Galaktionov V.A., Svirshchevskii S.R. Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics
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2 Invariant Subspaces: Mathematics in 1D 83
Table 2.1 Quadratic operators for W
5
= L{1, ..., x
4
} (y
5
= 0)
Basis of operators F[W
5
]
Q
0
Q
8
0
F
8
1
= (J
4
1
)
2
∼ (y
0
)
2
W
9
= L{1, ..., x
8
}
Q
1
Q
7
1
F
7
1
= DF
8
1
∼ y
1
y
0
W
8
= L{1, ..., x
7
}
Q
6
1
F
6
1
= (J
3
1
)
2
∼ (y
1
)
2
W
7
= L{1, ..., x
6
}
Q
2
Q
5
2
F
5
1
= DF
6
1
∼ y
2
y
1
W
6
= L{1, ..., x
5
}
Q
4
2
F
4
1
= (J
2
1
)
2
∼ (y
2
)
2
,
F
4
2
= J
2
1
J
2
3
− (J
2
2
)
2
∼ 4y
2
y
0
− 3(y
1
)
2
W
5
= L{1, ..., x
4
}
Q
3
Q
3
3
F
3
1
= DF
4
1
∼ y
3
y
2
,
F
3
2
= DF
4
2
∼ 2y
3
y
0
− y
2
y
1
W
4
= L{1, ..., x
3
}
Q
2
3
F
2
1
= (J
1
1
)
2
∼ (y
3
)
2
,
F
2
2
= J
1
1
J
1
3
− (J
1
2
)
2
∼ 3y
3
y
1
− 2(y
2
)
2
W
3
= L{1, ..., x
2
}
Q
4
Q
1
4
F
1
1
= DF
2
1
∼ y
4
y
3
,
F
1
2
= DF
2
2
∼ 3y
4
y
1
− y
3
y
2
W
2
= L{1, x }
Q
0
4
F
0
1
= (J
0
1
)
2
∼ (y
4
)
2
,
F
0
2
= J
0
1
J
0
3
− (J
0
2
)
2
∼ 2y
4
y
2
− (y
3
)
2
,
F
0
3
= J
0
1
J
0
5
− 4J
0
2
J
0
4
+ 3(J
0
3
)
2
∼ 2y
4
y
0
− 2y
3
y
1
+ (y
2
)
2
W
1
= L{1}
operators preserving the subspace (2.134) (it is of the maximal dimension for such
operators) form the subspace Q
4
2
.
Remark 2.38 If F[W
n
] = W
k
for k < n, the evolution equation
u
t
= F[u]
admits exact solutions on W
n
,
u(x , t) =
n−1
i=0
C
i
(t)x
i
,
where the correspondingDS contains n − k trivial equations
C
i
= 0fori = k + 1, ..., n.
For an illustration, see Example 1.13.
© 2007 by Taylor & Francis Group, LLC
84 Exact Solutions and Invariant Subspaces
Table 2.2 Quadratic operators for W
4
= L{1, ..., x
3
} (y
4
= 0)
Basis of operators F[W
4
]
Q
0
Q
6
0
F
6
1
= (J
3
1
)
2
∼ (y
0
)
2
W
7
= L{1, ..., x
6
}
Q
1
Q
5
1
F
5
1
= DF
6
1
∼ y
1
y
0
W
6
= L{1, ..., x
5
}
Q
4
1
F
4
1
= (J
2
1
)
2
∼ (y
1
)
2
W
5
= L{1, ..., x
4
}
Q
2
Q
3
2
F
3
1
= DF
4
1
∼ y
2
y
1
W
4
= L{1, ..., x
3
}
Q
2
2
F
2
1
= (J
1
1
)
2
∼ (y
2
)
2
,
F
2
2
= J
1
1
J
1
3
− (J
1
2
)
2
∼ 3y
2
y
0
− 2(y
1
)
2
W
3
= L{1, x, x
2
}
Q
3
Q
1
3
F
1
1
= DF
2
1
∼ y
3
y
2
,
F
1
2
= DF
2
2
∼ 3y
3
y
0
− y
2
y
1
W
2
= L{1, x }
Q
0
3
F
0
1
= (J
0
1
)
2
∼ (y
3
)
2
,
F
0
2
= J
0
1
J
0
3
− (J
0
2
)
2
∼ 2y
3
y
1
− (y
2
)
2
W
1
= L{1}
Using Table 2.1, it is easy to obtain a similar description of the set of quadratic op-
erators for the subspace (2.96) (equation (2.97)) with any n < 5. Denote expressions
(2.119), related to the equation (2.97), by J
n,k
i
[y]fori = 1, ..., n −k.LetJ
n+1,k
i
[y]
for i = 1, ..., n + 1 − k be analogous expressions for the equation y
n+1
= 0. It
directly follows from (2.119) that the first n −k of those are obtained from {J
n,k
i
[y]}
by the change y → y
1
, i.e., J
n+1,k
i
[y] = J
n,k
i
[y
1
]fori = 1, ..., n − k, and that only
the last one, J
n+1,k
n+1−k
[y] = k!x
n+1
(x
−1−n
y)
(n−k)
, contains y
0
.
