U], so on integration
1
U
˜
F[U] = d
i
e
µ
i
ξ
, (7.150)
where d
i
are arbitrary constants. Finally, the ODE (7.149) takes the form
D
=
I
i=1
d
i
e
µ
i
D
. (7.151)
Determining U(ξ ) from (7.150) and D(t) from (7.151) yields the exact solutions
(7.148) of the equation (7.134).
Another representation of solutions (7.148)is as follows:
u(x , t) = V (ζ ), ζ = xs(t), (7.152)
where ζ = e
ξ
and s(t) = e
D(t )
. Then, instead of (7.150) and (7.151), we find
˜
1
V
˜
F
i
[V ] = d
i
ζ
µ
i
+1
, where
˜
F
i
[V ] = ζ
µ
i
F
i
(ζ, V, V
, ..., V
(m)
)
and
s
=
I
i=1
d
i
s
µ
i
+1
.
Both the representations are equivalent. Notice that the second form (7.152) is con-
nected with the invariant subspace L{x}.
Example 7.35 (General quasilinear heat equation) As a generalization to be also
treated later on, consider a parabolic PDE of reaction-diffusion type,
u
t
= F
2
[u] ≡
φ(u)u
xx
+ ψ(u)(u
x
)
2
+ f (u)
(7.153)
that contains three functionsφ(u) ≥ 0, ψ(u),and f (u). Bearing in mind an arbitrary
smooth change u = R(v), which leaves the structure of the PDE (7.153) invariant,
we are going to find solutions in the separable form
u = e(x)g(t) (7.154)
that are associated with the 1D subspace L{e} with an unknown function e. Plugging
(7.154) into (7.153) and dividing by eg yields
g
g
= φ
e
e
+ uψ
e
e
2
+
e
u
. (7.155)
Denote
e
(x)
e(x )
= h(x ) ⇒
e
e
−
(e
)
2
e
2
= h
and
e
e
= h
+ h
2
.
Then (7.155) reads
g
g
= φ(h
+ h
2
) + uψh
2
+
e
u
= h
+ (φ + uψ)h
2
+
e
u
. (7.156)
© 2007 by Taylor & Francis Group, LLC
7 Partially Invariant Subspaces and GSV 367
Differentiating in x vanishes the left-hand side, so
φh
+ [φ
u + 2(φ + uψ)]hh
+ (k + uψ)
uh
3
+
e
u
uh = 0. (7.157)
As in the GSV, it follows from (7.157) that h(x) must satisfy, for some constants α
1
,
α
2
,andα
3
, the ODE
h
= α
1
h
3
+ α
2
h + α
3
hh
. (7.158)
Then (7.157) reduces to the following system:
(φ + uψ) + α
1
φ = 0,
(
e
u
)
u + α
2
φ = 0,
φ
u + 2(φ + uψ) + α
3
φ = 0.
(7.159)
We will integrate it later on.
7.5.3 Second-order parabolic PDEs
Let us return to the original treatmentof PDEs via evolutionaryinvariantsets (7.135).
Example 7.36 (Quasilinear heat equation, continued) Consider again equation
(7.153). The scaling orders of two operators in (7.153) are µ
1
= 2andµ
2
= 0, so
that the scaling index is I (F
2
) = 2. Substituting the differential rule (7.138),one can
calculate that, on M
0
,
F
2
[u] = H
1
x
2
[φ(H
− 1) + ψ H ] +
f
H
≡ H
C
1
[H ] +
1
x
2
C
2
[H ]
. (7.160)
By Theorem 7.33, the invariant conditions (7.141) take the form
H
f
H
= 0, H [φ(H
− 1) + ψh]
= 2[φ(H
− 1) + ψ H ], or (7.161)
f = ν H,
HH
− 2H
+ 2 +
φ
φ
H (H
− 1) +
ψ
φ
H (H
− 2) +
ψ
φ
H
2
= 0,
(7.162)
where ν ∈ IR is a constant. The equation (7.153) on M
0
reduces to the following
ODE for the function D(t) in the exact solution (7.142):
D
= d
1
+ d
2
e
2D
,
where d
1
= (
f
H
)(1) = ν and d
2
= [φ(H
− 1) + ψ H ](1). Notice that (7.161) is a
system of two equations with four arbitrary unknown functions φ, ψ, f ,andH .
Example 7.37 (Radial N-dimensional equations) Consider the quasilinear heat
equation in IR
N
u
t
= [φ(u)u + ψ(u)|∇u|
2
] + f (u).
In the radial case, where x > 0 is the radial spatial variable and u = u
xx
+
N−1
x
u
x
,
|∇u|
2
= (u
x
)
2
, this PDE has the same scaling orders as (7.153). Then, in (7.160),
C
1
[H ] = φ(H
+ N − 2) + ψ H.
