Galaktionov V.A., Svirshchevskii S.R. Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics
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6 Applications to Nonlinear PDEs in IR
N
323
If γ>1, i.e., N > 12, C
1
(t) blows up in finite time with the behavior
C
1
(t) =
(γ − 1)aD
2
(T − t)
−
N
N−12
(1 + o(1)).
In terms of the original solution v(x , t), we obtain the finite-time extinction pattern
v(x , t) ≈ (T − t)
N(N +2)
2(N −12)
ν
1
+ ν
2
|ζ |
2
−
N+2
2
,ζ= x(T − t)
N+6
2(N −12)
, (6.170)
where ν
1,2
> 0 are some constants. This extinction is due to the singular character of
the higher-orderdiffusion operator,where the negative exponent n =−
6
N+2
mimics
a very fast diffusion with non-zero flux at x =∞. This phenomenon is common for
the second-order quasilinear fast-diffusion equations (see Remarks), but, for higher-
order TFEs, it is not well understood and creates many
OPEN PROBLEMS.
If γ = 1 in (6.169), i.e., N = 12, we have
C
1
(t) ∼ e
aD
2
t
for t 1,
and obtain exponentially decaying patterns v(x, t) ∼ e
−7aD
2
t
as t →+∞.For
γ<1, i.e., N ∈ (6, 12),
C
1
(t) ∼ t
N
12−N
⇒ v(x, t) ∼ t
−
N(N +2)
2(12−N )
as t →∞,
so the solution v(x, t) has infinite-time extinction with a power-like decay.
(III) Separatrix behavior for N > 6. This new, most probable, unstable behav-
ior corresponds to the separatrix in (6.166). Then from the second ODE in (6.164),
C
2
(t) =
√
a−b
b
1
t
,andfinally we find the following new explicit solution of the TFE
(6.163):
v(x , t) = t
−
N+2
2
6
N−6
+
1
6
N−6
2(N+2)(N+4)
|ξ|
2
−
N+2
2
, where ξ =
x
t
.
6.6.4 On some extensions of polynomial subspaces
Quadratic operators. Extended subspaces that are composedof higher-degreepoly-
nomials exist for more general operators of the thin film type, such as
F[u] = αu
2
u + β∇u ·∇u + γ(u)
2
, (6.171)
which is considered in radial geometry in IR
N
.
Proposition 6.79
Operator
(6.171)
preserves the following subspaces:
(i) W
r
4
= L{1, |x|
2
, |x |
4
, |x |
6
}
iff
2(N + 2)α + 12β + 3(N +4)γ = 0;
(6.172)
(ii) W
r
5
= L{1, |x|
2
, |x |
4
, |x |
6
, |x |
8
}
iff
(6.173)
(N + 4)(3N + 14)α + 4(5N +26)β + 4(N +4)(N + 6)γ = 0,
3(N + 4)α + 24β + 4(N +6)γ = 0,
(6.174)
i.e., for
α =
2(N−2)
N+4
β
and
γ =−
3(N+2)
2(N+6)
β
.
© 2007 by Taylor & Francis Group, LLC
324 Exact Solutions and Invariant Subspaces
Proof. (i) The condition in (6.172) annuls the terms containing |x|
8
in F[u]. (ii)
(6.174) annuls all terms in F[u] containing |x |
10
and |x |
12
.
It is easy to check that (6.171) does not preserve the 6D subspace
W
r
6
= L{1, |x |
2
, |x |
4
, |x |
6
, |x |
8
, |x |
10
}
unless α = β = γ = 0. Using W
r
6
yields the terms containing |x|
12
, |x |
14
, |x |
16
and,
for annulling it, we obtain a homogeneous system of linear algebraic equations for
α, β,andγ that has a trivial solution only.
Example 6.80 Consider the fourth-order quasilinear parabolic PDE
u
t
= αu
2
u + β∇u ·∇u + γ(u)
2
for which the condition in (6.172) holds. Using subspace (6.172) with
u(x , t) = C
1
(t) + C
2
(t)|x |
2
+ C
3
(t)|x |
4
+ C
4
(t)|x |
6
yields the following DS:
C
1
= 8αN (N + 2)C
1
C
3
+ 4γ N
2
C
2
2
,
C
2
= 8(N + 2)(Nα + 2β + 2Nγ)C
2
C
3
+ 24α(N + 2)(N + 4)C
1
C
4
,
C
3
= 8(N + 2)
Nα + 4β + 2(N + 2)γ
C
2
3
+24(N + 4)
(N + 2)α + 2β + Nγ
C
2
C
4
,
C
4
= 8
4α(N + 2)(N +3) + 6β(3N +10) + 6γ(N + 2)(N + 4)
C
3
C
4
.