Example 2.39 (n = 4) This observation makes it possible,startingfrom the basis of
operators for equation y
n+1
= 0, to obtain a similar basis for equation (2.97). To this
end, it is necessary to delete operators that contain y
0
and to perform the substitution
y
i
→ y
i−1
in the operators that are left. Doing so, from Table 2.1, we obtain Table
2.2 that represents the set of operators, corresponding to the equation
y
4
= 0. (2.135)
Remark 2.40 We make some comments concerning the set F of operators admitted
by a given linear ODE L[y] = 0.
1). If F
1
, F
2
∈ F ,thenF
1
F
2
∈ F , with the standard definition of the superposition,
(F
1
F
2
)[y] = F
1
[F
2
[y]].
© 2007 by Taylor & Francis Group, LLC
2 Invariant Subspaces: Mathematics in 1D 85
2). Let L have constant coefficients. Then:
(i) The operator D of the total derivative in x belongs to F ;
(ii) If F ∈ F, both DF and FD belong to F and have the order that is larger by one
than the order of F;and
(iii) If F[y] is a homogeneous polynomial of y and its derivatives, then DF[y]and
FD[y] are homogeneous polynomials of the same degree.
Using these remarks, it is possible to obtain a basis of the linear space
ˆ
F
q
n−1
of
translation-invariant operators admitted by the ODE (2.97), starting with the opera-
tors of the lowest orders that form the basis of Q
n−1
n−1
2
for odd n and Q
n
2
for even n.
For example, in the case of the equation (2.134), the basis of Q
4
2
,
F
4
1
[y] ∼ (y
2
)
2
and F
4
2
[y] ∼ y
2
y
0
+ ...
(only the leading-order terms are indicated), and operators
F
4
1
D[y] ∼ (y
3
)
2
, DF
4
1
[y] ∼ y
3
y
2
, F
4
2
D[y] ∼ y
3
y
1
+ ..., DF
4
2
[y] ∼ y
3
y
0
+ ... ,
F
4
1
D
2
[y] ∼ (y
4
)
2
, DF
4
1
D[y] ∼ y
4
y
3
+ ... , D
2
F
4
1
[y] ∼ y
4
y
2
+ ... ,
DF
4
2
D[y] ∼ y
4
y
1
+ ... , D
2
F
4
2
[y] ∼ y
4
y
0
+ ...
form a basis of
ˆ
F
q
4
.
In the case of the equation (2.135),the basis of Q
2
,
F
2
1
[y] ∼ (y
2
)
2
, F
3
1
[y] ∼ y
2
y
1
, and F
2
2
[y] ∼ y
2
y
0
+ ... ,
and the additional operators
F
2
1
D[y] ∼ (y
3
)
2
, DF
2
1
[y] ∼ y
3
y
2
, F
2
2
D[y] ∼ y
3
y
1
+ ... , DF
2
2
[y] ∼ y
3
y
0
+ ... ,
form a basis in
ˆ
F
q
3
. In general, such bases differ from those in Tables 2.1 and 2.2.