Therefore, the second equation in (7.162) is replaced by the following one:
HH
− 2H
− 2(N −2) +
φ
φ
H (H
+ N −2)
+
ψ
φ
H (H
− 2) +
ψ
φ
H
2
= 0.
© 2007 by Taylor & Francis Group, LLC
368 Exact Solutions and Invariant Subspaces
Example 7.38 (Equations with gradient-dependent diffusivity) Consider a gen-
eralization of the p-Laplacian equations studied in Section 6.4,
u
t
= φ(u)(u
x
)
α
u
xx
+ f (u)(u
x
)
β
≡ F
1
[u] + F
2
[u]. (7.163)
Here, µ
1
= α + 2andµ
2
= β.Ifα + 2 = β, then (7.163) is invariant under the
scaling group (7.129).Let α +2 = β. In this case, on M
0
,
F
1
[u] + F
2
[u] = H
1
x
α+2
φ H
α
(H
− 1) +
1
x
β
H
β−1
f
,
and the invariant conditions are
H [φ H
α
(H
− 1)]
= (α +2)φ H
α
(H
− 1), H (H
β−1
f )
= β H
β−1
f, (7.164)
or, which is the same,
HH
+ α(H
)
2
− 2(α +1)H
+ (α +2) +
φ
φ
H (H
− 1) = 0,
Hf
+ [(β − 1)H
− β] f = 0.
The ODE for D(t) is
D
= d
1
e
(α+2)D
+ d
2
e
β D
,
with constants d
1
= [φH
α
(H
− 1)](1) and d
2
= (H
β−1
f )(1).
Example 7.39 (Fully nonlinear parabolic PDEs) Consider the equation
u
t
= φ(u)(u
x
)
α
(u
xx
)
γ
+ f (u)(u
x
)
β
≡ F
1
[u] + F
2
[u].
The term (u
xx
)
γ
is typical for the dual porous medium equation,whichwasdis-
cussed in Example 1.18. Then µ
1
= α + 2γ and µ
2
= β, and we assume that
µ
1
= µ
2
,sothatI = 2. On M
0
,
F
1
[u] + F
2
[u] = H
1
x
α+2
γφH
α+γ −1
(H
− 1)
γ
+
1
x
β
H
β−1
f
.
Therefore, in the invariant conditions (7.164), the first equation is replaced by
H [φ H
α+γ −1
(H
− 1)
γ
]
= (α +2γ)φH
α+γ −1
(H
− 1)
γ
,
and the ODE for D(t) becomes
D
= d
1
e
(α+2γ)D
+ d
2
e
β H
, with d
1
= [φH
α+γ −1
(H
− 1)
γ
](1).
7.5.4 Fourth-order generalized thin film equations
Example 7.40 Consider a general fourth-order PDE
u
t
= F
4
[u] ≡ q(u)u
xxx
+ ρ(u)u
xxxx
+ F
2
[u], (7.165)
where F
2
is the second-order operator given in (7.153). In Section 3.1, we dealt with
a number of such nonlinear thin film models possessing various exact solutions. The
analysis of (7.165) is similar to that for the second-order PDE (7.153). The third and
fourth-ordertermin (7.165)havethe scalingorders µ
3
= 3andµ
4
= 4, respectively,
and I (F
4
) = 3. Hence, on M
0
, we add two extra terms to the right-hand side of
© 2007 by Taylor & Francis Group, LLC
7 Partially Invariant Subspaces and GSV 369
(7.160),
F
4
[u] = F
2
[u] + H
1
x
3
qC
3
[H ]
+ H
1
x
4
ρC
4
[H ]
,
C
3
[H ] =
1
H
G
3
[H ], C
4
[H ] =
1
H
G
4
[H ],
where G
3,4
are given in (7.139). In this case, we add two equations to the system
(7.162),
H (qC
3
)
= 3qC
3
and H (ρC
4
)
= 4ρC
4
. (7.166)
The ODE for D(t) takes the form
H
= d
1
+ d
2
e
2H
+ d
3
e
3H
+ d
4
e
4H
,
where d
3
= (qC
3
[H ])(1) and d
4
= (ρC
4
[H ])(1). The TFE (7.165) contains five
arbitrary functions {φ,ψ, f, q,ρ}, so, together with H , four ODEs (7.162), (7.166)
for six functions are obtained. Hence, such solutions exist for PDEs (7.165) with two
arbitrary coefficients (say, ρ(u) and q(u) of the higher-order operators).
