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

Подождите немного. Документ загружается.

1 Linear Invariant Subspaces: Examples 33

Using this subspace (1.100) gives

u(x , t) = C

(t) + C

(t) cosγ x +C

(t) sin γ x,

with the following system of an algebraic and two differential equations:







g =−γ



+ C





=−εγ

− γ C



=−εγ

+ γ C

Concerningpolynomialsubspaces,for the third-orderquadratic operator in (1.98),

the basic subspace is 4D, W

= L{1, x, x

, x

}. The 5D, W

= L{1, x, x

, x

is available for the modiﬁed operator

F[u] =−uu

xxx

: W

→ W

Example 1.37 (Partial invariance in time-dependent cases) Let us return to the

original RPJ equation (1.98). Clearly, the operator there preserves the subspace of

linear functions W

lin

= L{1, x }, as well as the trigonometric subspace (1.100) for

arbitrary γ ∈ IR . These two properties make it possible to construct solutions of

(1.98) of the following form:

u(x , t) = C

(t) + C

(t)x +C

(t) cos(γ (t)x ) + C

(t) sin(γ (t)x), (1.101)

where now γ(t) is a function to be determined. Such solutions admit simple invariant

motivations; see also Remarks on relations to positons of integrable PDEs. Here, the

basis functions depend on the independent variable t, so a proper setting includes

partially invariant modules to be treated in the next chapter, Section 2.8.

Now, using a natural scaling idea, we consider the whole differential operator

F[u] = u

xxt

− F[u] ≡ u

xxt

+ uu

xxx

− u

, (1.102)

which is deﬁned in the space of smooth functions u = u(x , t) of both independent

variables. First, we get rid of the γ(t)-dependence in the spatial variable in (1.101)

by introducing the new independentvariable

ˆx = γ(t)x.

Then (1.102) reads

F[u] = γγ



ˆxu

ˆx ˆx ˆx

+ 2γγ



ˆx ˆx

+ γ

ˆx ˆxt

− γ

ˆx

ˆx ˆx

− uu

ˆx ˆx ˆx

while the exact solutions (1.101) take the form

u( ˆx, t) = C

ˆx +C

cos ˆx + C

sin ˆx. (1.103)

Second, for convenience,let us introduce the new function v by

u = v +

ˆx ,

so that

F[u] →

F[v] = γ





+ γ C



ˆx v

ˆx ˆx ˆx

+ γ



2γ



− γ C



ˆx ˆx

+γ



ˆx ˆxt

− γ(v

ˆx

ˆx ˆx

− vv

ˆx ˆx ˆx

)



34 Exact Solutions and Invariant Subspaces

F[v] preserves the standard trigonometric subspace

= L{1, cos ˆx, sin ˆx }, if γ



+ γ C

= 0, (1.104)

and the following action of the operator is obtained:

F[v] = 3γγ



ˆx ˆx

+ γ

ˆx ˆxt

− γ

ˆx

ˆx ˆx

− vv

ˆx ˆx ˆx

). (1.105)

Actually,

is invariant under all three operators on the right-hand side of (1.105).

This is seen from the identity in the original independent variables,

F[u] =



−







− C

+ C



cosγ x



−







+ C



sinγ x

+γ





+ γ C



x sinγ x −C

x cosγ x



(1.106)

where the last bad term not belonging to the subspace vanishes by (1.104).

Using the expansion(1.106), we obtain for solutions(1.101) of (1.98) three ODEs

for ﬁve coefﬁcients









=−εγ

+ 3C

− γ C



=−εγ

+ 3C

+ γ C



=−C

γ.

(1.107)

This undetermined DS allows us to control the pressure and pose suitable boundary

conditions. For instance, the homogeneous Dirichlet conditions,

u(±1, t) = 0fort > 0,

imply by (1.101) that

=−C

cosγ and C

=−C

sinγ.

Plunging these into (1.107) yields a well-posed DS for three coefﬁcients,









=−εγ

+ (γ cosγ − 3sinγ)C



=−εγ

− 3C

sinγ − γ C

cosγ,



= γ sinγ C

Similar computations can be performed for hyperbolic functions cosh(γ (t)x) and

sinh(γ (t)x ) in (1.101); see further examples below.