2.7 Extensions to
∂
∂t
-dependent operators
Here, we describe a generalization of the main results to operators, including the
time derivative u
t
,
F[u] = F(x, u, u
1
, ..., u
k
, u
t
, u
1,t
, ..., u
k,t
), where u
i
=
∂
i
u
∂x
i
(2.136)
and u
0
= u. Such applications are associated with the GT equation in Example 1.20
having the quadratic operator of the form
F
1
[u] = u
x
u
tx
− u
t
u
xx
, (2.137)
and with several other models.
For a given set of functions f
1
(x ), ..., f
n
(x ) that form a fundamental set of solu-
tions of the ODE (2.2), we consider the set of their linear combinations,
W
n
=
n
i=1
C
i
(t) f
i
(x )
, (2.138)
which is actually a module [373, Ch. III]. Without fear of confusion,we also keepthe
notation (2.1) for modules. The coefficients {C
i
(t)}, as well as functions {f
i
(x )},are
© 2007 by Taylor & Francis Group, LLC
86 Exact Solutions and Invariant Subspaces
assumedtobesufficiently smooth.The module (2.138) is supposed to be a collection
of solutions of the linear equation
L[u] ≡ u
n
+
n
i=1
a
i
(x )u
n−i
= 0 (2.139)
for u = u(x , t). The condition for W
n
to be invariant with respect to operator F,
F[W
n
] ⊆ W
n
, has the form of the identity
L[F[u]] ≡ 0 for every solution of (2.139). (2.140)
We use the notation F
n−1
(W
n
) for the set of operators (2.136) with k ≤ n − 1that
leavethemodule(2.138)(or equation(2.139))invariant.The formulationof the Main
Theoremthat describes this moduleand its proof are similar to those of Theorem2.1.
2.7.1 Main Theorem
Theorem 2.41 (“Main Theorem”)
The set
F
n−1
(W
n
)
of operators
(2.136)
that
leave module
(2.138)
invariant is given by
F[u] =
n
i=1
A
i
(I [u], I[u
t
]) f
i
(x ), (2.141)
where
I [u] = (I
1
[u], ... , I
n
[u])
is the complete set of first integrals of the ODE
(2.139)
and
A
i
for
i = 1, ..., n
are arbitrary smooth functions.
Proof. The invariance condition (2.140) is written down as
D
n
F + ... + a
n−1
(x )DF + a
n
(x )F = 0, with
D =
∂
∂x
+
n−2
i=0
u
i+1
∂
∂u
i
−
n
i=1
a
i
u
n−i
∂
∂u
n−1
+
n−2
i=0
u
i+1,t
∂
∂u
i,t
−
n
i=1
a
i
u
n−i,t
∂
∂u
n−1,t
.
(2.142)
Note that I[u
t
] is a complete set of first integrals of the equation L[u
t
] = 0. There-
fore, DI
i
[u] = 0andDI
i
[u
t
] = 0foralli = 1, ..., n. Passing from (x, u, u
1
, ...,
u
n−1
, u
t
, u
1,t
, ..., u
n−1,t
) to the new variables
˜x = x, I
i
= I
i
[u], and J
i
= I
i
[u
t
]fori = 1, ..., n,
we transform D into
˜
D =
∂
∂ ˜x
and equation (2.139) into
˜
D
n
˜
F +
n
i=1
a
i
(x )
˜
D
n−i
˜
F = 0.
Then the general solution is
˜
F[u] =
n
i=1
A
i
(I
1
, ..., I
n
, J
1
, ..., J
n
) f
i
( ˜x),
where A
i
are arbitrary functions. Returning to the original variables yields (2.141)
that completes the proof.
