Galaktionov V.A., Svirshchevskii S.R. Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics
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5 Quasilinear Wave and Boussinesq Models. Systems 263
Plugging this expression into (5.96) gives
C
1
f
1
+ C
2
f
2
=−C
1
f
2
− C
2
f
1
+ ν(C
1
f
2
− C
2
f
1
)
C
2
1
+ C
2
2
, or
C
1
= C
2
− νC
2
C
2
1
+ C
2
2
,
C
2
=−C
1
+ νC
1
C
2
1
+ C
2
2
.
Solving the DS and substituting into (5.97) yields the explicit solution
$
u
v
%
= C
1
$
cos(x +(D
2
1
ν − 1)t + D
2
)
sin(x + (D
2
1
ν − 1)t + D
2
)
%
,
where D
1
and D
2
are arbitrary constants. Indeed, this is the traveling wave
z = u + iv = D
1
e
i[x +(D
2
1
ν−1)t+D
2
]
.
The fourth-order NLS equation from nonlinear optics and quantum mechanics iz
t
+
z
xxxx
+ 2z
xx
+ ν |z|
2
z = 0 (see physics and references in Ablowitz–Segur [4])
possesses the same solution since z
xxxx
+ z
xx
= 0onW
2
.
5.4.3 Invariant subspaces for quadratic systems
Further examples use simple invariant subspaces introduced in Sections 1.3–1.5 for
the case of real-valued operators. In general,systems of nonlinearPDEs do not enjoy
such a variety of subspaces of higher dimensions, but some results admit straightfor-
ward extensions. In what follows, we do not concentrate on establishing the opti-
mal results on maximal dimensions of subspaces, a full classification of quadratic
translation-invariant operators and other related delicate topics, but mainly just ex-
plain how some of the subspaces and solutions can be constructed.
On the one hand, in order to use our previous results for vector-valued operators
F, we may deal with vector sets defined in terms of the “span” of given linearly
independent real-valued functions
W
n,m
= L{f
1
, ..., f
n
}≡{C
1
f
1
+ ... + C
n
f
n
: C
1
, ..., C
n
∈ IR
m
},
which is calculated over IR
m
. Then we will look for solutions
U(x , t) = C
1
(t) f
1
(x ) + ... + C
n
(t) f
n
(x ) (5.98)
for some vector-functions C
1
(t),...,C
n
(t) ∈ IR
m
for any t ≥ 0. W
n,m
has di-
mension mn and (5.98) means that each component belongs to the scalar subspace
W
n
= W
n,1
. We will use basic subspaces from Sections 1.3–1.5. We begin with
some introductory examples of semilinear systems, and, as above, will discuss sys-
tems of two PDEs taking m = 2. Extensions of most of the results to larger m are
straightforward.Here and later on, a, b, c,..., a
1,2
, b
1,2
, c
1,2
,..., and variousGreek let-
ters α, β, γ , ... with or without subscripts denote different real constants or constant
vectors in IR
m
when necessary.
On the other hand, the majority of the results can be obtained by seeking invariant
subspaces for standard real valued operators. We will use both approaches.
© 2007 by Taylor & Francis Group, LLC
264 Exact Solutions and Invariant Subspaces
Example 5.21 (Semilinear systems) Consider the following system of semilinear
heat equations:
u
t
= F
1
[u,v] ≡ a
1
u
xx
+ b
1
(u
x
)
2
+ c
1
(v
x
)
2
+ d
1
u + e
1
v + f
1
,
v
t
= F
2
[u,v] ≡ a
2
v
xx
+ b
2
(u
x
)
2
+ c
2
(v
x
)
2
+ d
2
u + e
2
v + f
2
.
