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 343
and a DS for the remaining two coefficients C
1,2
is obtained,
C
1
= P(0, 0)C
2
1
+
2P(1, −1)A
3
+ 2P(
1
3
, −
1
3
)A
4
A
5
C
1+ρ
3
2
,
C
2
= 2P(0, 1)C
1
C
2
.
This can be integrated in quadratures and possesses blow-up solutions. For the thin
film operator (7.23), we obtain six extra conditions that imply existence of a four-
parameter family of TFEs possessing such exact solutions.
For the second-order PDE (7.11), the DS (7.27) with C
k
= ... admit, in general,
either exponential particular solutions C
k
(t) = A
k
e
ρ
k
t
with ρ
1
= 0, or the algebraic
ones (7.22), where ρ
1
=−2. If P(0, 1) = P(0, −1) = P(0,
1
3
) = P(0, −
1
3
),there
exist solutions C
k
(t) = A
k
C
2
(t) for k = 3, 4, 5, where C
1
(t) and C
2
(t) solve a DS,
and (7.26) give two extra conditions on {A
k
}.
7.1.2 On further extensions
1.
∂
∂t
-dependent operators. We considered such operators in Sections 1.5.2 (see
tially invariant modules), a similar computationalanalysis can be performed, though
the overdeterminedDSs becomemore cumbersome and difficult. We must admit that
there is a higher probability to obtain purely, in both x and t variables, exponential
solutions that sometimes can be more efficiently manipulated by using Baker–Hirota
derivatives and bilinear operators, or by other methods from theory of integrable
PDEs.
Example 7.8 (PDE with nonstationary part from the KdV equation) Let us be-
gin with the following rather artificial PDE:
u
x
u
t
− uu
xt
= F[u], (7.28)
where the quadratic left-hand side has been borrowed from the bilinear form of the
KdV equation (0.31) in the Introduction. Here F[u] is the general operator (7.1).
First, consider the expansion (7.4) with the same assumptions on the exponents.Not
specifying the DS, we claim that the only possible solutions are exponential ones,
C
k
(t) = A
k
e
ρ
k
t
for all k = 1, 2, 3, 4, (7.29)
where {ρ
k
} satisfies a number of conditions associated with the values of the poly-
nomial P at p
k
. One can think this rigid exponential solution structure as being as-
sociated with the specific form of the left-hand side of (7.28), which is a full Baker–
Hirota derivative −2Du · u
t
,defined as, [22, 284]:
Dv · w =
d
dz
v(x +z)w(x − z)
z=0
= v
x
w − vw
x
. (7.30)
Therefore,the Baker–Hirota method applies ensuring that the right solutions are sup-
posed to be of exponential soliton-type only.
Furthermore, for the PDE with the first term only on the left-hand side
u
x
u
t
= F[u], (7.31)
© 2007 by Taylor & Francis Group, LLC
(1.135)) and 2.7, where invariant modules were dealt with. For invariant sets (par-
344 Exact Solutions and Invariant Subspaces
the conclusion remains the same: the overdetermined DS has the exponential solu-
tions (7.29) only. The exponential character of solutions is true generic for sets on
W
5
. For instance, using expansion (7.16) in (7.31) yields an overdetermined DS,
consisting of eleven equations,
3C
4
C
4
= P(3, 3)C
2
4
,
3C
4
C
5
+
5
2
C
5
C
4
= 2P(3,
5
2
)C
4
C
5
,
2C
3
C
5
+
5
2
C
5
C
3
= 2P(2,
5
2
)C
3
C
5
,
C
2
C
5
+
5
2
C
5
C
2
= 2P(1,
5
2
)C
1
C
5
,
2C
3
C
3
+ 3C
4
C
2
+ C
2
C
4
= 2P(1, 3)C
2
C
4
+ P(2, 2)C
2
3
,
2C
3
C
4
+ 3C
4
C
3
+
5
2
C
5
C
5
= 2P(2, 3)C
3
C
4
+ P(
5
2
,
5
2
)C
2
5
,
0 = P(0, 0)C
2
1
,
C
2
C
1
= 2P(0, 1)C
1
C
2
,
2C
3
C
1
+ C
2
C
2
= P(1, 1)C
2
2
+ 2P(0, 3)C
1
C
3
,
3C
4
C
1
+ C
2
C
3
= 2P(0, 3)C
1
C
4
+ 2P(1, 2)C
2
C
3
,
5
2
C
5
C
1
= 2P(0,
5
2
)C
1
C
5
.