7.5.5 Extensions
1. Other evolutionary invariant sets. In Section 1.4, we revealed several evolution
PDEs, in particular, of type (7.153) that possessed solutions on the trigonometric
subspace W = L{cos x}, or on the exponential subspace W = L{coshx}.These
solutions can be obtained by constructing sets of a different structure,
M
1
=
u : u
x
= tan xH(u)
.
The algebraic differentiation in M
1
is different from that in M
0
. Setting
w(x) =
1
cos
2
x
,
we obtain
u
2
≡ u
xx
= wH (H
+ 1) − HH
≡ H (wP
21
+ P
20
).
Denoting
Q[H ] = H [(HH
)
+ 3H
+ 2]
yields
u
4
= H (w
2
P
42
+ wP
41
+ P
40
), where
P
42
= 3Q + (H Q)
((·)
=
d
du
),
P
41
=−2Q − (H Q)
− (HH
)
− (H (HH
)
)
,
P
40
= (H (HH
)
)
.
In particular, the even derivatives u
2k
belong to a (k+1)-dimensional subspace,
u
2k
∈ L{1, w, ..., w
k
}.
Example 7.41 (Thin film-type equations) Consider the fourth-order PDE
u
t
= F
4
[u] ≡ ρ(u)u
xxxx
+ φ(u)u
xx
+ ψ(u)(u
x
)
2
. (7.167)
Since for u ∈ M
1
,
(u
x
)
2
= wH
2
− H
2
,
© 2007 by Taylor & Francis Group, LLC
370 Exact Solutions and Invariant Subspaces
it follows that, on M
1
,
F
4
[u] = H
w
2
(ρP
42
) + w(ρP
41
+ φP
21
+ ψ H )
+ρP
40
+ φP
20
− ψ H
≡ H (w
2
2
+ w
1
+
0
) ∈ L{1,w,w
2
}.
(7.168)
The index of F
4
(the number of linearly independent terms in (7.168)) is equal to 3.
Integrating u
x
= tan xH(u) yields
v ≡
u
1
dz
H(z)
=−lncos x + D(t)(cos x > 0).
Substituting this expression into (7.167) and using (7.168), we obtain
D
(t) = w
2
(x )
2
+ w(x)
1
+
0
. (7.169)
In view of the linear independence, functions
2
(v),
1
(v),and
0
(v) must be
exponential and equal to d
2
e
4v
, d
1
e
2v
,andd
0
, respectively. This yields the system of
three equations for H ,
H
2
= 4
2
, H
1
= 2
1
, and H
0
= 0. (7.170)
Then (7.169) becomes
H
= d
2
e
4H
+ d
1
e
2H
+ d
0
,
with the coefficients d
2
=
2
|
u=1
, d
1
=
1
|
u=1
,andd
0
=
0
|
u=1
. Here (7.170)
is a system of three ODEs for four arbitrary functions {φ,ψ,ρ, H }. Hence, such
solutions exist for a family of equations (7.167) with a single arbitrary coefficient,
e.g., ρ(u). Adding another thin film-type monomial, κ(u)u
x
u
xxx
, to the right-hand
side does not change the scaling index, since on M
1
,
u
1
u
3
∈ L{1,w,w
2
}.
Hence, no extra ODE is added to the system (7.170). A similar analysis is performed
for the sets given by the constraint
u
x
= tanh xH(u).
On more general approaches. We now clarify the origin of all the evolutionary
invariant sets studied above; cf. Example 7.35. Namely, consider the set
M : u
x
= g(x)H (u), (7.171)
and determine all possible functions g and H that guarantee the invariance in time.
Of course, integrating this constraint yields the following exact solution:
u(x , t) = U(e(x) + D(t)),
with three unknown functions e, D,andU. Such solution structures, which we have
revealed above in particular cases, in general, are related to St¨ackel’s form (1893)
and generalized separation of variables; see Remarks to Section 8.4.
We now borrow the following result from Section 8.3 (cf. identity (8.50)): (7.171)
is evolutionary invariant under the flow induced by the quasilinear heat equation
u
t
= (k(u)u
x
)
x
+ f (u), provided that (7.172)
© 2007 by Taylor & Francis Group, LLC
7 Partially Invariant Subspaces and GSV 371
F
1
≡ g
(x )kH + g
3
(x ) H
2
(kH)
+g(x)( f
H − fH
) + gg
(x )(3k
H
2
+ 2kHH
) ≡ 0.
(7.173)
The right-hand side of (7.173) belongs to the subspace
W = L
g
, g
3
, g, gg
(7.174)
and so, if it is 4D, we obtain four ODEs for three functions {k, f, H }, which are not
expected to admit any interesting solution.