Invariant analysis is obvious, since each equation contains the quadratic operators
preserving the polynomial subspace
W
3
= L{1, x, x
2
}, (5.99)
in the sense that F
1,2
: W
3
× W
3
→ W
3
. Looking for both components on W
3
,
u(x , t) = C
1
(t) + C
2
(t)x +C
3
(t)x
2
,
v(x , t) = D
1
(t) + D
2
(t)x + D
3
(t)x
2
,
(5.100)
yields a nonlinear DS for these six expansion coefficients,
C
1
= 2a
1
C
3
+ b
1
C
2
2
+ c
1
D
2
2
+ d
1
C
1
+ e
1
D
1
+ f
1
,
C
2
= 4b
1
C
2
C
3
+ 4c
1
D
2
D
3
+ d
1
C
2
+ e
1
D
2
,
C
3
= 4b
1
C
2
3
+ 4c
1
D
2
3
+ d
1
C
3
+ e
1
D
3
,
D
1
= 2a
2
D
2
+ b
2
C
2
2
+ c
2
D
2
2
+ d
2
C
1
+ e
2
D
1
+ f
2
,
D
2
= 4b
2
C
2
C
3
+ 4c
2
D
2
D
3
+ d
2
C
2
+ e
2
D
2
,
D
3
= 4b
2
C
2
3
+ 4c
2
D
2
3
+ d
2
C
3
+ e
2
D
3
.
Subspace (5.99) remains invariant for 1D operators in a system with extra quadratic
terms γ
1,2
u
x
v
x
added to both equations. Other linear operators with the subspace
(5.99) can be put into equations. In further examples we will omit linear terms and
operators and mainly deal with quadratic nonlinearities. For the corresponding sys-
tem of hyperbolic PDEs
u
tt
= F
1
[u,v],
v
tt
= F
2
[u,v],
the DS for solutions (5.100) takes the same form with
d
dt
replaced by
d
2
dt
2
on the
left-hand side, so the DS is now of twelfth order.
Example 5.22 (Trigonometric subspaces) In order to use trigonometricand expo-
nential subspaces, we need other semilinear quadratic models, e.g.,
u
t
= F
1
[u,v] ≡ u
xx
+ b
1
(v
x
)
2
+ c
1
v
2
,
v
t
= F
2
[u,v] ≡ v
xx
+ b
2
(u
x
)
2
+ c
2
u
2
.
(5.101)
The quadraticoperators on the right-hand side are not coupled, and therefore, we can
use the subspace
W
2
= L{1, cos x }, (5.102)
or its 3D extension W
3
= L{1, cos x, sin x }. Plugging
u(x , t) = C
1
(t) + C
2
(t) cos x,v(x, t) = D
1
(t) + D
2
(t) cos x (5.103)
into (5.101) yields the following invariance conditions and the DS:
b
1
= c
1
and b
2
= c
2
,
© 2007 by Taylor & Francis Group, LLC
5 Quasilinear Wave and Boussinesq Models. Systems 265
C
1
= b
1
C
2
D
2
+ b
1
D
2
1
,
C
2
=−C
2
+ 2b
1
D
1
D
2
,
D
1
= b
2
C
2
D
2
+ b
2
C
2
1
,
D
2
=−D
2
+ 2b
2
C
1
C
2
.
(5.104)
In the opposite case,
b
1
=−c
1
and b
2
=−c
2
,
we use the subspaces composed of exponential functions, W
2
= L{1, coshx}, with
some obvious changes in the DS. Second-order derivatives D
2
x
in these PDEs can be
replaced by any 2mth-order ones. Hyperbolic systems can also be treated similarly.
Example 5.23 (KdV-type systems) Integrable systems of coupled KdV-typeequa-
tions have been intensively studied from the beginning of the 1980s. First examples
include the Hirota–Satsuma equations [288]
u
t
= u
xxx
+ uu
x
+ vv
x
,
v
t
=−2v
xxx
− uv
x
,
possessing multi-soliton solutions (constructed by reduction to the bilinear form by
setting u = 2(ln f )
xx
and v =
g
f
)andaninfinite number of conservation laws; and
the Ito equations [302]
u
t
= u
xxx
+ 3uu
x
+ 3vv
x
,
v
t
= vu
x
+ uv
x
,
again possessing infinitely many conservation laws and a recursion operator.
Consider a system of the third-order equations of the KdV-type with quadratic
operators from the previous example,
u
t
= F
1
[u,v] ≡ u
xxx
+ b
1
(v
x
)
2
+ c
1
v
2
,
v
t
= F
2
[u,v] ≡ v
xxx
+ b
2
(u
x
)
2
+ c
2
u
2
.