This system is solved, giving exponential solutions (7.29). The same holds for the
related second-order PDE
u
x
u
tt
= F[u].
Example 7.9 Consider the expansion (7.13) for PDE (7.31). Then, using the same
action formula for both quadratic operators therein, we derive the DS, consisting of
eight equations, where the invariance conditions, such as (7.14), also become ODEs
(cf. (7.15)),
0 = P(0, 0)C
2
1
, 2C
3
C
3
= P(2, 2)C
2
3
,
C
2
C
3
+ 2C
3
C
2
+
3
2
C
4
C
4
= 2P(1, 2)C
2
C
3
+ P(
3
2
,
3
2
)C
2
4
,
C
2
C
2
+ 2C
3
C
1
= 2P(0, 2)C
1
C
3
+ P(1, 1)C
2
2
,
3
2
C
4
C
2
+ C
2
C
4
= 2P(1,
3
2
)C
1
C
4
, C
2
C
1
= 2P(0, 1)C
1
C
2
,
3
2
C
4
C
3
+ 2C
3
C
4
= 2P(2,
3
2
)C
3
C
4
,
3
2
C
4
C
1
= 2P(0,
3
2
)C
1
C
4
.
It is easy to check that this DS has exponential solutions.
2. Complex exponents. Taking p
k
= a
k
+ ib
k
with b
k
= 0 yields quadratic sys-
subspaces.
struction becomes more technical and can be performed for classes of operators and
linear subspaces by using codes for computer-supportedalgebraic manipulations.
7.2 Quadratic Kuramoto–Sivashinsky equations
We now study partially invariant subspaces for another family of quadratic fourth-
order operators
F[u] = uu
t
+ αuu
x
+ βuu
xx
+ γ(u
x
)
2
+δuu
xxxx
+ εu
x
u
xxx
+ µ(u
xx
)
2
+ νuu
xxx
.
© 2007 by Taylor & Francis Group, LLC
3. Cubic operators. For cubic (see (1.136)) and higher-degree operators, the con-
tems of ODEs; see (1.134). Such systems can possess solutions on partially invariant
7 Partially Invariant Subspaces and GSV 345
The corresponding PDE
F[u] = 0 (7.32)
can be treated as a generalized Kuramoto–Sivashinskyequation;see Section 3.8. Our
goal is to describe the partially invariant trigonometric subspace
W
3
= L{1, cos x , sin x},
and find all consistent overdetermined DSs and the corresponding exact solutions.
This can be done analytically, though it leads to technical difficulties. The approach
applies to more general PDEs
F[u] = L[u] ≡
(k≤2m)
a
k
D
k
x
u,
where L is a linear 2mth-order elliptic operator with constant coefficients {a
k
}.
We look for the following solutions on W
3
for any t ≥ 0:
u(x , t) = C
1
(t) + C
2
(t) cos x + C
3
(t) sin x .
Denote for convenience
s = ε − γ +µ, p = β − δ − ε +γ − µ, q = α − ν, r = ε − γ.
Substituting into (7.32) yields the following overdetermined DS:
C
1
C
1
+ C
2
C
2
− ( p +r)C
2
2
+ qC
2
C
3
−rC
2
3
= 0,
C
2
C
1
+ C
1
C
2
− ( p + s)C
1
C
2
+ qC
1
C
3
= 0,
C
3
C
1
+ C
1
C
3
− qC
1
C
2
− ( p + s)C
1
C
3
= 0,
C
2
C
2
− C
3
C
3
− pC
2
2
+ 2qC
2
C
3
+ pC
2
3
= 0,
C
3
C
2
+ C
2
C
3
− qC
2
2
− 2pC
2
C
3
+ qC
2
3
= 0,
(7.33)
where the equations are projections of F[u] = 0 onto 1, cosx,sinx ,
1
2
cos2x,and
1
2
sin2x respectively. Assuming that
C
2
1
− C
2
2
= 0andC
2
2
+ C
2
3
= 0,
we take C
1
and C
2
from the first two equations of (7.33) and C
2
and C
3
from the last
two, and obtain the system
C
1
= sC
1
,
C
2
= pC
2
− qC
3
,
C
3
= qC
2
+ pC
3
,
sC
2
1
= r
C
2
2
+ C
2
3
.