1.4.2 Non-local models from Navier–Stokes equations in IR

Example 1.38 (Nonstationary von K

arm

an solutions)Consider the Navier–Stokes

equations in IR











+ uu

+ vu

=−

+ ν(u

+ u

+ uv

+ vv

=−

+ ν(v

+ v

= 0,

(1.108)

where (u,v) = u is the velocity ﬁeld, p is the pressure, ρ>0 is the constantdensity,

and ν>0 is the constant kinematic viscosity. We are looking for solutions

u =



f (z, t) dz,v=−yf(x , t), and p = h(x, t). (1.109)

1 Linear Invariant Subspaces: Examples 35

The structure of such solutions corresponds to a nonstationary plane version of the

von K

arm

an solutions [327], which have been applied to various problems of ﬂuid

Plugging (1.109) into (1.108) yields a semilinear non-local parabolic equation for

the function f (x, t), replaced now by u = u(x , t):

= u

− u

+ u



u(z, t) dz. (1.110)

Consider for (1.110) a free-boundary problem on (0, s(t)) × IR

with the symme-

try condition at the origin, u

(0, t) = 0, and the following conditions at the free

boundary x = s(t):

(s(t), t) = 0, s



=−



u(z, t) dz.

the same approach, after time-dependent scaling, as shown in (1.104), we obtain

solutions of the type (1.101), with γ(t) =

s(t)

u(x , t) = C

(t) + C

(t) cos



π x

s(t)











=−C

− C



=−2C

− (

)



=−sC

Example 1.39 (General solutions on W

) In connection with the previous exam-

ple, let us perform more general invariant analysis of the Navier–Stokes equations

(1.108). The spatial structure of von K´arm´an solutions (1.109) expressesthe fact that

(1.108) can posses solutions on the 2D subspace

u,v, p ∈ W

= L{1, y},

which is not invariant under all the nonlinear operators. So, we set



u = C

(x , t) + C

(x , t)y,

v = D

(x , t) + D

(x , t)y,

p = E

(x , t) + E

(x , t)y.

Substituting into (1.108) yields the following system of eight equations (projections

of the three PDEs onto the vectors 1, y,andy

respectively) for six coefﬁcients:











+ C

+ D

=−

+ νC

1xx

+ (C

)

+ C

=−

+ νC

2xx

= 0,

+ C

+ D

=−

+ ν D

1xx

+ C

+ D

= ν D

2xx

= 0,

+ D

= 0,

= 0.

The shortest equations imply that

either C

= 0, or C

= 0andD

= 0.

dynamics; see [42] and references in [13, Ch. 7].

For derivation, references, and further mathematical results, see [245, Ch. 9]. Using

36 Exact Solutions and Invariant Subspaces

Considerfor instance the ﬁrst case. The second equation then yields that E

= E

(t)

is arbitrary. Next, plugging

=−



from the seventh equation into the previous one (with C

= 0), we obtain the in-

dependent governing equation of the type (1.110) for the coefﬁcient D

. Solving a

suitable FBP for this parabolic equation yields D

(x , t), and then the fourth equation

becomes a linear parabolic PDE for D

with an arbitrary function E

(t). Finally, the

ﬁrst equation can be treated as an ODE for E

(x , t) with a parameter t.

Example 1.40 (Polynomial subspace) Thefollowingnon-localsemilinear heat equa-

tion on (0, 1) × IR

is also associated with the Navier–Stokes equations in IR

= u

+ β(u

)

+ u

− µu



u(z, t) dz − (1 +µ)



(z, t) dz.

For µ =

, it admits solutions on the subspace of quadratic polynomials,

u(x , t) = C

(t) + C

(t)x +C

(t)x













=−

−







β −



−

+ 2C



=−

+ 4βC



=−C

+ 4βC

1.4.3 Criterion of invariance for polynomial operators

Consider a general quadratic operator of the form

F[u] =



(i, j )

i, j

u, (1.111)

where a =#a

i, j

# (i, j = 0, ..., k) is a given constant real symmetric (k+1)×(k+1)

matrix. By P, we denote the corresponding polynomial,

P(X, Y ) =



i, j

, so F[u] = P(D

, D

)u(x , t)u(y, t)



y=x

. (1.112)

Consider the function

u =



(µ∈N )



cos(µx ) + D

sin(µx)



, (1.113)

where N is a subset of nonnegative integers. If N ={0, 1, ..., n},thenu belongs to

thestandardtrigonometricsubspace W

2n+1

= L{1, cos x , sin x, ..., cos(nx), sin(nx)}.