Then, unlike (5.103), the solutions are periodic moving with the constant speed 1,
u(x , t) = C
1
+ C
2
cos(x − t), v(x, t) = D
1
+ D
2
cos(x − t).
The DS is (5.104), where −C
2
is excluded from the second equation, and −D
2
from
the last one. Solutions exist for systems of semilinear (2m+1)th-order PDEs.
Example 5.24 (Quasilinear systems) The system with homogeneousquadraticop-
erators
u
t
= F
1
[u,v] ≡ a
1
(vu
x
)
x
+ b
1
(v
x
)
2
,
v
t
= F
2
[u,v] ≡ a
2
(uv
x
)
x
+ b
2
(u
x
)
2
,
possesses solutions on W
2
= L{1, x
2
},
u(x , t) = C
1
(t) + C
2
(t)x
2
,v(x , t) = D
1
(t) + D
2
(t)x
2
,
C
1
= 2a
1
C
2
D
1
,
C
2
= 6a
1
C
2
D
2
+ 4b
2
1
D
2
2
,
D
1
= 2a
2
C
1
D
2
,
D
2
= 6a
2
C
2
D
2
+ 4b
2
C
2
2
.
© 2007 by Taylor & Francis Group, LLC
266 Exact Solutions and Invariant Subspaces
One can construct more general solutions on the subspace (5.99).
Example 5.25 For some higher-order quasilinear equations, other polynomial sub-
spaces can be used. For instance, the subspace W
4
= L{1, x, x
2
, x
3
} is suitable for
the third-order PDEs
u
t
= a
1
(vu
xx
)
x
+ b
1
(uu
xx
)
x
+ (lower-order terms),
v
t
= a
2
(uv
xx
)
x
+ b
2
(vv
xx
)
x
+ (lower-order terms),
where the lower-order (differential) terms are assumed to admit W
4
.
Using systems of that type, we introduce an example of a higher-dimensionalsub-
space. A more general analysis will be presented later on.
Example 5.26 (Extended polynomial subspace) Consider
u
t
= F
1
[u,v] ≡ a
1
(vu
x
)
x
+ b
1
u
x
v
x
,
v
t
= F
2
[u,v] ≡ a
2
(uv
x
)
x
+ b
2
u
x
v
x
.
(5.105)
Let us look for components on W
4
= L{1, x , x
2
, x
3
} and set
u(x , t) = C
1
(t) + C
2
(t)x + C
3
(t)x
2
+ C
4
(t)x
3
,
v(x , t) = D
1
(t) + D
2
(t)x + D
3
(t)x
2
+ D
4
(t)x
3
.
(5.106)
In general, quadratic operators in (5.105) do not leave W
4
invariant and map it onto
W
5
containing the extra vector x
4
. Substituting (5.106) into (5.105) yields that the
coefficient of x
4
vanishes identically in both equations, provided that
a
1
b
1
=
a
2
b
2
=−
3
5
.
This is the invariance condition of the subspace W
4
, under which (5.105) on W
4
reduces to an eighth-order DS for the coefficients {C
i
, D
i
}.
Let us show that further extensions of the polynomial subspace W
4
is not possi-
ble. Unlike in the case of a single equation with quadratic nonlinearities (see Exam-
W
5
= L{1, x , x
2
, x
3
, x
4
}=W
4
⊕ L{x
4
}
cannot be invariant. Indeed, the quadratic operators map W
5
onto the extended sub-
space W
7
= W
5
⊕ L{x
5
, x
6
}. Therefore, for functions
u = C
1
+ C
2
x + C
3
x
2
+ C
4
x
3
+ C
5
x
4
,
v = D
1
+ D
2
x + D
3
x
2
+ D
4
x
3
+ D
5
x
4
,
too many invariant conditions are obtained. The first equation in (5.105) gives
6
2(2a
1
+ b
1
)C
5
D
4
+ (3a
1
+ 2b
1
)C
4
D
5
x
5
+ 4(7a
1
+ 4b
1
)C
5
D
5
x
6
≡ 0,
from which we get three conditions2a
1
+b
1
= 0, 3a
1
+2b
1
= 0, and 7a
1
+4b
1
= 0,
so a
1
= b
1
= 0. Similarly, a
2
and b
2
are annulled by the second equation,
6
2(2a
2
+ b
2
)C
4
D
5
+ (3a
2
+ 2b
2
)C
5
D
4
x
5
+ 4(7a
2
+ 4b
2
)C
5
D
5
x
6
≡ 0.
