Galaktionov V.A., Svirshchevskii S.R. Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics

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1 Linear Invariant Subspaces: Examples 43

For solutions on the general subspace (1.123), the maximal number of invariant

conditions in (1.124) is

n(n−1)

(recall that p

= 0isﬁxed apriori). For proper oper-

ators F with higher-degree polynomials, there exist subspaces of higher dimension,

and hence, solutions via DSs can be constructed for corresponding evolution PDEs.

On exponential-trigonometric subspaces. If complex exponents p

= a

+ ib

= 0, are involved in (1.123), this gives the exact solutions for a system of PDEs

stated for the real and imaginary parts of u = U + iV ,



= F[U ] − F[V ],

= 2P[U, V ].

(1.134)

The real case V = 0 yields the single PDE U

= F[U]. Since

= e

(cosb

x + isinb

x),

the expansion on W

consists of exponential-trigonometric functions, and algebraic

manipulations become more technical.

On quadratic

∂

∂t

-dependent operators. Consider more general operators that in-

clude differentiating in t,

F[u] =



(i, j,r,s)

i, j,r,s







, (1.135)

where, for each pair of ﬁxed {r, s}, the real matrix #a

i, j,r,s

# is symmetric relative to

indices {i, j }. Such operators occur in the theory of fully integrable and other equa-

is similar,

F[u] =



(m,l)





(r,s)



(i, j )

i, j,r,s

(r)

(s)



where the internal sum can be viewed as the action of the polynomial operator

r,s

[u] ≡ P

r,s

( p

, p

(r)

(s)

Assuming that all the derivatives {C

(r)

} are arbitrary, the subspace (module) W

invariant if all terms like that vanish for any p

+ p

∈ 



, so that Proposition 1.47

remains valid. We postpone analysis of such solutions until Section 7.1, where using

partially invariant linear subspaces will provide us with more ﬂexibility.

On cubic and other operators. Consider cubic operators with polynomials

F[u] =



(i, j,k)

i, j,k

u, P(X, Y, Z ) =



i, j,k

, (1.136)

where {a

i, j,k

} is an absolutely symmetric (i.e., relative to any pair of indices) tensor

of rank 3. In this case, the analogy of formula (1.126) includes all triple products

forming the terms e

, with invariance conditions

P( p

, p

) = 0, if p

+ p

∈ 



The approach is extended to polynomial operators of higher algebraic degree.

tions; cf. the quadratic representation (0.31) of the KdV equation in the Introduction

and the GT equation (1.64). The action of F to functions (1.125) on W

44 Exact Solutions and Invariant Subspaces

1.5.3 On a relation between exponential and polynomial subspaces

Consider a class of PDEs possessing solutions of the form

¯u(x, t) = C(t)e

ϕ(t)x

, (1.137)

with two expansion coefﬁcients C(t)>0andϕ(t). The basic nonlinearity is now

g(u) = u ln u (u > 0), ⇒ g( ¯u) = (C lnC) e

ϕx

+ (Cϕ) x e

ϕx

The time-derivative also satisﬁes u

: L{e

ϕx

}→L{e

ϕx

, xe

ϕx

}. Moreover, differen-

tial operators D

(lnu) with any k ≥ 2 are annihilators of the subspace L{e

ϕx

} for

any ϕ. Therefore, solutions (1.137) are available for the PDEs



( j ≥1)



u ln u + b

(lnu)



+ a

u ln u, with the DS (1.138)







( j ≥0)

j +1





( j ≥0)

( j +lnC)C.

Setting u = e

in (1.138) (so, in view of (1.137), ¯v = ln C + ϕx) yields a PDE with

the leading polynomial operator,

= e

−v



( j ≥1)

v) + e

−v



( j ≥1)

j +1

v + a

The right-hand side admits the subspace of linear functions L{1, x}, from whence

come the equivalent solutions

¯v(x , t) = C

(t) + C

(t)x

(the second term vanishes).

Example 1.52 (Cahn–Hilliard-type equation) The following parabolic PDE:

=−u

xxxx



α lnu +

+ γ





+ δu ln u (1.139)

admits solutions (1.137), where





= δϕ + αϕ



= [−ϕ

+ (α lnC + αC + γ C)ϕ

+ δ lnC]C.