(7.34)
Solving the first three equations of (7.34) yields
C
1
(t) = D
1
e
st
,
C
2
(t) = e
pt
(D
2
cosqt − D
3
sinqt),
C
3
(t) = e
pt
(D
2
sinqt + D
3
cosqt),
(7.35)
where D
1
, D
2
,andD
3
are arbitrary constants. Substituting these functions into the
last algebraic equation in (7.34) gives the following equality:
sD
2
1
e
2st
= re
2pt
D
2
2
+ D
2
3
. (7.36)
Analyzing (7.36) yields the following:
© 2007 by Taylor & Francis Group, LLC
346 Exact Solutions and Invariant Subspaces
Proposition 7.10
There exist three cases:
(i) s = r = 0
, i.e.,
ε = γ
and
µ = 0.
Then the explicit solution of
(7.32)
is
u(x , t) = D
1
+ e
pt
[D
2
cos(x − qt) + D
3
sin(x − qt)],
where
D
1
,
D
2
,and
D
3
are arbitrary.
(ii) r = 0
and
s = 0
.Then
u(x , t) = e
pt
[D
2
cos(x − qt) + D
3
sin(x − qt)],
where
D
2
and
D
3
are arbitrary.
(iii) r = 0
and
s = 0
.Then
(7.36)
is equivalent to two conditions
s = p
and
sD
2
1
= r
D
2
2
+ D
2
3
.
and the solution takes the form
u(x , t) = e
pt
[D
1
+ D
2
cos(x − qt) + D
3
sin(x − qt)].
The only reasonable compacton-like solution exists in (iii), where we set D
1
=
D
2
=
1
2
and D
3
= 0toget
u
c
(x , t) = e
pt
cos
2
[
1
2
(x − qt)]. (7.37)
This implies the extra condition s = r , so solution (7.37) exists for s = p = r = 0,
or, in the original notation,
β = δ +2(ε − γ) and µ = 0.
We do not check if, being extended by zero in {|x − qt|≥π}, (7.37) will be a
solution of the Cauchy problem (most probably not, for such PDEs). In any case,
the compacton (7.37) localized in the domain {|x − qt| <π} with the exponentially
varying amplitude is a solution of an FBP with necessary free-boundary conditions,
including the zero contact angle one. Usually, a correct setting of such FBPs is a
difficult
OPEN PROBLEM that was discussed in Section 3.2 for some TFEs.
7.3 Method of generalized separation of variables
In this section, another extension of notions and techniques related to (partially) in-
variant subspaces is presented.
7.3.1 The general scheme for GSV
Consider a class of evolution PDEs of the form
T [u] = F[u], (7.38)
where T = T (
∂
∂t
) and F = F(
∂
∂x
) are some nonlinear differential operators, in
most applications, of the polynomial type. The method of generalized separation of
© 2007 by Taylor & Francis Group, LLC
7 Partially Invariant Subspaces and GSV 347
variables (GSV) consists of looking for solutions in the form of finite sums
u = U (x , t) ≡
n
i=1
a
i
(t) f
i
(x ) = a(t)f
T
(x ), (7.39)
where unknown sufficiently smooth vector functions a(t) = (a
1
(t), ..., a
n
(t)) and
f(x) = ( f
1
(x ), ..., f
n
(x )) are assumed to have linearly independent components.
The mathematical basis of the GSV is as follows. By A
i
[a(t)] = A
i
(a(t), a
(t), ...)
and B
i
[f(x )] = B
i
(f(x), f
(x ), ...) we denote some differential operators.