There is another version of extending invariant subspaces for systems, where the
components are taken from different subspaces; see King’s solutions in Example
© 2007 by Taylor & Francis Group, LLC
ple 1.14 and Section 2.4), for system (5.105), the subspace
5 Quasilinear Wave and Boussinesq Models. Systems 267
1.12. These systems are supposed to contain different operators in PDEs for each
component, unlike a completely symmetric equations (5.105).
Example 5.27 Take u(x, t) and v(x, t) from different subspaces,
u(x , t) = C
1
(t) + C
2
(t)x +C
3
(t)x
2
+ C
4
(t)x
3
+ C
5
(t)x
4
∈ W
5
,
v(x , t) = D
1
(t) + D
2
(t)x + D
3
(t)x
2
∈ W
3
.
It is easy to reconstruct a number of operators supporting these subspaces in the
corresponding component equations,
u
t
= a
1
vu
xxxx
+ b
1
vu
xx
+ c
1
uv
xx
+ d
1
u
xx
(v
x
)
2
+ e
1
(v
3
)
xx
+ ... ,
v
t
= a
2
(vu)
xxxx
+ b
2
v
xx
+ c
2
u
xx
+ d
2
(v
x
)
2
+ (v
3
)
xxxx
+ ... .
Example 5.28 More-dimensional subspaces appear for higher-order systems. For
instance, the coupled quasilinear third-order equations
u
t
= a
1
(vu
xx
)
x
+ b
1
u
xx
v
x
,
v
t
= a
2
(uv
xx
)
x
+ b
2
u
xx
v
x
,
possess solutions on W
5
= L{1, x, x
2
, x
3
, x
4
}, provided that
a
1
b
1
=
a
2
b
2
=−
2
3
.
Example 5.29 (Trigonometric subspaces) We will look for solutions of system
u
t
= F
1
[u,v] ≡ a
1
(vu
x
)
x
+ b
1
u
x
v
x
+ c
1
uv,
v
t
= F
2
[u,v] ≡ a
2
(uv
x
)
x
+ b
2
u
x
v
x
+ c
2
uv,
on the subspace (5.102). Substituting solutions (5.103) into the system yields the
invariance condition 2a
1
+ b
1
= c
1
,2a
2
+ b
2
= c
2
,andtheDS
C
1
= (a
1
+ b
1
)C
2
D
2
+ c
1
C
1
D
1
,
C
2
=−a
1
C
2
D
1
+ c
1
(C
2
D
1
+ C
1
D
2
),
D
1
= (a
2
+ b
2
)C
2
D
2
+ c
2
C
1
D
1
,
D
2
=−a
2
C
1
D
2
+ c
2
(C
2
D
1
+ C
1
D
2
).
Extensions of this example to systems of third and higher-order PDEs are similar to
those for polynomial subspaces.
Example 5.30 Combining semilinear and quasilinear equations, a special invari-
ance condition appears. The first PDE in the parabolic system
u
t
= u
xx
+ b
1
(u
x
v
x
+ uv),
v
t
= αvv
xx
+ β(v
x
)
2
+ γv
2
+ δu,
suggests using the subspace W
2
= L{1, cos x}, i.e., solutions (5.103). The quadratic
operator in the second equation admits this subspace if γ − α − β = 0, and the
coefficients of solutions (5.103) satisfy the DS
C
1
= b
1
(C
1
D
1
+ C
2
D
2
),
C
2
=−C
2
+ b
1
(C
1
D
2
+ D
1
C
2
),
D
1
= βC
2
D
2
+ γ D
2
1
+ b
2
(C
1
D
1
+ C
2
D
2
) + δC
1
,
D
2
=−α D
1
C
2
+ 2γ D
1
D
2
+ b
2
(D
1
C
2
+ C
1
D
2
) + δC
2
.