Hence,

ϕ(t) = e

δt

[

(A − e

2δt

)]

−

for δ = 0,

and

ϕ(t) = [2α(−t)]

−

for δ = 0.

1.5.4 On subspaces of irrational functions: dipole-type solutions

The quadratic operator from PME theory

[u] = uu

+ γ(u

)

, where γ = −1, (1.140)

admits a simple 2D subspace (here x > 0)

= L{x

, x

}, with α =

1+γ

. (1.141)

1 Linear Invariant Subspaces: Examples 45

The origin of such a remarkable subspace is the well-known Barenblatt–Zel’dovich

(BZ) dipole solution (1957) [28] of the PME

= (v

)

in IR

× IR

and v(0, t) ≡ 0 (σ > 0).

The pressure variable u = v

then solves u

= F

[u] containing operator (1.140)

with γ =

= 1) and is in-

duced by a group of scalings. The explicit integrability of the resulting ODE follows

from the conservation law of the ﬁrst moment of solutions of the PME, meaning that



∞

xv(x , t) dx is preserved with time if it is ﬁnite at t = 0. See more details on

such solutions in Section 3.6 devoted to higher-order diffusion equations. Consider

another related example.

Example 1.53 (Quadratic wave equation) The quasilinear quadratic “forced” (an

extra term µu on the right-hand side) and “damped” (the friction term βu

on the

left-hand side with β>0) wave equation

+ βu

= (uu

)

+ µu ≡ F

[u] + µu (γ = 1) (1.142)

admits exact dipole-type solutions on the subspace (1.141),

u(x , t) = C

(t)

√

x + C

(t)x





+ βC



+ µC



+ βC



= 6C

+ µC

These solutions are not connected with a scaling group, leaving (1.142) invariant.

1.5.5 Another special subspace

Various non-standard invariant subspaces will be studied in the next chapter, but we

present a simple example here.

Example 1.54 (Quasilinear wave equation with source) The hyperbolic PDE

= u

+ u

≥ 0) (1.143)

admits solutions

u(x , t) = C

(t) + C

(t) f (x) ∈ L{1, f (x)},

where f satisﬁes the ODE for elliptic functions





+ f

= µf + ν,

with arbitrary ﬁxed parameters µ and ν. The DS is (see applications in Section 5.2)





= C

+ νC



= µC

+ 2C

1.5.6 On reduction to quadratic equations

It is no accident that the main classiﬁcation results on invariant subspaces are as-

sociated with evolution equations having quadratic or cubic nonlinearities. First of

. The BZ solution is self-similar (see (3.146) for σ

46 Exact Solutions and Invariant Subspaces

all, the majority of the nonlinear PDEs of mathematical physics fall into this class.

Secondly, it will be shown in Section 2.2 that, in several important cases, linear in-

variant subspaces of the maximal dimension appear precisely for quadratic operators

only. Therefore, it is important to specify classes of nonlinear evolution PDEs (say,

of parabolic type) which can be reduced to those having quadratic operators.

Here, we consider a class of 1D quasilinear heat or reaction-diffusion equations

= k(u)u

+ q(u), (1.144)

which are widely used in many areas of mechanics, combustion theory, and biology.

Let us perform a smooth change

u = g(v)

and describe those PDEs (1.144) which can be reduced to equations with quadratic

operators. It is not difﬁcult to see that, up to a linear change, there exist three types

of such equations:

(I) u

µu

+ν

δu

+κ+λu

−n

µu

+ν

(II) u

µe

+ν

δe

+κ+λe

−u

µe

+ν

(III) u

µ lnu+ν

δ ln

u+κ ln u+λ

µ lnu+ν

where µ and ν are constants satisfying µ

+ ν

= 0andn = 0. In the ﬁrst PDE (I),

the change u = v

1/n

reduces it to

(I) (µv + ν)v

= vv



− 1



)

+ n(δv

+ κv + λ),

where the quadratic operator also arises on the left-hand side. This class of PDEs will

be more systematically used in Chapter 7 dealing with partially invariant subspaces.

In the second equation (II), the change is logarithmic, u = lnv,wherev solves

(II) (µv + ν)v

= vv

− (v

)

+ δv

+ κv + λ.