These solutions can be used for detecting interface equations for the corresponding
FBPs, and for revealing various quenching and extinction patterns (
OPEN PROB-
LEMS).
On a cubic operator in IR
N
. As an example of another type, consider a cubic oper-
ator of the form
F[u] =−
2l
(au
3
+ bu
2
+ cu) for l = 1, 2, ... . (6.175)
Proposition 6.81
Operator
(6.175)
admits
W
r
l+1
= L{1, r
2
, ..., r
2l
}
.
In particular, solutionson W
r
l+1
describe the fine structure of singularityformation
for the quasilinear parabolic equation with absorption
u
t
=−
2l
(u
3
) − 1,
where a suitable FBP should be posed.
6.6.5 On remarkable thin film operators in IR
2
In the second-order case in Section 6.3, we have studied the remarkable operator
(6.57), exhibiting a variety of invariant properties. Such related remarkable operators
exist in thin film theory.
First remarkable operator. Let us begin with special properties of the following
thin film operator:
F
1
[u] =−u
2
u − (u)
2
+ 2∇u ·∇u for (x, y) ∈ IR
2
. (6.176)
© 2007 by Taylor & Francis Group, LLC
6 Applications to Nonlinear PDEs in IR
N
325
This is a 2D version of the special operator (3.125) with β =−1. According to
Proposition 3.19, we expect that (6.176) still inherits extended invariantproperties of
the 1D operator, so that invariant subspaces can be composed of both trigonometric
and hyperbolicfunctions.
Proposition 6.82
Operator
(6.176)
admits
W
4
= L{1, cosh2x , cos2y, coshx cos y}.
Proof. The following holds:
u = C
1
+ C
2
cosh2x + C
3
cos2y + C
4
coshx cos y ⇒ (6.177)
F
1
[u] =−32
C
2
2
+ C
2
3
− 16C
1
C
2
cosh2x
−16C
1
C
3
cos2y − 16(C
2
+ C
3
)C
4
coshx cos y.
(6.178)
Example 6.83 (TFE with absorption) Exact solutions (6.177) of the TFE
u
t
=−u
2
u − (u)
2
+ 2∇u ·∇u − 1inIR
2
× IR
+
describe extinction or quenching phenomena. The DS follows from (6.178).
Second remarkable operator. The thin film operator
F
2
[u] =−u
2
u + (u)
2
is associated with the same second-order one (6.57), but in a different manner. To be
precise, there exist solutions (6.86) associated with invariant modules for
ˆ
F
2
[u] = u
t
− F
2
[u].
Example 6.84 Consider the fourth-order parabolic PDE
u
t
= α[−u
2
u + (u)
2
] + β(uu −|∇u|
2
) in IR
2
× IR
+
.
Then there exist solutions (6.86), where the DS takes the form
C
1
=−β
C
2
2
+ C
2
3
),
C
2
= C
3
= 0,
C
4
=
−α
γ
2
1
+ γ
2
2
2
+ β
γ
2
1
+ γ
2
2
C
1
C
4
− 2β(γ
1
C
2
+ γ
2
C
3
)C
4
,
γ
1
=−α
γ
2
1
+ γ
2
2
2
C
2
+ β
γ
2
1
+ γ
2
2
C
2
,
γ
2
=−α
γ
2
1
+ γ
2
2
2
C
3
+ β
γ
2
1
+ γ
2
2
C
3
.
(6.179)
The DS for γ
1,2
is integrated in quadratures, but solutions are not explicit.
Example 6.85 (2mth-order remarkable operators) For higher-order operators,
similar invariant properties in IR
2
are exhibited by
F
2
[u] =−u
2m
u + (
m
u)
2
and F
2
[u] = u
2m−1
u −|∇
m−1
u|
2
,
which, for any m = 1, 2, ..., are elliptic in the positivity domain of u. Parabolic PDEs
with such operators generate the DSs, such as (6.179). The resulting solutions are
convenient for studying the interface propagation in FBPs, as well as for describing
quenching and extinction singularity formation phenomena.