Remark 2.42 If operators (2.136) also contain higher-order derivatives
∂
i
u
∂t
i
for i =
2, ..., m, instead of (2.141), we find
F[u] =
n
i=1
A
i
I [u], I [u
t
], ..., I [
∂
m
u
∂t
m
]
f
i
(x ). (2.143)
© 2007 by Taylor & Francis Group, LLC
2 Invariant Subspaces: Mathematics in 1D 87
Remark 2.43 If t = (t
1
, ..., t
p
) ∈ IR
p
, we set in (2.143)
u
t
={u
t
i
}, I [u
t
] ={I [u
t
i
]};
u
tt
={u
t
i
t
j
}, I [u
tt
] ={I [u
t
i
t
j
]} (i, j = 1, ..., p); etc.
2.7.2 Theorem on the maximal dimension
Theorem 2.8 is also extended to operators (2.136).
Theorem 2.44 (“Theorem on maximal dimension”)
If the module
(2.138)
is in-
variant under the operator
(2.136)
or its generalizations in Remarks
2.42
,
2.43
,and
the operator is nonlinear, then
(2.19)
holds.
Proof. As in the proof of Theorem 2.8, it is convenient to use slightly different equa-
tions of the W
n
,
u
n
= a
1
u
n−1
+ ... + a
n−1
u
1
+ a
n
u, (2.144)
and the corresponding invariance condition,
D
n
F ≡ a
1
D
n−1
F + ... + a
n−1
DF + a
n
F on (2.144), (2.145)
where D is the operator of the total derivative in x. We replace
u
t
→ v, so that u
j,t
→ v
j
.
Arguing by contradiction, let n ≥ 2k + 2. Following the lines of the proof of
Theorem 2.8, for functions F(x, u, u
1
, ..., u
k
,v,v
1
, ..., v
k
),wefind the formula that
is similar to (2.22),
D
p
F = u
k+p
F
u
k
+ v
k+p
F
v
k
+
[
p
2
]−1
i=1
C
i
p
u
k+p−i
u
k+i
+νC
[
p
2
]
p
u
k+p−[
p
2
]
u
k+[
p
2
]
F
u
k
u
k
+
[
p
2
]−1
i=1
C
i
p
v
k+p−i
v
k+i
+νC
[
p
2
]
p
v
k+p−[
p
2
]
v
k+[
p
2
]
F
v
k
v
k
+
p−1
i=1
C
i
p
u
k+p−i
v
k+i
F
u
k
v
k
+ ... ,
(2.146)
where we keep the linear termscontaininghigher-orderderivativesof the order k +p,
and also the quadratic terms of the total order 2k + p.
Applying the arguments used in the proof of Theorem 2.8 first to F as a function
of the variables u, u
1
, ..., u
k
, and second of v, v
1
, ..., v
k
, immediately gives
F
u
i
u
j
= 0andF
v
i
v
j
= 0foralli, j = 0, 1, ..., k.
Therefore, omitting the linear terms, (2.146) takes the form
D
p
F =
p−1
i=1
C
i
p
u
k+p−i
v
k+i
F
u
k
v
k
+ ... . (2.147)
For n ≥ 2k +2, we select in the square brackets in (2.147) the derivatives in u of the
order not less than n − 1, so
D
n
F =
k+1
i=1
α
i
u
k+n−i
v
k+i
F
u
k
v
k
+ ... . (2.148)
Using the linear equation (2.144) of W
n
, we express all the derivatives u
k+n−i
for
i = 1, 2, ..., k in terms of u
n−1
, ..., u. Then (2.148) will contain a single quadratic
term with u
n−1
v
2k+1
,i.e.,C
k+1
n
u
n−1
v
2k+1
F
u
k
v
k
, that has the maximal total order
© 2007 by Taylor & Francis Group, LLC
88 Exact Solutions and Invariant Subspaces
2k + p. Such terms cannot appear in the derivatives D
p
F for p < n. Hence, in
order to satisfy the invariance condition, (2.145) we need F
u
k
v
k
= 0. Taking this into
account, similar to (2.148), we find
D
n
F =
k+1
i=1
β
i
u
k+n−i
v
k−1+i
F
u
k
v
k−1
+ ... ,
where β
i
> 0. All the indicated summands have the total order of the derivatives
2k + n − 1. Excluding the derivatives u
k+n−i
for i = 1, 2, ..., k gives the unique
quadratic term with u
n−1
v
2k
, namely
β
k+1
u
n−1
v
2k
F
u
k
v
k−1
,
which verifies the maximal total order 2k + n − 1 (such summands cannot occur in
D
p
F for p < n). Equating this coefficient to zero yields F
u
k
v
k−1
= 0. Assuming
next that
F
u
k
v
k
= F
u
k
v
k−1
= ... = F
u
k
v
k−( j −1)
= 0,
we obtain
D
n
F =
k+1
i=1
γ
i
u
k+n−i
v
k−j +i
F
u
k
v
k−j
+ ... , where γ
i
> 0.