In the third PDE (III), we use the exponential transformation u = e

, yielding

(III) (µv + ν)v

= v

+ (v

)

+ δv

+ κv + λ.

For µ = 0, the left-hand sides become linear and the exact solutions of these three

types of quadratic PDEs can be found by using invariant subspaces.

Example 1.55 (Semilinear cubic heat equation) Consider equation (I) with pa-

rameters µ = 1, ν = κ = 0, and n =−1,

= u

+ δu + λu

Setting u = v

1/n

yields the following quadratic PDE (see Section 7.4.1):

= vv

− 2(v

)

− δv

− λ.

Remarks and comments on the literature

§1.1.In all chapters we consider various quasilinear second-order parabolic PDEs. Such

models occur in diffusion and combustion theory; see descriptions and earlier references in

1 Linear Invariant Subspaces: Examples 47

the classic book [594]. Modern ﬁltration theory goes back to the beginning of the twentieth

century to works by N.Ye. Zhukovskii, who is better known for his fundamental research in

aerodynamics, hydrodynamics, and ODE theory (on his non-oscillation test in 1892, see [226,

p. 19]). His contribution to “theory of ground waters” is explained in Kochina’s paper [350].

Parabolic PDE models in ﬁltration theory of liquids and gases in porous media were already

derived by Leibenzon in the 1920s and 1930s [378], by Richard’s (1931) [489], and Muskat

(1937) [429]. More information on parabolic PDEs, systems, andvarious aspects of existence-

uniqueness and blow-up singularity theory can be found in [9, 148, 149, 164, 205, 206, 226,

245, 403, 509, 530, 533, 578].

§1.2.First systematic studies of invariant subspaces for nonlinear operators were performed

in [217, 220, 232], where some earlier references can be found. Theory and applications of

functional differential equations (Section 1.2.3) are available in many books; see e.g., [276,

351]. Concerning Lie-group symmetry analysis for delay-differential equations, see [549].

General aspects of the GSV for evolution PDEs (Section 1.2.4) were developed in [238]

(a similar term “nonlinear separation of variables” was suggested in [445]). Finite rings were

used in [555], where a full description of 2D commutative subrings for the product

u ∗ v = (uv)

associated with the PME operator in

= (u

)

was given. This reiterated the well-known solutions, including the dipole Barenblatt–Zel’dovich

√

x, x

}

(i.e., (1.141) for γ = 1) of the operator (u

)

For linear differential operators, the problem of characterizing those operators that leave a

given ﬁnite-dimensional subspace of polynomials invariant has been known as the general-

ized Bochner problem [67]. Some partial solutions have connections with Burnside’s theorem

on polynomial generator representations of endomorphisms of an irreducible module for Lie

algebra. Differential operators (i.e., Schr¨odinger operators) preserving a ﬁnite-dimensional

subspace W

of a given space of smooth functions are said to be partially (or quasi-exactly)

solvable. See basic ideas in Turbiner [563], [188], references in [312], and discussion in the

more recent paper [257]. In mathematical physics, there are several self-adjoint 1D operators

admitted countable sets of polynomial (monomial) eigenfunctions; e.g. the classic second-

order self-adjoint one

= D

−

in IR ,

with eigenfunctions being Hermite polynomials, or its higher-order non-self-adjoint analogies

= (−1)

m+1

−

in IR for any m = 2, 3, ...

for which polynomial eigenfunctions are known to be complete and closed in a weighted L

space [163] (in fact, any order, 2m → k ≥ 3, ﬁts, with more delicate topology for odd k).

In general, the problem on ﬁnite-dimensional invariant subspaces for such linear operators,

especially for operators in IR

, is known as intractable.