© 2007 by Taylor & Francis Group, LLC
326 Exact Solutions and Invariant Subspaces
6.7 Moving compact structures in nonlinear dispersion equations
Continuing the study of compactons in IR
N
that was began in Section 4.5, we extend
the above results to PDEs of nonlinear dispersion types. Let F[u] be a higher-order
operator admitting the subspace of quadratic and linear functions,
W
2N+1
= L
1, x
2
1
, ..., x
2
N
, x
1
, ..., x
N
, (6.180)
e.g., F = F
3
given in (6.150). Let
ˆ
F[u] be a simpler operator admitting W
2N+1
,for
instance,
ˆ
F[u] = γ u with a constant γ = 0.
Looking for compact structures moving in the x
1
-directions, consider the PDE
u
t
=
F[u] +
ˆ
F[u]
x
1
, (6.181)
where the “inhomogeneous” lower-order term with
ˆ
F is necessary to initiate such
a drift. For the operator in (6.150), this is a fifth-order nonlinear dispersion equa-
tion, while choosing the second-orderoperator (6.1) yields a third-orderPDE similar
to those studied in Section 4.2 in 1D. In general, F can be composed from vari-
ous second and higher-order operators admitting the necessary subspace. Using the
2mth-order operator from Proposition 6.70 needs another subspace
ˆ
W = L
1, x
2m
1
, ..., x
2m
N
, x
2m−1
1
, ..., x
2m−1
N
, ...
.
For the subspace (6.180), we look for solutions
u(x , t) = C(t) + a
1
(t)x
2
1
+ ... + a
N
(t)x
2
N
+ d
1
(t)x
1
+ ... + d
N
(t)x
N
, (6.182)
and obtain a standard DS for expansioncoefficients.Note that the differentiation D
x
1
on the right-hand side of (6.181) implies that
a
1
= a
2
= ... = a
N
= 0,
d
2
= ... = d
N
= 0.
This suggests that solutions (6.182) can describe dynamics around the radial TW
compactons moving in the x
1
-direction,
u
c
(x , t) = f (y), where y =
(x
1
− λt)
2
+ x
2
2
+ ... + x
2
N
. (6.183)
Plugging (6.183) into (6.181), and integrating once in y with the zero constant of
integration, yields the radial ODE for f ,
−λf = F[ f ] +
ˆ
F[ f ], (6.184)
where ,
2
and other differential forms in F and
ˆ
F are calculated in radial y-
geometry. Equation (6.184) possesses stationary, independent of t solutions (6.182).
More complicated dynamics on W
2N+1
or
ˆ
W (not necessarily associated with TWs)
and compact patterns exist for the correspondingsecond-order PDE
u
tt
=
F[u] +
ˆ
F[u]
x
1
.
© 2007 by Taylor & Francis Group, LLC
6 Applications to Nonlinear PDEs in IR
N
327
Example 6.86 (Compacton TWs in IR
N
) For instance, take F = F
3
as in (6.150)
and the simplest
ˆ
F[u] = γ u, i.e., the PDE
u
t
=
−
1
2
u
2
u
2
−∇u ·∇u
2
+ γ u
x
1
. (6.185)
Looking for solutions moving in the x
1
-direction (cf. (6.151)),
u(x , t) = C + a
1
(x
1
− λt)
2
+ a
2
x
2
2
+ ... + a
N
x
2
N
,
(6.153), we obtain the following projection of the PDE onto 2(x
1
− λt):
−λa
1
=−
1
2
(
2
u
2
)a
1
− 16(tr A)a
2
1
− 32a
3
1
+ γ a
1
.
Assuming for simplicity that a
1
= a
2
= ... = a
N
= a (obviously, there exist other
not that symmetric solutions) yields by (6.152)
2
u
2
= 8(tr A)
2
+ 16
a
2
i
= 8N (N + 2)a
2
, so
γ + λ = 4N(N + 2)a
2
+ 16Na
2
+ 32a
2
= 4(N + 2)(N +4)a
2
.