Hence, F
u
k
v
k−j
= 0foranyj = 0, 1, ..., k. Since the variablesu and v are equivalent,
by a similarity argument, this implies that F
u
k−j
v
k
= 0forj = 0, 1, ..., k.
It follows that
F[y] = f
k
(x )u
k
+ g
k
(x )v
k
+
F(x , u, ... , u
k−1
, v, ..., v
k−1
).
Repeating the same speculations for the function
F yields
F = f
k−1
(x )u
k−1
+ g
k
(x )v
k−1
+
F(x , u, ... , u
k−2
, v, ..., v
k−2
),
etc. Finally,
F =
k
i=1
f
i
(x )u
i
+ g
i
(x )v
i
+ h(x),
i.e., F[u] is a linear operator.
2.7.3 Examples
Example 2.45 (W
3
for the operator (2.137)) According to Theorem 2.44, the op-
erator (2.137) from the GT equation can admit W
n
of dimension n not exceeding
five. The analysis of the invariance conditions implies that the maximal dimension is
in fact three. Up to scalings in x, all 3D modules are described by the equation
u
+ au
= 0, with a =−1, 0, and 1,
and have the form
W
3
= L{1, e
x
, e
−x
}, W
3
= L{1, x , x
2
}, and W
3
= L{1, cos x , sin x}.
Symmetric restrictions of operators. Given a
∂
∂t
-dependent operator (2.136), we
introduce its symmetric restriction
ˆ
F[u] = F(x, u, u
1
, ..., u
k
, u, u
1
, ..., u
k
), (2.149)
© 2007 by Taylor & Francis Group, LLC
2 Invariant Subspaces: Mathematics in 1D 89
which is obtained from (2.136) by replacing u
t
→ u. For example, the GT-operator
(2.137) has the symmetric restriction
ˆ
F
1
[u] = (u
x
)
2
− uu
xx
≡−F
rem
[u],
where the remarkable operator (1.99) appears. The relation between operators and
their symmetric restrictions is as follows:
Proposition 2.46
If the module
(2.138)
is invariantunder the operator
(2.136)
,then
it is invariant under its symmetric restriction
(2.149)
.
In general, nonlinear operators can have linear symmetric restrictions, or even the
null-restriction
ˆ
F = 0.
Example 2.47 (Quadratic operators with polynomial W
3
, W
4
,andW
5
)Let
us describe all the quadratic operators with constant coefficients of the second order
in x and of the first order in t (as in (2.137)), preserving polynomial modules of
dimensions 3, 4, and 5. Analyzing the invariance conditions yields the following
bases of the linear space of such operators:
(i) W
3
= L{1, x, x
2
}:
F
1
[u] = (u
xx
)
2
, F
2
[u] = (u
txx
)
2
, F
3
[u] = u
xx
u
txx
;
F
4
[u] = u
x
u
xx
, F
5
[u] = u
tx
u
txx
, F
6
[u] = u
tx
u
xx
, F
7
[u] = u
x
u
txx
;
F
8
[u] = uu
xx
, F
9
[u] = u
t
u
txx
, F
10
[u] = u
t
u
xx
, F
11
[u] = uu
txx
;
F
12
[u] = (u
x
)
2
, F
13
[u] = (u
tx
)
2
, F
14
[u] = u
x
u
tx
;
F
15
[u] = uu
tx
− u
t
u
x
.