§ 1.3–1.5. Several results are given in [239]. Galilean invariant PDEs in Example 1.23 were

introduced in [213], symmetries of some nonlinear heat conduction models were studied in

[112]; see also references therein. (The mathematics of such fully nonlinear models, including

existence and nonuniqueness is not clear, so some conclusions there are not justiﬁed.) Exact

solution [28], see (3.146), on the invariant subspace

48 Exact Solutions and Invariant Subspaces

solutions of (1.64) on 3D polynomial and exponential subspaces were obtained in [323] by

detecting the so-called deﬁning equations and linear differential constraints. Exact solutions

on invariant subspaces for the quasilinear heat equation with source and convection,

= (D(u)u

)

+ P(u)u

+ Q(u),

were studied in [476]; see also [481, p. 145] for extensions. Proposition 1.24 was proved

in [553]. The exact solutions of some particular equations, such as (1.85) and (1.139), were

derived in [246] via nonclassical symmetries. It is well known that, for b = 0, equation (1.96)

admits a ﬁve-parameter Lie group of transformations and several explicit invariant solutions;

see [150] and [228]. Solutions in Example 1.36 were constructed in [220]. See [160, 159]

for some other applications of invariant subspaces to parabolic PDEs. Solutions in Example

1.37 were studied in detail in [346], where further information on the Berman problem can be

found. Solutions on subspaces of the mixed polynomial-trigonometric or exponential type (cf.

L{1, x, cos x, sin x}, or L{1, x, cosh x, sinh x}

or KP equations; see Remarks to Section 4.1.

Derivation of 1D non-local parabolic models in Example 1.38 (and partially in Example

1.40) from the Navier–Stokes equations in IR

can be found in [13, Ch. 7]. Solutions in Exam-

ple 1.38 were constructed in [475] (blow-up singularities were rigorously described in [244]).

Example 1.40 is given in [220].

Solutions (1.121) in Example 1.43 were detected in [110] by using a linear ordinary differ-

ential constraint with coefﬁcients, depending on t (no other new exact solutions of quadratic

parabolic PDEs were obtained there by this approach). For evolution PDEs (1.127) with gen-

eral quadratic operators (1.111), some solutions on exponential partially invariant subspaces

were obtained in [397]; see further extensions and references in Section 7.1.

Notice another classical quadratic Prandtl’s equation occurring in Prandtl’s boundary layer

theory (proposed in 1904, [470]; the ﬁrst exact similarity solution in IR

is due to Blasius

(1908) [63]) for incompressible ﬂuids,

yyy

+ ψ

− ψ

+uu

= 0, (1.145)

where ψ = ψ(x, y, t ) is the stream function and U (x, t ) is the given external far-ﬁeld (at

y =∞) velocity distribution; see further mathematical details in Oleinik–Samokhin [444].

This PDE has the same quadratic operator as that in the Gibbons–Tsarev equation (1.64).

Exact solutions of (1.145) are studied in [402], where further references can be found. Another

quadratic operator arises in the symmetric regularized long wave equation

+ u

+ uu

+ u

xxtt

= 0;

see exact solutions in [180] constructed by an interesting algebraic method using some linear

polynomial structures.

Some other solutions on invariant subspaces can be explained by using various differential

= (u

)

+ f (u),

appear in applications to gas dynamics and shallow water wave theory; see [18], where group

constraints and determining equations; see [322, 447, 592] and [13, Ch. 3]. More general

properties are studied, and Chapter 5 for more results.

quasilinear wave models (cf. (1.143)),

are typical for the positons or the negatons solutions of the integrable PDEs, such as the KdV

(1.101))

CHAPTER 2

Invariant Subspaces and Modules:

Mathematics in One Dimension

This chapter is devoted to mathematical theory of invariant subspaces for ordinary differen-

tial operators. We establish a general representation of operators preserving a given linear

subspace

(

the Main Theorem

)

, and prove other properties that are necessary for further ap-

plication to nonlinear PDEs in the subsequent chapters. In particular, we obtain the sharp

estimate of the dimension of linear subspaces invariant under a nonlinear operator of a given

differential order

(

the Theorem on maximal dimension

)

2.1 Main Theorem on invariant subspaces

There exist two main problems in the context of invariant subspaces:

Problem I, F → {W

}: given operator F, ﬁnd all invariant subspaces W

,and

Problem II, W

→ {F}: given invariant subspace W

, ﬁnd all operators F.