This gives the explicit moving compacton for λ>−γ ,
u
c
(x , t) =
B −
1
2
γ +λ
(N+2)(N+4)
(x
1
− λt)
2
+ x
2
2
+ ... + x
2
N
+
,
where B = C(0)>0. Thus, solutions on the invariant subspace (6.182) can describe
more complicated non-radial dynamics around the TW u
c
(x , t).
As in the TFE model (6.150) and in Example 6.74, the transformation of such
nonnegative solutions on W
2N+1
(or on
ˆ
W ) into solutions with a zero contact angle
on interfaces is done by setting v = u
2
, so that (6.181) reads
v
t
= 2
√
v
F[
√
v] +
ˆ
F[
√
v]
x
1
. (6.186)
In view of our analysis of oscillatory behavior of compacton equations in Sections
4.2 and 4.3, such solutions of the (2m+1)th-order PDE (6.186) on polynomial invari-
ant subspaces
ˆ
W satisfy the Cauchy problem for m = 1 only, while for m ≥ 2, these
need a suitable FBP setting.
For the C
N
(m, a + b) equation that describes the sedimentation of particles in a
dilute dispersion (see [494] for further references and applications)
v
t
+ (v
m
)
x
+
1
b
v
a
(v
b
)
x
1
= 0 (v ≥ 0),
there exist almost explicit TWs (6.183) in IR × IR
3
for m = a = 1 + b and m = 2,
a + b = 3; see [497]. For such third-order PDEs, these solutions do not belong to
polynomial invariant subspaces.
6.8 From invariant polynomial subspaces in IR
N
to invariant trigonometric
subspaces in IR
N−1
In this section, we deal with trigonometric subspaces in the multi-dimensional case.
We show that, using simple polynomial subspaces for quadratic operators in IR
N
© 2007 by Taylor & Francis Group, LLC
substituting into (6.185),and using the right-hand side of the first equation in the DS
328 Exact Solutions and Invariant Subspaces
in spherical coordinates, it is possible to generate various trigonometric subspaces
for classes of nonlinear operators in IR
N−1
. Actually, excluding a few examples,
trigonometric subspaces in IR
N
for N ≥ 2 have not been treated before in a sys-
tematic manner, and their existence was associated with rather technical algebraic
manipulations. Now, we link such trigonometric subspaces with simpler polynomial
ones for quadratic second and higher-order operators.
6.8.1 Second-order operators
Operators in IR
2
. We begin with a simple example. Consider the quadratic hyper-
bolic PDE
u
tt
= F[u] ≡ uu + γ |∇u|
2
in IR
2
× IR . (6.187)
By Proposition 6.1(ii), there exist solutions given by a general quadratic form (the
free coefficient C(t) ≡ 0)
u(x , t) = x
T
A(t)x ≡ a
1,1
(t)x
2
1
+ 2a
1,2
(t)x
1
x
2
+ a
2,2
(t)x
2
2
,
ordinates on the plane
x
1
= r cosθ,
x
2
= r sinθ,
(6.188)
gives the finite trigonometric expansion
u(x , t) = r
2
a
1,1
cos
2
θ +2a
1,2
cosθ sinθ + a
2,2
sin
2
θ
≡ r
2
(C
1
+ C
2
cos2θ + C
3
sin2θ),
where C
1
=
1
2
(a
1,1
+ a
2,2
), C
2
=
1
2
(a
1,1
− a
2,2
),andC
3
= a
1,2
. Therefore,
u(x , t) = r
2
U(θ, t), where U(θ, t) ∈ W
3
= L{1, cos2θ,sin 2θ} (6.189)
belongs to a standard trigonometric subspace for ordinary differential operators; cf.
Proposition 1.29. Let us derive a PDE for the function U by using that
=
r
+
1
r
2
σ
, (6.190)
where
r
is the radial part of the Laplacian in IR
2
,and
σ
is the Laplace–Beltrami
operator on the circle S
1
,
r
=
∂
2
∂r
2
+
1
r
∂
∂r
and
σ
=
∂
2
∂θ
2
.