The underlined operators are the symmetric restrictions of the next ones. The last
operator, F
15
, has the null-projection, i.e.,
ˆ
F
15
= 0.
(ii) In a similar fashion, for W
4
= L{1, x , x
2
, x
3
}:
F
1
[u] = (u
xx
)
2
, F
2
[u] = (u
txx
)
2
, F
3
[u] = u
xx
u
txx
, F
4
[u] = u
tx
u
txx
;
F
5
[u] = u
x
u
xx
, F
6
[u] = u
x
u
txx
, F
7
[u] = u
tx
u
xx
;
F
8
[u] = uu
xx
−
2
3
(u
x
)
2
, F
9
[u] = u
t
u
txx
−
2
3
(u
tx
)
2
,
F
10
[u] = u
t
u
xx
−
2
3
u
x
u
tx
, F
11
[u] = uu
txx
−
2
3
u
x
u
tx
.
(iii) For the 5D subspace W
5
= L{1, x, x
2
, x
3
, x
4
}, we find the following operators:
F
1
[u] = (u
xx
)
2
, F
2
[u] = (u
txx
)
2
, F
3
[u] = u
xx
u
txx
;
F
4
[u] = uu
xx
−
3
4
(u
x
)
2
, F
5
[u] = u
t
u
txx
−
3
4
(u
tx
)
2
,
F
6
[u] = uu
txx
+ u
t
u
xx
−
3
2
u
x
u
tx
; F
7
[u] = u
x
u
txx
− u
tx
u
xx
.
Note that
ˆ
F
7
= 0. As an illustration, let us present the computations concerning the
last three operators for functions
u = C
1
+ C
2
x + C
3
x
2
+ C
4
x
3
+ C
5
x
4
∈ W
5
.
© 2007 by Taylor & Francis Group, LLC
90 Exact Solutions and Invariant Subspaces
The following holds:
F
5
[u] = 12(C
4
C
5
− C
4
C
5
)x
4
+ 16(C
3
C
5
− C
3
C
5
)x
3
+ 6(2C
2
C
5
−2C
2
C
5
+ C
3
C
4
− C
3
C
4
)x
2
+ 6(C
2
C
4
− C
2
C
4
)x +2(C
2
C
3
− C
2
C
3
);
F
6
[u] =
1
4
8C
3
C
5
− 3
C
4
2
]x
4
+
6C
2
C
5
− C
3
C
4
x
3
+
1
2
24C
1
C
5
+3C
2
C
4
− 2
C
3
2
x
2
+
6C
1
C
4
− C
2
C
3
x + 4C
1
C
3
−
3
4
C
2
2
;
F
7
[u] =
1
2
4C
3
C
5
+ 4C
3
C
5
− 3C
4
C
4
x
4
+
6C
2
C
5
+ 6C
2
C
5
− C
3
C
4
−C
3
C
4
x
3
+
1
2
24C
1
C
5
+ 24C
1
C
5
+ 3C
2
C
4
+ 3C
2
C
4
− 4C
3
C
3
x
2
+2
6C
1
C
4
+ 6C
1
C
4
− C
2
C
3
− C
2
C
3
x + 4C
1
C
3
+ 4C
1
C
3
− 3C
2
C
2
.
By these expressions, the PDEs F
k
[u] = g ∈ W
5
reduce to ODE systems.
2.7.4 Operators F : W
n
→
˜
W
m
It is convenient to use the class of
∂
∂t
-dependent operators (2.136) to describe the
next generalization, where we move away from the notion of invariant modules, but
keep the main mathematical tools and results.