© 2007 by Taylor & Francis Group, LLC
268 Exact Solutions and Invariant Subspaces
5.4.4 On restricted invariance: reductions to real-valued operators
For systems, consisting of similar quadratic or cubic PDEs for u and v,thereisalso
the possibility to use some restricted invariant properties, assuming that the com-
ponents are algebraically related to each other. In this case, we can directly use the
results of classification of real-valued operators from Chapters 1 and 2. The follow-
ing simple computations illustrate such an approach.
Example 5.31 Consider the following system
m = 2, U = (u,v)
T
of hyperbolic
PDEs with general quadratic and linear operators:
U
tt
= F[U] ≡ a
1
uu
xx
+ a
2
(u
x
)
2
+ a
3
u
2
+ b
1
vu
xx
+ b
2
v
x
u
x
+b
3
uv + c
1
vv
xx
+ c
2
(v
x
)
2
+ c
3
v
2
+ au
xx
+ bu + cv
xx
+ dv +e,
(5.107)
containing fourteen parameters from IR
2
denoted, as usual, by the boldface Latin
letters, i.e., 28 arbitrary scalar parameters altogether. Looking for solutions of the
PDEs (5.107) in the form
v = Au + B, (5.108)
where A and B are two extra scalar unknowns, yields two quadratic PDEs for u
u
tt
= P
i1
uu
xx
+ P
i2
(u
x
)
2
+ P
i3
u
2
+ P
i4
u
xx
+ P
i5
u + P
i6
, i = 1, 2, (5.109)
with twelve constants {P
ij
}, depending on all the parameters. For instance,
P
11
= a
11
+ b
11
A + c
11
A
2
, P
21
=
1
A
a
12
+ b
12
A + c
12
A
2
, etc.
Solutions (5.108) exist, provided that the following six conditions hold:
P
1j
= P
2j
for j = 1, 2, ..., 6.
This system is consistent, since the total number of parameters including A and B is
28 + 2 = 30.
Thus,we can use the usual subspaces W
n
fromChapters1 and 2 for a singlequadratic
equation, such as (5.109), to obtain exact solutions for the system of PDEs (5.107).
This correspondsto restricted invariance analysis, since theinvarianceconditions are
checked for u ∈ W
n
on the affine manifold (5.108) in terms of v.
5.4.5 On extensions of basic invariant subspaces
Extensions of invariant subspaces are restricted by the following result, which is
formulated for m = 2:
Theorem 5.32
(
“Theorem on maximal dimension”
)Let
F[u,v]
be a nonlinear
ordinary differential operator of the order
k
that admits the invariant subspace
W
n
,
i.e.,
F : W
n
× W
n
→ W
n
.Then
n ≤ 2k + 1
.
The proof is contained in the proof of Theorem 2.44.
Extending polynomial subspaces. We now use the scalar form of operators for
a more systematic extension analysis concerning the following quadratic operator
© 2007 by Taylor & Francis Group, LLC
5 Quasilinear Wave and Boussinesq Models. Systems 269
defined for the vector function U = (u,v)
T
:
F[u,v] = αu
xx
v + βu
x
v
x
+ γ uv
xx
, (5.110)
where α, β,andγ are scalar parameters. Clearly, W
3
= L{1, x, x
2
} is invariant, and
next we formulate the conditions of its extensions.
Proposition 5.33
Operator
(5.110)
preserves:
(i) W
4
= L{1, x , x
2
, x
3
}
iff
2α +3β + 2γ = 0;
and
(ii) W
5
= L{1, x, x
2
, x
3
, x
4
}
iff
3α +4β + 3γ = 0.
(iii) F = 0
does not admit polynomial subspaces of dimension six or more
.