By the polar representation of the gradient
u
x
1
= u
r
cosθ +u
θ
sinθ
r
,
u
x
2
= u
r
sinθ − u
θ
cosθ
r
,
(6.191)
it is easy to see that |∇u|
2
= (u
r
)
2
+
1
r
2
(u
θ
)
2
. Substituting (6.189) into the original
equation (6.187) yields the following hyperbolic PDE:
U
tt
=
ˆ
F[U] ≡ UU
θθ
+ γ(U
θ
)
2
+ 4(1 + γ)U
2
,
© 2007 by Taylor & Francis Group, LLC
where the expansion coefficients satisfy a DS; see (6.11). Introducing the polar co-
6 Applications to Nonlinear PDEs in IR
N
329
so the standard operator
ˆ
F preserving the subspace W
3
in (6.189) is detected.
Operators in IR
3
and IR
N
. Similarly, for such second-orderparabolic or hyperbolic
PDEs in IR
N
, using the spherical coordinates in N-dimensional geometry explains
howthe polynomialsubspaces,such as (6.5) or (6.4), generate the multi-dimensional
trigonometric subspaces spanned by the cos, sin, and the products of such functions
of all the N-1 spherical angles. An example of such subspaces has been considered
in Proposition 6.42 for the remarkable quadratic operator. Let us present another
example in IR
3
, showing how the trigonometric subspaces in IR
2
can be constructed
for more general operators.
Example 6.87 (Deriving PME with source) Consider the quadratic PME
u
t
= (u
2
) in IR
3
× IR
+
, (6.192)
for which the main computations are quite simple. On the 6D polynomial subspace
(6.5), exact solutions are
u(x , t) = a
1,1
x
2
1
+a
2,2
x
2
2
+a
3,3
x
2
3
+2a
1,2
x
1
x
2
+2a
1,3
x
1
x
3
+2a
2,3
x
2
x
3
. (6.193)
Introducing the spherical coordinates in IR
3
with two angles σ ={θ,ϑ},
x
1
= r sinϑ cosθ,
x
2
= r sinϑ sinθ,
x
3
= r cosϑ,
(6.194)
and plugging into (6.193) yields
u(x , t) = r
2
U(σ, t) ≡ r
2
(a
1,1
sin
2
ϑ cos
2
θ
+a
2,2
sin
2
ϑ sin
2
θ + a
3,3
cos
2
ϑ + 2a
1,2
sin
2
ϑ sin θ cosθ
+2a
1,3
sinϑ cosϑ cosθ +2a
2,3
sinϑ cosϑ sinθ).
Here U belongs to the 8D trigonometric subspace
W
tr
8
= L{1, cos2θ,sin2θ,cos2ϑ, sin 2ϑ cosθ,
sin2ϑ sin θ,cos2ϑ cos2θ,sin2θ cos2ϑ}.
Let us next derive the corresponding equation with a quadratic operator preserv-
ing such a subspace. Using (6.190) with the Laplace–Beltrami operator on the unit
sphere S
2
in IR
3
σ
=
1
sinϑ
∂
∂ϑ
sinϑ
∂
∂ϑ
+
1
sin
2
ϑ
∂
2
∂θ
2
. (6.195)
Substituting u = r
2
U into (6.192) gives
r
2
U
t
= r
2
σ
U
2
+U
2
r
r
4
,
which yields the following PME with source:
U
t
=
σ
U
2
+ 20U
2
in S
2
× IR
+
.
This PDE generates blow-up, so exact solutions describe interesting features of such
localized patterns. The blow-up is regional here and these solutions explain proper-
ties of a non-symmetric, non-radial singular behavior.
© 2007 by Taylor & Francis Group, LLC
330 Exact Solutions and Invariant Subspaces
6.8.2 Higher-order operators
Space IR
2
. We begin with the quadratic thin film equation
u
t
= F[u] ≡−∇·(u∇u) in IR
2
× IR
+
.
Taking the polynomial subspace as in (6.137) yields solutions
u(x , t) = (x
2
)
T
A(t)x
2
≡ a
1,1
(t)x
4
1
+ 2a
1,2
(t)x
2
1
x
2
2
+ a
2,2
(t)x
4
2
.