Lemma 7.11 Let for functions (7.39)
T [U] =
m
i=1
A
i
[a(t)]
˜
f
i
(x ),
F[U] =
k
i=1
B
i
[f(x )]˜a
i
(t),
(7.40)
where the sets {
˜
f
i
(x )} and {˜a
i
(t)} are linearly independent. Then, equation (7.38)
on solutions (7.39) reduces to two systems
(I) (A
1
[a(t)], ..., A
m
[a(t)]) = ( ˜a
1
(t), ..., ˜a
k
(t))C,
(II) (B
1
[f(x )], ..., B
k
[f(x )]) = (
˜
f
1
(x ), ...,
˜
f
m
(x ))C
T
,
(7.41)
where C =#C
j
i
# is a constant k × mmatrix.
Proof. Plugging (7.39) into (7.38) yields
m
i=1
A
i
[a(t)]
˜
f
i
(x ) =
k
j =1
B
j
[f(x )]˜a
j
(t).
Differentiating in t leads to the system
m
i=1
(A
i
[a(t)])
(p)
˜
f
i
(x ) =
k
j =1
B
j
[f(x )]˜a
(p)
j
(t) for p = 0, ..., k − 1,
or, in the matrix form,
A[a(t)]
˜
f
T
(x ) = α(t)B
T
[f(x )], (7.42)
where A[a(t)] =#A
(p)
i
[a(t)]# and α(t) =#˜a
(p)
j
(t)# are matrices, and we denote
˜
f = (
˜
f
1
, ...,
˜
f
m
), B = (
˜
B
1
, ...,
˜
B
k
). Since the set ( ˜a
1
(t), ..., ˜a
k
(t)) is linearly in-
dependent, the corresponding Wronskian detα(t) is non-zero, so there exists the
inverse matrix α
−1
(t), [132, Ch. III]. Multiplying (7.42) by α
−1
(t) yields
α
−1
(t) A[a(t)]
˜
f
T
(x ) = B
T
[f(x )].
On differentiation in t,wefind that
d
dt
α
−1
(t) A[a(t)]
˜
f
T
(x ) = 0. (7.43)
Since (
˜
f
1
(x ), ...,
˜
f
m
(x )) is also a linearly independent set, it follows from (7.43)
that α
−1
A = C must be a constant matrix. Hence, A[a(t)] = α(t)C and B[f (x )] =
˜
f(x)C
T
. That completes the proof.
It follows from this proof that the GSV does not assume any specific form of the
solutions such as (7.39), and deals with operator equalities (7.40) only.
© 2007 by Taylor & Francis Group, LLC
348 Exact Solutions and Invariant Subspaces
7.3.2 The GSV in application: systems (I) and (II)
Example 7.12 (GSV for a parabolic equation) Consider the operator
F[u] = αuu
xx
+ β(u
x
)
2
+ γ u
2
+ εuu
x
+ µu
x
+ νu + δ. (7.44)
Let T [u] = uu
t
, so that (7.38) is a quasilinear parabolic equation of the reaction-
diffusion-absorption type. Fix n = 2 and look for solutions
u = U (x, t) = g(x ) + f (x )a(t),
where a(t) ≡ constant and f (x ) and g(x ) are linearly independent. In this case,
(7.40) reads
T [U] = aa
f
2
+ a
fg,
F[U] = a
2
α ff
+ β( f
)
2
+ γ f
2
+ ε ff
+a
α(gf
+ fg
) + 2β f
g
+ 2γ fg+ ε( fg)
+ µf
+ ν f
+
αgg
+ β(g
)
2
+ γ g
2
+ εgg
+ µg
+ νg + δ
.
According to Lemma 7.11, the PDE T [U ] = F[U ] reduces to two systems
(I)
aa
= c
1
a
2
+ c
2
a + c
3
,
a
=˜c
1
a
2
+˜c
2
a +˜c
3
,
and
(II)
α ff
+ β( f
)
2
+ γ f
2
+ ε ff
= c
1
f
2
+˜c
1
fg,
α(gf
+ fg
) + 2β f
g
+ 2γ fg+ ε( fg)
+ µf
+ ν f = c
2
f
2
+˜c
2
fg,
αgg
+ β(g
)
2
+ γ g
2
+ εgg
+ µg
+ νg + δ = c
3
f
2
+˜c
3
fg.