Proof. By easy computations of F[u], the above hypotheses annul all terms contain-
ing: (i) x
4
, and (ii) x
5
and x
6
. (iii) For n = 6, there is a direct proof by taking the
expansion on W
6
,
U = C
1
+ C
2
x + C
3
x
2
+ C
4
x
3
+ C
5
x
4
+ C
6
x
5
with six coefficients C
i
= (C
1i
, C
2i
)
T
∈ IR
2
, the terms with x
6
, x
7
,andx
8
are
annulled iff {α, β, γ } satisfy the linear system
20α + 15β + 6γ = 0, 3α +4β + 3γ = 0, 6α +15β + 20γ = 0,
3α +5β + 5γ = 0, 5α + 5β + 3γ = 0, 4α +5β + 4γ = 0.
The first three equations annul the terms with C
16
C
24
x
6
, C
15
C
25
x
6
,andC
14
C
26
x
6
,
while the last three annul the terms with C
15
C
26
x
7
, C
16
C
25
x
7
,andC
16
C
26
x
8
,re-
spectively. The first three equations have the trivial solution only.
For arbitrary n ≥ 6, the negative result follows from Theorem 5.32.
Extending trigonometric subspaces. For this type of subspaces, consider operators
with an extra reaction term,
F[u,v] = αu
xx
v + βu
x
v
x
+ γ uv
xx
+ δuv. (5.111)
Proposition 5.34
Operator
(5.111)
preserves:
(i) W
3
= L{1, cos x , sin x}
iff
α +β + γ − δ = 0;
(ii) W
5
= L{1, cos x , sin x, cos2x , sin 2x}
for
α = 2
,
β =−3
,
γ = 2
,and
δ = 4
, where the operator is
F[u,v] = 2u
xx
v − 3u
x
v
x
+ 2uv
xx
+ 4uv.
(iii) F = 0
does not admit trigonometric subspaces of more than five dimensions.
Proof. (i) is straightforward by substituting
U = C
1
+ C
2
cosx + C
3
sin x ∈ W
3
and observing that the invariance condition annuls the terms with cos2x and sin2x
that contain four members having C
21
C
22
, C
21
C
32
, C
22
C
31
,andC
31
C
32
.
© 2007 by Taylor & Francis Group, LLC
270 Exact Solutions and Invariant Subspaces
(ii) Takingthe cos-expansionU = C
1
+C
2
cos x +C
3
cos2x yieldsthe following
homogeneous linear system:
4α +4β + 4γ = δ,
α +2β + 4γ = δ,
4α +2β + γ = δ,
where the first equation annuls the terms, containingC
13
C
23
cos4x. Two other linear
equations,written in the form of −α−4γ +δ = 2β and −4α−γ +δ = 2β, guarantee
that the terms
(−α −4γ + δ) cos x cos2x + 2β sin x sin 2x
C
12
C
23
+
(−4α − γ + δ) cos x cos2x + 2β sin x sin 2x
C
22
C
13
belong to L{cos x}. Using Reduce shows that the whole subspace W
5
is invariant.
(iii) See Theorem 5.32.
Remark 5.35 (Open problem: gap completing) In the proof of (ii), the cos-expan-
sion correctly determines the operator. We conjecture that this is the case for any
suitable quadratic operator F, such as (1.86) with, say, even k ≡ i + j . Namely, if
F preserves the invariant cos-subspace, it does the full cos/sin-subspace. (For poly-
nomial subspaces, see a related conjecture in Remark 3.28.) Most of problems of
“non-uniform” distributions of gaps are also
OPEN.
Remark 5.36 (Symmetric restriction) Another approach to nonexistence in (iii) is
basedonusingthesymmetric restriction of F (cf. Section 2.7.3),
ˆ
F[u] = F[u, u] = (α +γ)uu
xx
+ β(u
x
)
2
+ δu
2
.
Hence, if F admits W
n
,then
ˆ
F = 0 (and is not linear) must also admit W
n
. Hence
n > 2k + 1 is impossible by Theorem 2.8 on maximal dimension. This proof does
not cover the case of the null projection, α =−γ and β = δ = 0.