Differentiatingin each termand performingsimple manipulations,we ﬁnd the action

F[u] =



(µ,ν)





(i, j )

i, j



cos(µx +

i) + D

sin(µx +





cos(νx +

j) + D

sin(νx +







(µ,ν)





(i, j )

i, j





cos((µ + ν)x +

(i + j)) + cos((µ − ν)x +

(i − j ))



(µ

+ ν

)



sin((µ + ν)x +

(i + j)) − sin((µ − ν)x +

(i − j ))



+ D



cos((µ − ν)x +

(i − j )) − cos((µ + ν)x +

(i + j ))





1 Linear Invariant Subspaces: Examples 37

In the second sum with C

, we obtained two similar terms with C

and C

and in the latter, we changed the summation (µ, ν) → (ν, µ) to get a single term.

The ﬁrst term with C

has the coefﬁcient



(i+j iseven)

i, j



(−1)

i+j

cos((µ + ν)x) + (−1)

i−j

cos((µ − ν)x)





(i+j isodd)

i, j



(−1)

i+j +1

sin((µ + ν)x) + (−1)

i−j +1

cos((µ − ν)x)



Since all the products C

, C

,andD

(latter two generate similar sums)

are linearly independent, taking into account cos and sin functions of (µ + ν)x,we

obtain ﬁrst two groups of invariance conditions:



(i+j iseven)

i, j

(−1)

i+j

= 0,



(i+j isodd)

i, j

(−1)

i+j +1

= 0

(1.114)

for any µ, ν ∈ N with µ ≤ ν, such that µ + ν ∈ N .

Similarly, the terms with cos and sin of the argument (µ − ν)x yield two more

groups of invariance conditions (1.114) in which, everywhere in both sums, i + j

should be replaced by i − j. These apply for ν − µ ∈ N , µ ≤ ν. For the standard

subspace W

2n+1

without “gaps”, these two groups of conditions do not appear.

For operators (1.86) with ﬁxed i + j = k and the subspace W

2n+1

, the invariance

conditions (1.114) are simpliﬁed and involve the polynomial (1.112),

P(µ, ν) = 0forallµ ≤ ν such that µ + ν>n. (1.115)

We will study such linear systems on the coefﬁcients {a

i, j

} in Section 1.5.2 devoted

to exponential subspaces.

Similar invariance analysis can be performed for the cubic operators (1.90) with

analogous trigonometric manipulations and conclusions.

1.5 Examples: exponential subspaces

1.5.1 Classiﬁcation and examples

We begin with the 3D subspace

= L{1, e

, e

−x

}. (1.116)

Proposition 1.41

Subspace

(1.116)

is invariant under operator

(1.56)

iff



+ b

= 0,

− b

+ b

− b

+ b

= 0.

The set of such operators is a 4D linear space spanned by

[u] = u(u

− u), F

[u] = u

− u),

[u] = u(u

− u), F

[u] = (u

)

− u

Example 1.42 (Blow-up for fast diffusion equation with source) The quasilinear

fast diffusion equation with a quadratic reaction term





+ v

in IR × IR

(1.117)

admits blow-up solutions for sufﬁciently large initial functions v

(x )>0. By the

38 Exact Solutions and Invariant Subspaces

pressure transformation u =

, (1.117) reduces to the quadratic PDE

= F

rem

[u] − 1 ≡ uu

− (u

)

− 1. (1.118)

By Proposition 1.41, the remarkable operator F

rem

in (1.118) (note that F

rem

equals

to F

− F

) preserves subspace (1.116) or, which is the same, the subspace of hyper-

bolic functions

= L{1, cosh x, sinh x}. (1.119)

Studyingthe evolutionof the symmetric in x functionsyields the solutionsof (1.117)

u(x , t) = C

(t) + C

(t) cosh x ∈ W

= L{1, cosh x},





= C

− 1,



= C

(1.120)

The DS admits the ﬁrst integral

− C

− 2ln|C

|=constant,

and can be integrated in quadratures. The exact blow-up asymptotics of the DS

(1.120) describe some interesting features of localized, single point blow-up patterns

for the original PDE (1.117).