Here c
i
and ˜c
i
are some constants (elements of the 2×3matrixC). It is worth men-
tioning that the GSV analysis can be naturally associated with a partially invariant
module, W
2
= L{1, a(t)}, for the given operators, that establishes links with the
previous context.
Concerning the first system (I), it is not difficult to show that, up to translations
and scalings, there exist just two essentially different cases
a(t) = t and a(t) = e
t
.
For other T [u] operators to be studied, more sophisticated functions can appear.
The second overdetermined system (II) is more difficult to handle, but it admits a
complete classification. Later on, we study various overdetermined systems that are
particular cases of (II).
Beforeintroducingother parabolicand hyperbolicexamples,it is worthdiscussing
some common aspects of such systems. In the general case, where T is an arbitrary
quadratic homogeneous differential operator such that T [1] = 0(i.e.,b
00
= 0),
T [u] =
(i, j, i+j =0)
b
i, j
D
i
t
uD
j
t
u, (7.45)
the following holds:
T [g(x ) + f (x)a(t)] = A
1
[a(t)] f
2
(x ) + A
2
[a(t)] f (x )g(x).
© 2007 by Taylor & Francis Group, LLC
7 Partially Invariant Subspaces and GSV 349
Here A
1
[a] = T [a]andA
2
[a] = T [a + 1] − T [a]. Then system (I) takes the form
(I)
A
1
[a] = c
1
a
2
+ c
2
a + c
3
,
A
2
[a] =˜c
1
a
2
+˜c
2
a +˜c
3
.
The following two cases are mainly treated later on:
1)
A
1
[a] = Aa
2
,
A
2
[a] = Ba,
and 2)
A
1
[a] =
˜
Aa,
A
2
[a] =
˜
B,
with various constants A, B and
˜
A,
˜
B. The overdetermined system (II) remains
practically the same in both cases and admits a unified treatment.
7.4 Generalized separation and partially invariant modules
Here a slightly different version of the GSV is used. We present another case of clas-
sification of overdetermined DSs occurring in the analysis of nonlinear inhomoge-
neous PDEs for which invariant subspaces or sets are not prescribed by polynomial,
trigonometric,or exponentialfunctions. For convenience,unlike (7.44), for F[u]we
now denote a purely quadratic second-order operator,
F[u] = αuu
xx
+ β(u
x
)
2
+ γ u
2
+ εuu
x
, (7.46)
and introduce linear terms separately. Here T [u] is still given by (7.45), and we set
A[u] = T [u] − F[u]. (7.47)
In fact, F can be taken in the general form (7.1), though, as will be shown, for 2mth-
order ordinary differential operators F with m ≥ 2, the solvability and consistency
analysis of overdetermined systems become illusive.
Again, for convenience of notation, we use symmetric bilinear forms of quadratic
operators (7.45) and (7.46),
T [a, b] = T [a + b] − T [a] − T [b],
F[ f, g] = F[ f + g] − F[ f ] − F[g],
(7.48)
and the corresponding polynomials, so that F[u] =
1
2
F[u, u]andT [u] =
1
2
T [u, u].
Let us next introduce two linear operators (in the previous GSV analysis Q[u]was
included into F)
P(D
t
)[u] =
b
k
D
k
t
u and Q(D
x
)[u] =
d
l
D
l
x
u,
with real constant coefficients {b
k
} and {d
l
}. Denote, for convenience,
R = Q(D
x
) − P(D
t
). (7.49)
Consider a quadratic inhomogeneous PDE of the form
A[u] = R[u] + p, (7.50)
where p = p(x, t) is a given function. As a new example, this includes the following
hyperbolic equation:
uu
tt
= αuu
xx
+ β(u
x
)
2
+ γ u
2
+ εuu
x
+ µu
x
+ νu + δ.
© 2007 by Taylor & Francis Group, LLC
350 Exact Solutions and Invariant Subspaces
7.4.1 Partially invariant 2D modules
According to the GSV method, we study properties of the operator (7.47) on
W
2
= L{1, a(t)}, (7.51)
with a smooth function a(t) to be determined.