Example 5.37 The following system of Boussinesq-type equations:
U
tt
=−U
xxxx
+ aU
xx
+
2vu
xx
− 3v
x
u
x
+ 2uv
xx
+ 4uv
$
α
β
%
admits exacts solutions
U(x , t) = C
1
(t) + C
2
(t) cos x + C
3
(t) sin x +C
4
(t) cos2x + C
5
(t) sin 2x,
where ten coefficients {C
ij
, i = 1, 2, j = 1, 2, 3, 4, 5} solve a 20D DS.
Extending exponential subspaces. As in Sections 1.3–1.5, the exponential sub-
space for (5.111) is W
3
= L{1, e
x
, e
−x
} with the straightforward invariance con-
dition α + β +γ + δ = 0. The analysis of the corresponding extended 5D subspace
W
5
= L{1, e
x
, e
−x
, e
2x
, e
−2x
} is performed in a similar fashion. Some other exam-
ples of systems with invariant subspaces will be presented in the next chapter, where
operators in IR
N
are studied.
© 2007 by Taylor & Francis Group, LLC
5 Quasilinear Wave and Boussinesq Models. Systems 271
Remarks and comments on the literature
§ 5.1–5.3. Another area of applications, where various quasilinear wave PDEsappear, is theory
of relativity and gravitational instantons dealing with models
v
xx
+ v
yy
= (ψ (v))
tt
,
where ψ(v) is a nonlinear coefficient. Setting u = ψ(v), in the case where ψ
≥ 0, yields a
quasilinear wave equation in IR
2
× IR . In particular, the exponential coefficient ψ(v) = e
v
leads to the “heavenly” equation (a continuous version u
tt
= (e
u
)
xx
of the Toda lattice). This
plays a role in the theory of gravitational instantons and describes self-dual Einstein spaces
with Euclidean signature and a rotational Killing vector, areapreserving diffeomorphisms, and
is a completely integrable system; see references in [413] and new exact analytical solutions
therein. The integrable Pleba´nski second heavenly equation
v
tx
+ v
zy
+ v
xx
v
yy
− (v
xy
)
2
= 0inIR
3
× IR (5.112)
is descriptive of self-dual Einstein spaces with Ricci-flat metrics, [463]. More precisely, the
Einstein field equations that govern self-dual gravitational fields are known to be reduced to
single scalar-valued equations which are the first and second heavenly equations. Equation
(5.112) belongs to the M-A-type and admits solutions on polynomial subspaces (see Example
6.55),v= C
1
+C
2
x +C
3
x
2
, where the coefficients {C
k
(y, z, t )} satisfy a PDE system which
can be simplified on other subspaces.
The classical Boussinesq equation
u
tt
=−u
xxxx
+u
xx
− (u
2
)
xx
(5.113)
first appeared in 1871 [74, p. 258]. It arises in many physical applications, such as propagation
of long waves in shallow water, 1D nonlinear lattice-waves, vibration in nonlinear strings, and
ion sound waves in plasma; see a survey in [127], where various exact solutions and related
references are presented. The improved Boussinesq equation (5.17) (without the quadratic
perturbation 2u
2
) is a model of vibrations in elastic rods and of DNA dynamics; see references
in[127,Sect.6].N -soliton solutions of (5.113) were given by Hirota
†
[285]; see also [320] for
other solutions that describe, for instance, an elastic soliton-breather interaction. Boussinesq
equations, such as (5.113), admit interactingsoliton solutions creating finite-time singularities;
see [68], where earlier references are traced back to the beginning of the 1980s.
For the quasilinear hyperbolic-elliptic PDEs similar to those in Section 5.1, some evolution
properties of solutions on various invariant subspaces and the corresponding DSs were studied
in [162, 161]. The results of the group classification of such PDEs can be found in [10, Ch. 12],
[299] for PDEs
u
tt
= f (x, u
x
)u
xx
+ g(x, u
x
),
and in [366] for u
tt
= u
xx
+ F(x, t, u, u
x
), where a detailed list of references can be found.
The quadratic wave equation (5.1),
u
tt
= (uu
x
)
x
,
which arises in different other physical contexts (e.g., longitudinal wave propagation on a
moving threadline and electromagnetic transmission), possesses interesting exact solutions. In
[199], where conditional symmetries of(5.1) were studied, the following solution was derived:
u(x, t) = f (x −
1
2
t
2
) + t
2
,
†
The first exact soliton-type solutions of the Boussinesq equation were constructed by Baker in 1903
[22] via hyperelliptic ℘ function of genus two; see [19].