In polar coordinates (6.188), this gives the trigonometric expansion
u(x , t) = r
4
a
1,1
cos
4
θ + 2a
1,2
cos
2
θ sin
2
θ +a
2,2
sin
4
θ
≡ r
4
(C
1
+ C
2
cos2θ +C
3
cos4θ) ≡ r
4
U(θ, t),
(6.196)
so U ∈ W
3
= L{1, cos2θ,cos4θ}. Using formulae (6.190) and (6.191) yields the
following quadratic PDE for the function U in (6.196):
U
tt
=−UU
θθθθ
− U
θ
U
θθθ
− 28UU
θθ
− 16(U
θ
)
2
− 192U
2
,
with a more general quadratic operator than those in Proposition 3.16. Using other
polynomial subspaces in Proposition 6.68 leads to extended trigonometric ones.
Example 6.88 (2mth-order equations) This approach applies to higher-order op-
erators. For instance, consider the 2mth-order quadratic wave equation
u
tt
= (−1)
m+1
m
(u
2
) in IR
2
× IR , (6.197)
which is hyperbolic in the positivity domain {u > 0}. As in Proposition 6.70, we
take the polynomial solutions
u(x , t) =
(i+j =m)
a
i, j
(t) x
2i
1
x
2j
2
.
In polar coordinates, this yields the trigonometricexpansion
u = r
2m
U ≡ r
2m
a
i, j
(t) cos
2i
θ sin
2j
θ,
so that U belongs to the trigonometric subspace
L{1, cos2θ,cos4θ, ..., cos2mθ }.
The operator preserving this subspace is obtained by plugging u = r
2m
U into the
PDE (6.197),
U
tt
= F
m
[U] ≡
1
r
2m
(−1)
m+1
r
+
1
r
2
σ
m
(r
4m
U
2
).
For instance, for m = 3,
F
3
[U] =
3
σ
(U
2
) + 308
2
σ
(U
2
) + 30016
σ
(U
2
) + 921600U
2
,
with the trigonometric subspace
W
4
= L{1, cos 2θ,cos4θ,cos6θ }.
This subspace can be extended to 7D by adding necessary sin functions.
© 2007 by Taylor & Francis Group, LLC
6 Applications to Nonlinear PDEs in IR
N
331
Space IR
N
. As in the second-order case, polynomial subspaces for operators in IR
N
detect the trigonometric ones in IR
N−1
. Consider an example in IR
3
.
Example 6.89 The fourth-order equation which is parabolic in {u > 0},
u
t
=−
2
(u
2
) in IR
3
× IR
+
, (6.198)
admits polynomial solutions
u(x , t) =
(i+j +k=2)
a
i, j,k
(t) x
2i
1
x
2j
2
x
2k
3
(6.199)
u = r
4
U ≡ r
4
a
i, j,k
(sinϑ cosθ)
2i
(sinϑ sinθ)
2j
cos
2k
ϑ,
where U belongs to the subspace
L{cos(2i θ)cos(2jϑ), 0 ≤ i, j ≤ 2}.
Substituting u = r
4
U into (6.198) gives a quadratic equation for U(σ, t),
U
t
=−
2
σ
(U
2
) − 114
σ
(U
2
) − 3024U
2
,
where
σ
is the Laplace–Beltrami operator (6.195).
Remarks and comments on the literature
As has been mentioned, the general theory of annihilating operators and annihilators of finite-
dimensional invariant modules in IR
N
was developed in [312]. “The formulae for the affine
annihilators and annihilators are often extremely complicated, even for relatively simple sub-
spaces” [312, p. 316], so general results are often difficult to apply in practice.
§6.1.Some of the results are taken from [220] and [239]. For the PME, polynomial exact
solutions were obtained in [342], where the resulting DS was carefully studied. A general the-
orem on finite-dimensional subspaces of analytic functions of several variables was obtained in
[312] (similar to the Main Theorem in Section 2.1 in 1D), where a complete description of the
set of operators preserving a given subspace was obtained. The Ernst equation (6.13) in Exam-
ple 6.5 is connected with the sdYM equation; see [1] and comments below. Non-symmetric
blow-up and extinction on multi-dimensional polynomial subspaces for the quasilinear heat
equation
u
t
=∇·(u
σ
∇u) ± u
1−σ
(as well as for (6.41)) were studied in [235]. On group classification of semilinear and quasi-
linear heat equations in IR
N
× IR
+
, see [10, pp. 144-163], [121] for
u
t
= u
xx
+ u
yy
+ F(t, x, y, u, u
x
, u
y
),
and references therein for more recent results.