Considertwodifferentmodules, W
n
and
˜
W
m
, givenby two differentODEs, written
in the form of
W
n
=
n
i=1
α
i
(t) f
i
(x )
, u
n
=
n
j =1
a
j
(x )u
n−j
;
˜
W
m
=
u =
m
i=1
β
i
(t)
˜
f
i
(x )
, u
m
=
m
j =1
b
j
(x )u
m−j
,
(2.150)
where u
j
=
∂
j
u
∂x
j
. We are looking for operators F : W
n
→
˜
W
m
and denote by
F
n−1
(W
n
,
˜
W
m
) the whole set of such operators. In the case where W
n
=
˜
W
m
,we
return to the concept of invariant modules. If F ∈ F
n−1
(W
n
,
˜
W
m
),then
for any u =
n
i=1
C
i
(t) f
i
(x ), F[u] =
m
i=1
i
(C
1
, ..., C
n
, C
1
, ..., C
n
)
˜
f
i
(x ).
In particular, the PDE
F[u] = 0onW
n
is equivalent to the system of ODEs
1
(C
1
, ..., C
n
, C
1
, ..., C
n
) = 0,
... ... ...
m
(C
1
, ..., C
n
, C
1
, ..., C
n
) = 0.
The necessary and sufficient condition for operator F to map any solution of the
first ODE, L[u] = 0, in (2.150) into a solution of the second one,
˜
L[u] = 0, is
˜
L[F[u]]
L[u]=0
≡ 0,
which looks similar to (2.4). By the same analysis, we arrive at the following result,
where the same notation, as in Theorem 2.41, is kept.
Theorem 2.48 (“Main Theorem”)
The set
F
n−1
(W
n
,
˜
W
m
)
consists of operators
F[u] =
m
i=1
A
i
(I [u], I[u
t
])
˜
f
i
(x ),
© 2007 by Taylor & Francis Group, LLC
2 Invariant Subspaces: Mathematics in 1D 91
where
I [u] = (I
1
[u], ... , I
n
[u])
is the complete set of first integrals of the first ODE
in
(2.150)
,
A
i
for
i = 1, ..., m
are any smooth functions, and
{
˜
f
1
(x ), ...,
˜
f
m
(x )}
is a
fundamental set of solutions of the second ODE in
(2.150)
.
Remarks 2.42 and 2.43 are also applied to mappings W
n
→
˜
W
m
.
Theorem 2.49 (“Theorem on maximal dimension”)
Let
F : W
n
→
˜
W
m
,where
m ≤ n
, be a nonlinear operator
(2.136)
of the differential order
k
.Then
n ≤ 2k +1
.
Proof. Let m = n and W
n
and
˜
W
n
be prescribed by the linear equations
u
n
= a
1
u
n−1
+ ... + a
n−1
u
1
+ a
n
u, (2.151)
u
n
= b
1
u
n−1
+ ... + b
n−1
u
1
+ b
n
u,
respectively. Then, for F, the following holds:
D
n
F ≡ b
1
D
n−1
F + ... + b
n−1
DF + b
n
F on (2.151),
where D is the operator of the full derivative in x. The result of the theorem follows
from the arguments that have been used in the proof of Theorem 2.44.
In the case of m < n,
˜
W
m
can be extended to
˜
W
n
by adding n − m arbitrary
functions to the basis that form, together with {
˜
f
i
(x ), i = 1, ..., m}, a linearly inde-
pendent set. Then F : W
n
→
˜
W
n
and the previous argument works.
Corollary 2.50
A nonlinear
k
th-order differential operator
F : W
n
→
˜
W
m
,where
n ≥ 2k + 2
, can exist only if
m > n
.
These results are valid in the cases indicated in Remarks 2.42 and 2.43, as well as
for the operators that do not contain derivatives in t.
In the following example, we consider the operator (1.94),
F[u] = u
x
u
tx
− βu
t
u
xx
+ γ uu
txx
+ δuu
t
.
For some particular choices of the coefficients, we find 3D modules W
3
and
˜
W
3
such
that F : W
3
→
˜
W
3
.