1. Partial invariance. We first establish a formal existence of a set M ⊂ W
2
,such
that, for any u(x, t) = g(x) + f (x) a(t) (i.e., u(x , t) ∈ M for any x ∈ IR ),
A[g + fa] ∈ W
2
. (7.52)
This leads to the system 1) in Section 7.3.2.
Lemma 7.13
Let
a(t)
satisfy the system
T [a] = Aa
2
,
T [a, 1] = Ba,
(7.53)
where
A
and
B
are some constants. Then there exists the invariant set
M
A
={u = g + fa ∈ W
2
: f
satisfies
F[ f ] = Af
2
}. (7.54)
Proof. By (7.48),
T [g + fa] = f
2
T [a] + gf T[a, 1],
F[g + fa] = F[g] + F[g, f ]a + F[ f ] a
2
,
(7.55)
so that
A[g + fa] ≡−F[g] +{Bgf − F[g, f ]}a +{Af
2
− F[ f ]}a
2
. (7.56)
Hence, (7.52) is valid for all v ∈ M
A
.
It follows from (7.55) that (7.52) holds in a more general case, where
T [a] = A
2
a
2
+ A
1
a + A
0
and T [a, 1] = B
2
a
2
+ B
1
a + B
0
.
In Lemma 7.13, we set A
1
= A
0
= B
2
= B
0
= 0. It follows from (7.45) that a(t)
can be taken in the exponential form
a(t) = e
λt
, with a constant λ ∈ IR . (7.57)
Then (7.53) holds with
A =
(i, j, i=j )
b
i, j
λ
i
λ
j
, where B =
( j =0)
b
0, j
λ
j
+
(i=0)
b
i,0
λ
i
.
2. PDEs on partially invariant modules. Lemma7.13 givesa family {M
A
, A ∈ IR }
of sets which are invariant on W
2
under the operator(7.47). Let us now show that the
PDE (7.50) can be restricted to such sets M
A
. We assume that
P(D
t
)[a(t)] = 0, so a ∈ ker P. (7.58)
The case of an arbitrary eigenfunction, P[a(t)] = ρa(t) for some constant ρ ∈ IR ,
is treated in a similar fashion.
© 2007 by Taylor & Francis Group, LLC
7 Partially Invariant Subspaces and GSV 351
Proposition 7.14
Let
a(t)
satisfy
(7.53)
and
(7.58)
. Then, equation
(7.50)
on
M
A
,
with
p ∈ W
2
,
A[u] = R[u] + p
for
u = g + fa∈ M
A
, (7.59)
is equivalent to the system
F[g] =−Q[g] − p,
F[g, f ] − Bgf =−Q[ f ].
(7.60)
Proof. It follows from (7.49) that
R[g + fa] = Q[g] + Q[ f ]a.
Hence, (7.60) is a consequence of (7.59) and (7.56).
In the case of (7.57), denoting by
¯
P(λ) the polynomial of P, the following holds:
P[e
λt
] = ρe
λt
, with ρ =
¯
P(λ).
7.4.2 Existence of solutions via partial invariance
Let us now formulate a class of problems (7.59) having a nontrivial solution on M
A
.
Consider the case where Q(D
x
) is a first-order linear operator,
Q(D
x
)[u] = c
0
u + c
1
u
x
, where c
0
, c
1
∈ IR , and p(x) ≡ δ ∈ IR . (7.61)
Let σ ∈ IR be a real root of
βσ
2
− c
1
σ +δ = 0 (c
2
1
≥ 4βδ). (7.62)
Below D, D
1
, D
2
, ... denote different arbitrary constants. The type of the solution
u(x , t) of (7.59) on M
A
differs in the cases where A = 0andA = 0.