Concerning general matrix equations, such as (6.9), note that the quadratic first-order and
related second-order cubic equations
A
= A
2
+ B and A
= 2A
3
+ BA+ AB,
where B is a constant matrix, are integrable and indeed solvable [84, 301]. A related matrix
equation
A
=−(tr A)A +(adj A)A + A
T
A (adj A = (det A)A
−1
) (6.200)
© 2007 by Taylor & Francis Group, LLC
(cf. subspace (6.142)). Using in (6.199) the spherical coordinates (6.194) yields
332 Exact Solutions and Invariant Subspaces
appears [102] in the process of reduction of the self-dual Yang–Mills (sdYM) equations (a
system of equations for Lie algebra-valued functions on C
4
). In IR
3
, introducing simple fac-
torization, (6.200) can be transformed into a quadratic system [1]
ω
1
= ω
2
ω
3
− ω
1
(ω
2
+ ω
3
) + τ
2
,
ω
2
= ω
3
ω
1
− ω
2
(ω
3
+ ω
1
) + τ
2
,
ω
3
= ω
1
ω
2
− ω
3
(ω
1
+ ω
2
) + τ
2
,
(6.201)
where τ
2
= τ
2
1
+ τ
2
2
+ τ
2
3
and
τ
1
=−τ
1
(ω
2
+ ω
3
),
τ
2
=−τ
2
(ω
3
+ ω
1
),
τ
3
=−τ
3
(ω
1
+ ω
2
).
For τ = 0, (6.201) is the classical Darboux–Halphen system derived by Darboux (1878) in the
analysis of triple orthogonal surfaces [140], and later solved by Halphen (1881) [277] using
linearization in terms of Fuchsian differential equations with three regular singular points. In
the case where τ = 0, the function y =−2(ω
1
+ω
2
+ ω
3
) satisfies the Chazy equation
y
= 2yy
− 3(y
)
2
,
which is explicitly solved in terms of solutions of a linear hypergeometric equation [104].
Given a solution y(t), the three distinct roots ω
1
(t), ω
2
(t),andω
3
(t) of the cubic equation
ω
3
+
1
2
yω
2
+
1
2
y
ω +
1
12
y
= 0
solve the Darboux–Halphen system (6.201) with τ = 0. See [1] for a survey, multiple refer-
ences, and many reductions of the sdYM equations to integrable PDEs.
It seems that second-order quadratic equations for A in (6.9) occurring in the hyperbolic
case are not solvable even in the class of diagonal matrices. The first-order equations, such as
given in (6.12), are indeed solvable.
Proposition 6.13 and related examples were taken from [239]. Such solutions were first
observed in [453] for diffusion-absorption equations in IR (see also [435], where the pseudo-
symmetries were used), and further extensions to parabolic and wave PDEs were developed in
[341] (Example 6.17 deals with a slight modification of an equation from this paper). Example
6.19 explains the invariant subspace essence of particular exact solutions in 1D with L =
D
2
x
+ αI that were constructed in [325, p. 1409], by determining a third-order differential
constraint.
Exact solutions of the LRT equation (6.28) were studied in a number of papers; see [504,
505] and further references in [133, 267]. Invariant and partially invariant solutions were con-
sidered in [567, 566]. Local and global (for small initial data) existence results can be proved
for the viscosity LRT equation (6.32), [375]. A group classification and some exact solutions
of (6.33) are presented in [10, p. 301].
Notice a quadratic system from binormal flow
X
t
= X
s
× X
ss
≡
X
t
×X
tt
(s
)
3
for curves {X(·, t), t > 0} in IR
3
, which are parameterized by the arclength s. This equation
of the evolution of isolated vortexes in an inviscid liquids was proposed by Da Rios in 1906,
whose study established the first geometric link between soliton theory and the motion of
inextensible curves. Nowadays this equation is used as an approximate model for a vortex tube
of infinitesimal cross section X(·, t) described by Euler equations; see details and references
in [275], where similarity solutions were constructed.
§6.2.Proposition 6.24 is taken from [220]; see also [343]. The subspace in Proposition 6.29
© 2007 by Taylor & Francis Group, LLC