Example 2.51 (
˜
W
3
= W
3
)Letγ = β = 0andδ = 0, i.e.,
F[u] = u
x
u
tx
+ β(uu
txx
− u
t
u
xx
).
There exist two cases:
1. The modules are given by
W
3
= L{1, x , ln x}
u
=−
2
x
u
,
˜
W
3
= F[W
3
] = L
1,
1
x
,
1
x
2
u
=−
6
x
u
−
6
x
2
u
,
and the action of the operator is
F[C
1
+ C
2
x + C
3
ln x] = C
2
C
2
+
(C
2
C
3
)
+ β
C
3
C
2
− C
2
C
3
1
x
+
C
3
C
3
+ β
C
3
C
1
− C
1
C
3
1
x
2
.
As an application, taking the PDE
F[u] ≡ u
x
u
tx
+ β(uu
txx
− u
t
u
xx
) =
1
2
,
© 2007 by Taylor & Francis Group, LLC
92 Exact Solutions and Invariant Subspaces
we find the exact solution
u(x , t) = C
1
(t) + C
2
(t)x + C
3
(t) ln x ,
C
2
C
2
=
1
2
,
(C
2
C
3
)
+ β
C
3
C
2
− C
2
C
3
= 0,
C
3
C
3
+ β
C
3
C
1
− C
1
C
3
= 0.
The first equation of this DS yields C
2
(t) =
√
t, and the rest of equations are also
explicitly integrated, so, for β = 0, 1, the solutions are
u(x , t) =
1
β
(k
2
− µ lnt)k
1
t
µ
+
√
tx+ k
1
t
µ
ln x,
where µ =
β+1
2(β−1)
,andk
1,2
are arbitrary constants.
2. The module and its image are given by
W
3
= L{1, ln x, ln
2
x}
u
=−
3
x
u
−
1
x
2
u
,
˜
W
3
= F[W
3
] = L
1
x
2
,
ln x
x
2
,
ln
2
x
x
2
u
=−
9
x
u
−
19
x
2
u
−
8
x
3
u
.
The action is
F[C
1
+ C
2
ln x +C
3
ln
2
x] =
C
2
C
2
+ β
2C
1
C
3
− 2C
3
C
1
− C
1
C
2
+ C
2
C
1
1
x
2
+2
(C
2
C
3
)
+ β
C
2
C
3
− C
3
C
2
− C
1
C
3
+ C
3
C
1
lnx
x
2
+
4C
3
C
3
+ β
C
3
C
2
− C
2
C
3
ln
2
x
x
2
.
Example 2.52 (
˜
W
3
= W
3
)Letβ =−γ =
2
3
and δ =−
4
3
, i.e.,
F[u] = u
x
u
tx
−
2
3
u
t
u
xx
−
2
3
uu
txx
−
4
3
uu
t
. (2.152)
There are two cases:
1. The module and action of the operator are
W
3
= L{1, cos2x, sin 2x } (u
=−4u
),
F[C
1
+ C
2
cos2x + C
3
sin2x] = 4(C
2
C
2
+ C
3
C
3
)
−
4
3
C
1
C
1
+
4
3
(C
1
C
2
)
cos2x +
4
3
(C
1
C
2
)
sin2x.
2. The module and the action are
W
3
= L{1, cos x , cos 2x}
u
= 3cotxu
− (1 + 3cot
2
x)u
,
F[C
1
+ C
2
cos x + C
3
cos2x] =
2C
2
3
+
1
4
C
2
2
−
2
3
C
2
1
+2
C
2
C
3
−
1
3
C
1
cos x +
4
3
C
1
C
3
−
1
4
C
2
2
cos2x.
2.8 SUMMARY: Basic types of equations and solutions
Forthe reader’s convenience,we present here a summaryof equations,solutions,and
main schemes of our analysis to be used later on for constructing exact solutions in
the subsequent chapters. Almost all models and PDEs to be studied therein fall into
the scope of the following classification:
© 2007 by Taylor & Francis Group, LLC
Both 3D modules are sub-modules of the 5D one; see Proposition 1.34.