It is curious that “trigonometric” Proposition 1.29 also yields solutions

u(x , t) = C

(t) + C

(t) cos x ∈ W

= L{1, cos x },

where the DS is slightly different,





=−C

− 1,



=−C

Such solutions describe distinct features of blow-up of 2π-periodic solutions.

We now introduce the exponential analog of Example 1.37.

Example 1.43 (Partially invariant module). Consider the

∂

∂t

-dependent operator

F[u] = u

− F

rem

[u] ≡ u

−



− (u

)



for u = u(x, t) given by

u(x , t) = C

(t) + C

(t)x +C

(t)e

γ(t )x

. (1.121)

This case of “time-dependent” subspaces admits a natural invariant treatment after

scaling ˆx = γ(t)x, as shownin Example 1.37. We omit furtherrelated computations.

In the original variables, the expansion is

F[u] = C



+ C



x +





− C

+ 2C



γ x





− C



γ x

Forinstance, forthe semilinear Kuramoto–Sivashinsky-typeequation (see Section 3.8

for details)

=−u

xxxx

+ uu

− (u

)

+ αu + β,

1 Linear Invariant Subspaces: Examples 39

looking for solutions in the form (1.121) leads to the following DS:













=−C

+ αC

+ β,



= αC



= γ C

(γ C

− 2C

) + (α − γ



= C

The second ODE gives C

(t) = e

αt

if α = 0, and C

(t) = 1ifα = 0. Substituting

(t) into the last equation, we obtain two functions γ(t) for which such solutions

exist,

γ(t) =



α(e

αT

− e

αt

)

−1

for α = 0,

T −t

for α = 0,

where T is a ﬁxed blow-up time of the solution.

Omitting straightforward analysis of cubic operators (1.57) admitting (1.116),

consider the 5D exponential subspace

= L{1, e

, e

−x

, e

−2x

}. (1.122)

Proposition 1.44

Subspace

(1.122)

is invariant under

(1.56)

iff











16b

+ 8b

+ 4b

+ 2b

+ b

= 0,

16b

− 8b

+ 4b

− 2b

+ b

= 0,

+ 6b

+ 5b

+ 4b

+ 3b

+ 2b

= 0,

− 6b

+ 5b

− 4b

− 3b

+ 2b

= 0.

The set of such operators is a 2D linear space spanned by

[u] = uu

−

)

− u

and

[u] = (u

− 4u)

Proposition 1.45

No nontrivial cubic operators

(1.57)

admit the subspace

(1.122)

The proof is the same as that of Proposition 1.32. A full classiﬁcation of 5D in-

variant subspaces for quadratic second-order operators will be the main subject of

Theorem 2.22.

Example 1.46 In a similar manner, the operator of the GT equation (1.64) is associ-

ated with the hyperbolicsubspace (1.119). The extension to subspace (1.122) for the

operator family (1.94) is performed as in Proposition 1.34.

1.5.2 General polynomial operators

We now deal with the general quadratic kth-order operators (1.111) with the polyno-

mial (1.112).

Set p

= 0andﬁxothern − 1 real exponents, introducing the set

 ={p

, i = 1, ..., n}, where p

= p

for i = j.

Consider the n-dimensional linear subspace

= L{e

, i = 1, ..., n}, and denote





={p

+ p

: p

, p

∈ , p

+ p

∈ }.