3D reduction. The next step is to consider an extra reduction of the subspace (7.174)
by choosing g(x ) such that, for some constants α
1,2,3
, (7.158) holds. Assuming that
L{g
3
, g, gg
} is 3D yields the following system for three functions:
H (kH)
+ α
1
k = 0,
f
H − fH
+ α
2
kH = 0,
3k
H + 2kH
+ α
3
k = 0.
(7.175)
It is not difficult to characterize the whole family of such equations (7.172). Setting
kH = P, we derive from the first two equations
HH
=−
α
1
2
P
P
and
f
H
=
α
2
α
1
P
.
Integrating the first one,
H
2
=−α
1
P
P
+ A (A ∈ IR )
and substituting this H = H (P) and k =
P
H( p)
into the third equation in (7.175)
yields an autonomous third-order ODE for the single function P(u),
PP
= 2
α
3
α
1
(P
)
2
−α
1
P
P
+ A − 5P
P
(for A = 0, this can be reduced to a first-order equation).
Thus, in general, we obtain a four-parametric family of PDEs (7.172) possessing
such explicit solutions.
The algebraic differentiation on the set (7.171) depends on g and is not as perfect
as in the case where g(x ) =
1
x
. For instance,
u
xx
= g
2
HH
+ g
H,
u
xxx
= g
3
[α
1
H + H (HH
)
] + g(α
2
H ) + gg
(3HH
+ α
3
H ),
etc. It is not difficult to derive the ODE governingthe evolution on M for an arbitrary
admissible g(x). It follows from (7.171) that the function
v(x , t) = G(u) ≡
du
H(u)
=
x
0
g(y) dy + D(t) solves (7.176)
v
t
= k(u)v
xx
+ (kH)
(u)(v
x
)
2
+
f
H
(u).
Since v
x
= g(x) and v
xx
= g
(x ), setting x = 0 yields v(0, t) ≡ G(u(0, t)) = D(t),
and so u(0, t) = G
−1
(v(0, t)), from which comes the following ODE for D(t):
D
= k(G
−1
(D))g
(0) + (kH)
(G
−1
(D))g
2
(0) +
f
H
(G
−1
(D)). (7.177)
2D reductions. It is now easy to obtain from (7.158) further reductions leading to
© 2007 by Taylor & Francis Group, LLC
372 Exact Solutions and Invariant Subspaces
some specific functions g,
α
1
= α
2
= 0,α
3
=−2 ⇒ g(x) =
1
x
,
α
1
= α
3
= 0,α
2
=−1 ⇒ g(x) = cosx,
α
1
= α
3
= 0,α
2
= 1 ⇒ g(x ) = coshx .
(7.178)
The most transparent degenerate case is
α
1
= α
2
= α
3
= 0 ⇒ g(x ) = x. (7.179)
These cases will play a role for constructing sign-invariants for quasilinear parabolic
PDEs which is the main subject of the next chapter.
7.5.6 Evolutionary invariant sets for hyperbolic PDEs
Consider the set (7.171) for the second-order hyperbolic equation
u
tt
= (k(u)u
x
)
x
+ f (u). (7.180)
Equating u
xtt
given by (7.171) and (7.180), we obtain the invariance condition (cf.
(7.173))
F
2
≡−gH
(u
t
)
2
+ g
kH + g
3
(k
H
3
+ 2k
H
2
H
+ kH
2
H
)
+g( f
H − fH
) + gg
(2kHH
+ 3k
H
2
) ≡ 0.
(7.181)
If this identity is true, performing the same change (7.176) yields the PDE
v
tt
+ H
(u)(v
t
)
2
= k(u)v
xx
+ (kH)
(u)(v
x
)
2
+
f
H
(u),
so that, exactly as in (7.177), we obtain the ODE
D
+ H
(G
−1
(D))(D
)
2
= k(G
−1
(D))g
(0)
+(kH)
(G
−1
(D))g
2
(0) +
f
H
(G
−1
(D)).
There is an essential difference with the parabolic case that is related to the first
term on the right-hand side of (7.181), so that one needs H to be a linear function,
H
= 0.
The rest of theanalysis is equally based on the dimension of the linear space (7.174).
In particular,
g(x) =
1
x
⇒ W = L
1
x
,
1
x
3
,
and (7.181) yields two ODEs for the coefficients
f
H − fH
= 0,
k
H
2
+ 2k
HH
− 2kH
− 3k
H +2k = 0.
The first ODE implies
f (u) = µH (u).
In particular, choosing a linear function
H (u) = αu
yields, for k(u), second-orderEuler’s equation
α
2
u
2
k
+ α(2α −3)uk
+ 2(1 − α)k = 0.
© 2007 by Taylor & Francis Group, LLC