In this section, Problem II is studied. We assume that the subspace

= L{f

(x ), ..., f

(x )} (2.1)

is deﬁned as the space of solutions of a linear nth-order ODE,

L[y] ≡ y

(n)

+ a

(x )y

(n−1)

+ ... + a

n−1

(x )y



+ a

(x )y = 0, (2.2)

for which the functions {f

(x )} form a fundamental set of solutions, an FSS; see

e.g., [132, 434]. Let F

n−1

) (or simply F

n−1

) denote the whole set of differential

operators

F[y] = F(x, y, y



, ... , y

(k)

) (2.3)

of the order k not greater than n − 1, leaving the subspace (2.1), i.e., equation (2.2),

invariant.The function F(·) in (2.3) is assumed to be smooth enough.The invariance

condition of the subspace W

with respect to F takes the form

L[F[y]]



L[y]=0

≡ 0. (2.4)

We will use ﬁrst integrals of the equation (2.2) (i.e., expressions that are constant

on solutions). There exists the following set of ﬁrst integrals:

[y] = α

i,1

(x )y

(n−1)

+ ... + α

i,n−1

(x )y



+ α

i,n

(x )y for i = 1, ..., n, (2.5)

where the coefﬁcients α

i,1

(x ), ... , α

i,n

(x ) are deﬁned recursively by

i,1

= g

,α

i,2

= a

− (α

i,1

)



, ... , α

i,n

= a

n−1

− (α

i,n−1

)



(2.6)

via an arbitrary FSS {g

(x )} of the equation that is conjugated (adjoint) to (2.2).

50 Exact Solutions and Invariant Subspaces

We now present a general description of F

n−1

). This result is referred to as

the Main Theorem on invariant subspaces.

Theorem 2.1 (“Main Theorem”)

The set

n−1

)

consists of all operators of the

form

F[y] =



i=1

, ... , I

) f

(x ), (2.7)

where

, ... , I

)

for

i = 1, ..., n

are arbitrary smooth functions of the complete

set of ﬁrst integrals of equation

(2.2)

Proof. Consider the invariance condition (2.4),

F + a

(x )D

n−1

F + ... + a

n−1

(x )DF + a

(x )F = 0, (2.8)

where D is the operator of the full derivative by virtue of equation (2.2), i.e.,

D =



(2.2)

∂

∂x

+ y



∂

∂y

+ ... + y

(n−1)

∂

∂y

(n−2)

−



(n−1)

+ ... + a

n−1



+ a



∂

∂y

(n−1)

(2.9)

In the extended space of variables {x, y, y



, ... , y

(n−1)

}, we introduce the new vari-

ables {˜x, I

, ... , I

} by formulae

˜x = x and I

= I

(x , y, y



, ... , y

(n−1)

) for i = 1, ... , n.

Then operator D in (2.9) takes the form

D = ∂/∂ ˜x, and the invariance condition

(2.8) is transformed into

F + a

( ˜x)

n−1

F + ... + a

n−1

(x )

DF + a

( ˜x)F = 0. (2.10)

The general solution of (2.10) is given by the FSS {f

(x )} of (2.2),

F = A

, ... , I

) f

( ˜x) + ... + A

, ... , I

) f

( ˜x),

where A

, ... , I

) for i = 1, ..., n are arbitrary sufﬁciently smooth functions. In

the original variables, this is the equality (2.7).

Remark 2.2 Condition (2.4) is actually the invariance criterion for equation (2.2)

with respect to the Lie-B

acklund operator X = F[y]

∂

∂y

, and, therefore, all the results

can be interpreted in terms of symmetries of linear ODEs; see more details in [12,

297, 446] and [10, Ch. 5].

Remark 2.3 If the FSS {f

(x )} is known, the FSS {g

(x )} of the conjugate equation

can be obtained without integrating by the standard formulae

= (−1)

n+i

W [ f

, ... , f

i−1

, f

i+1

, ... , f

]

W [ f

, ... , f

]

for i = 1, ... , n, (2.11)

where

W [h

,...,h

] =



... h



... h



... ... ... ...