1. Case A = 0.
Theorem 7.15
Assume that
(7.53)
holds, where
A = B = 0. (7.63)
Then, if
β =−2α = 0,ε=−
αc
0
2ασ+c
1
, A =
γ(3ασ+c
1
)
2ασ+c
1
= 0, (7.64)
where
2ασ +c
1
= 0
, problem
(7.59)
has the nontrivial solution
u(x , t) = g(x ) + f (x) a(t). (7.65)
The functions
{g, f }
in
(7.65)
are determined as follows:
(i)
If
c
2
0
> 4σγ(2ασ + c
1
)
,then
g(x) =−
(2ασ+c
1
)
γ
µ
1
D
1
e
µ
2
x
+µ
2
D
2
e
µ
1
x
D
1
e
µ
2
x
+D
2
e
µ
1
x
−
c
0
γ
, f (x) =
De
(µ
1
+µ
2
)x
D
1
e
µ
2
x
+D
2
e
µ
1
x
, (7.66)
where
µ
1
and
µ
2
are different (real) roots of
(2ασ +c
1
)µ
2
+ c
0
µ + σγ = 0; (7.67)
(ii)
If
c
2
0
= 4σγ(2ασ + c
1
)
,then
(7.67)
has a unique root
µ
and
g(x) =−
(2ασ+c
1
)
γ
µ −
D
2
D
1
+D
2
x
−
c
0
γ
, f (x) =
De
µx
D
1
+D
2
x
; (7.68)
© 2007 by Taylor & Francis Group, LLC
352 Exact Solutions and Invariant Subspaces
(iii)
If
c
2
0
< 4σγ(2ασ + c
1
)
,then
g(x) =−
(2ασ+c
1
)
γ
[ν
2
tan(ν
2
(x + D
1
)) + ν
1
] −
c
0
γ
,
f (x) =
De
ν
1
x
cos(ν
2
(x+D
1
))
,
(7.69)
where
µ = ν
1
± iν
2
are complex roots of
(7.67)
.
Proof. We rewrite the equation of M
A
in (7.54) as
α ff
+ β( f
)
2
+ ε ff
= (A − γ)f
2
. (7.70)
Equations (7.60) can be written as follows:
αgg
+ β(g
)
2
+ γ g
2
+ εgg
+ c
0
g + c
1
g
+ δ = 0, (7.71)
α(gf
+ fg
) + 2βg
f
+ ε(gf)
+ c
0
f +c
1
f
− (A − 2γ)gf = 0. (7.72)
Here we use that, by (7.48),
F[g, f ] ≡ α(gf
+ fg
) + 2βg
f
+ ε(gf)
+ 2γ gf. (7.73)
We nowshowthat,undertheabovehypotheses,theoverdeterminedsystem (7.70)–
(7.72) has a nontrivialsolution {g(x), f (x)}. Substituting g
and f
from (7.70) and
(7.71) into (7.72) yields
β(gf
− fg
)
2
= c
1
f (gf
− fg
) − δ f
2
, so (7.74)
g
=
f
f
g − σ, (7.75)
where σ = σ
±
=
1
2β
c
1
±
c
2
1
− 4βδ
from (7.62). By (7.75),
g
=
f
f
g − σ
f
f
. (7.76)
Substituting (7.75) and (7.76) into (7.71) and using (7.70) implies
Ag = [(α +2β)σ −c
1
]
f
f
+ (εσ − c
0
). (7.77)
Thus, in general, problem (7.70)–(7.72) is equivalent to the overdetermined system
(7.70), (7.75), and (7.77). Let us next derive the identity satisfied by f (x),whichis
precisely a criterion for solvability of our equation on the invariant set.
Since A = 0, (7.75) can be rewritten as (Ag) f
− f (Ag)
= σ Af, so substituting
Ag from (7.77) yields
[(α +2β)σ − c
1
]
f
f
2
+ (εσ − c
0
)
f
f
− [(α + 2β)σ −c
1
]
f
f
= σ A. (7.78)
Since α = 0, it follows from (7.70) that
f
f
=−
α+β
α
f
f
2
−
ε
α
f
f
+
A−γ
α
, (7.79)
and, therefore, (7.78) implies that f (x) must satisfy the identity
[(α +2β)σ − c
1
]
β+2α
α
f
f
2
+ (εσ − c
0
)
f
f
−[(α + 2β)σ − c
1
]
−
ε
α
f
f
+
A−γ
α
≡ σ A.
(7.80)
© 2007 by Taylor & Francis Group, LLC