© 2007 by Taylor & Francis Group, LLC
272 Exact Solutions and Invariant Subspaces
with f solving the ODE ( ff
)
= 2− f
. This ansatz was originally proposed in 1949, [558],
for the nonlinear wave equation u
tt
= u
xx
+ β(u
2
)
xx
that arises in transonic gas flows. A
general Klein–Gordon equation, u
tt
= u
xx
+ g(u), admits the solution
u(x, t) = f (y), y = x
2
− t
2
, with the ODE ( f
y)
+
1
4
g( f ) = 0.
In the canonical form u
xt
= g(u), the solution is u(x, t) = f (xt),where( f
y)
= g( f ).In
[89], potential symmetries were used to show that (5.1) possesses implicit solutions
x
2
= ut
2
+ C(
t
u
)
4/5
, t
2
=
x
2
u
+C(
x
u
2
)
8/7
(C = constant).
There is a large amount of mathematical literature devoted to blow-up singularities in semi-
linear and quasilinear wave equations. Let us mention the classical papers [305, 328, 332] and
more recent, general results and surveys in [8, 90, 255, 382].
Compactons and compact breathers for the modified improved Boussinesq equation
u
tt
= hu
xxtt
+σ
(u
x
)u
xx
+ V
(u)
(and more realistic PDEs with damping mechanism, +νu
xxt
) are studied in [506], where
further references are given. Here, h = 0 leads to a nonlinear Klein–Gordon equation also
admitting compact structures.
In Section 5.3.3, sixth-order Boussinesq-type equations are considered, and the results are
easily extended to similar quadratic models of arbitrary order 2m. Dispersive fourth-order
Boussinesq equations, called B(m, n), i.e., (5.79) with γ = 0, were considered in [496] and
studied in [581], where explicit TW compactons were constructed.
Various results on exact solutions, classical and nonclassical symmetry reductions of the
ZK equation (5.83), and earlier references are given in [129], where, on p. 390 and p. 405,
solutions on the subspace L{1, x, x
2
} are considered.
§5.4.Parabolic systems of quasilinear PDEs occur in combustion theory [594, 33], in chemi-
cal reaction theory [187], in fluid mechanics [528], and many other areas of application. Exact
solutions on invariant subspaces for a number of systems of nonlinear PDEs were consid-
ered in [217, 232, 110, 113], where further references can be found. A systematic analysis of
invariant subspaces for general systems of two second-order parabolic PDEs
u
t
= f (u,v,u
x
,v
x
)u
xx
+ g(u,v,u
x
,v
x
),
v
t
= p(u,v,u
x
,v
x
)v
xx
+ q(u,v,u
x
,v
x
),
was performed in [477]. A Lie group classification for systems of reaction-diffusion equations
is available in [10, p. 171]. Explicit solutions for a particular quadratic system from the class
in Example 5.31 were constructed in [111], where the standard trigonometric and exponential
subspaces were used. For systems with first-order convection terms (not considered here) some
new non-Lie group-invariant exact solutions were obtained in [114].
Concerning integrable KdV-type systems (i.e., possessing infinitely many generalized sym-
metries), notice earlier examples introduced by Fuchssteiner [210]; see a general classification
in [197, 198, 562] and references therein. Some of the KdV-type systems and various exten-
sions possess exact solutions on standard basic invariant subspaces. Soliton solutions of more
general vector KdV and Ito equations were constructed in [287] by reduction to bilinear forms.
Explicit multi-soliton solutions exist for a number of sypersymmetric integrable equations,
which are systems of coupled equations for u(x, t) (a bosonic field) and ξ(x, t) (a fermionic
field). One of the supersymmetric KdV equations is the Manin–Radul super KdV equation
u
t
=−u
xxx
− 6uu
x
+ 3ξξ
xx
,
ξ
t
=−ξ
xxx
− 3ξ
x
u − 3ξ u
x
.
© 2007 by Taylor & Francis Group, LLC