(l−1)

... h

(l−1)



denotes the Wronskian of the given functions {h

, ... , h

2 Invariant Subspaces: Mathematics in 1D 51

Remark 2.4 (Application to PDEs)Fortheﬁrst integrals (2.5) constructed by the

FSS (2.11), the following duality condition holds: for all i, j = 1, ..., n,

] = δ



1fori = j,

0fori = j,



is Kronecker’s delta



. (2.12)

Therefore, given the evolution PDE

= F[u]

with the operator F deﬁned by (2.7) via ﬁrst integrals {I

} satisfying (2.12), setting

u(x , t) =



i=1

(t) f

(x )

leads to the DS



= A

, ..., C

), i = 1, ..., n.

The right-hand sides of these ODEs are precisely the coefﬁcients in the operator

representation (2.7).

Theorem 2.1 is illustrated by a few examples, where we take g

= f

if the conju-

gate operator L

∗

coincides with L or −L.

Example 2.5 (2D subspace) Choose the 2D subspace W

= L{e

, e

−x

} with the

ODE



− y = 0.

This equation coincides with its conjugate, so we set g

= f

= e

, g

= f

= e

−x

By (2.7), all the operators F admitting W

are

F[y] = A

, I

+ A

, I

−x

, (2.13)

where ﬁrst integrals are I

= (y



− y)e

and I

= (y



+ y)e

−x

We can use this expression for describing operators satisfying some additional

conditions. For instance, we now ﬁnd all the operators that do not depend explicitly

on x (these are especially important in applications). To this end, we use the notation

∂ A

∂ I

, A

∂

∂ I

, ... .

The translation-invariant operators satisfy

∂ F

∂x

≡ 0, or (2.14)



− I

+ A





− I

− A



−x

≡ 0.

Equating the coefﬁcients of e

and e

−x

to zero, and solving the resulting system

yields A

= I

) and A

= I

), so that by (2.13),

F[y] = (y



+ y)B

(J ) + (y



− y)B

(J ),

where J = I

= (y



)

− y

and B

(J ) and B

(J ) are arbitrary functions. Some

particular cases are as follows:

[y] =



−y

, F

[y] =



, F

[y] = (y



− y)(y



+ y)

[y] = y







)

− y



, and F

[y] = y





)

− y



52 Exact Solutions and Invariant Subspaces

These operators yield a number of evolution PDEs with solutions on W

u(x , t) = C

(t)e

+ C

(t)e

−x

; (2.15)

for example, the following reaction-diffusion equations:

= u

, u

= u

+ u[(u

)

− u

], u

= [(u

)

− u

etc. (Similarly, hyperbolic PDEs with u

= ... can be treated.) Clearly, W

is invari-

ant under the linear operator u

. For the non-polynomial operator F[u] =

the ﬁrst equation, W

is invariant, since



+ C

−x



−x

∈ W

= 0).

Therefore, solutions (2.15) generate the DS C



= C

, C



= C

Example 2.6 (3D subspace) Consider W

= L{1, e

, e

−x

} deﬁned by the ODE



− y



= 0.

Up to the sign, this equation coincides with its conjugate, so that we take the follow-

ing functions g

= f

= 1, g

= f

= e

,andg

= f

= e

−x

. By Theorem 2.1, the

general representation of operators F preserving W

F[y] = A

, I

) + A

, I

+ A

, I

−x

, (2.16)

where A

, A

,andA

are arbitrary smooth functions of the ﬁrst integrals

= y



− y, I

= (y



− y



, and I

= (y



+ y



−x

As above, let us detect the translational invariant operators F satisfying (2.14). This

leads to a system for functions A

, A

,andA

− I

= 0, I

− I

+ A

= 0, I

− I

− A

= 0.

Solving it and using (2.16) yields

F[y] = B

, J

) + (y



+ y



, J

) + (y



− y



, J

where J

= I

= y



− y, J

= I

= (y



)

− (y



)

, and B

, J

), B

, J

and B

, J

) are arbitrary functions.

If, in addition to (2.14), we assume that F is of the ﬁrst order, i.e.,

∂ F

∂ y



≡ 0,

we obtain the 4D linear space of operators spanned by

[y] = (y



)

− y

, F

[y] = y



, F

[y] = y, and F

[y] = 1.

These operators make it possible to construct evolution PDEs with solutions on W

u(x , t) = C

(t) + C

(t)e

+ C

(t)e

−x

In particular, this is the case for the semilinear heat equation with absorption

= u

+ (u

)

− u