(1.123)

40 Exact Solutions and Invariant Subspaces

Proposition 1.47

Subspace

(1.123)

is invariant under the operator

(1.111)

iff

P( p

, p

) = 0

for all pairs

( p

, p

)

such that

+ p

∈ 



. (1.124)

Proof. Taking arbitrary

u =



(i)

∈ W

(1.125)

and plugging into (1.111) yields

F[u] =



(i, j )



(m,l)

i, j

≡



(m,l)

P( p

, p

(1.126)

Since {C

} are independent, it follows from (1.126) that the coefﬁcients of e

must vanish for any p

+ p

∈ 



Dealing with invariant subspaces, in this proof, we do not take into account a pos-

sible correlation of similar terms in (1.126) that can give rise to cancellation of some

of the terms. Actually, for arbitrary values of expansion coefﬁcients {C

} involved,

this is not possible. Assuming existence of some algebraic relations between expan-

sion coefﬁcients {C

}, meaning partial invariance of W

, this topic will be under

scrutiny in Section 7.1. Evidently, the linear system (1.124) on coefﬁcients {a

i, j

}

means existence of an inﬁnite number of quadratic (non-integrable)PDEs with solu-

tions on exponential subspaces W

of arbitrary ﬁnite dimension n (e.g., soliton-type

solutions).

Corollary 1.48 Given an arbitrarily large integer l > 1, there exists a polynomial

operator (1.111) of the order k large enough, admitting at least l exponential invari-

ant subspaces W

, (1.123), of any dimension n = 1, 2, ..., l.

The corresponding ﬁrst-order evolution PDEs

= F[u] ≡



(i, j )

i, j

u on W

(1.127)

are n-dimensional DSs. The second-order PDEs, u

= F[u]onW

,are2nth-order

DSs. Such classes of PDEs can be treated as intermediate relative to known inte-

grable equations admitting inﬁnitely many (partially) invariant subspaces.

Let us return to systems (1.124). The general operator (1.111) contains

(k+1)(k+2)

free coefﬁcients {a

i, j

}. So the system (1.124) cannot have more than

(k+1)(k+2)

− 1

linearly independent equations to generate such nontrivial operators. The total num-

berofelementsin



(its cardinal number) satisﬁes !



≤

(k+1)(k+2)

− 1. In an

example below, we illustrate solvability of such systems in a particular setting.

Example 1.49 (Operators with monomials of the same total order) Consider op-

erators (1.86) that contain





+ 1 unknown coefﬁcients {a

i, j

}. Without loss of gen-

erality, we assume that k is even and !



= s =

. By ( p

, p

), ..., ( p

, p

we denote the pairs of exponents such that p

+ p

∈ 



for all j = 1, ..., s.The

system (1.124) is then composed of precisely s equations,

0,k



+ p



+ a

1,k−1



k−1

+ p

k−1



+ ...

= 0forj = 1, ..., s.

(1.128)

1 Linear Invariant Subspaces: Examples 41

(If k is odd, the last term is replaced by a

k−1

k+1

k−1

k+1

, and the analysis is sim-

ilar.) Dividing each equation in (1.128) by p

and denoting r

yields the

equivalent system

0,k



1 + r



+ a

1,k−1



k−1



+ ...

−1,



−1



+ 2a

= 0.

(1.129)

For mutually distinct ratios {r

}, we then obtain a unique operator (1.86), up to a

scalar factor. It is not very difﬁcult to solve the system (1.129),

l,k−l

= (−1)



, ..., r

, ...,



for l = 0, ...,

− 1;

(−1)

(·) for l = 0.

(1.130)

Here, σ

(·) are the elementary symmetric function of 2s = k arguments,

, ..., y

) =



(1≤i

<...<i

≤2s)

...y

(σ

(·) = 1).

For instance, for k = 4, the unique thin ﬁlm operator (1.86) has the form

xxxx

u −





xxx



+ 2 +



)

For the space W

= L{1, e

, e

},i.e.,p

= 0, p

= 1, and p

= 2, we have

two pares of exponents, (1, 2) and (2, 2), generating 



={3, 4}. Hence, we obtain

= 2andr

= 1, and the operator

F[u] = u

xxxx

u −

xxx

)

The corresponding parabolic and hyperbolic PDEs,

= F[u]andu

= F[u],

haveﬁnite-dimensionaldynamicson W

. An ODE reductionexists for the ﬁfth-order

nonlinear dispersion PDE u

= (F[u])

. We will study such models in Section 4.2.

Example 1.50 (TFEs) Consider a fourth-order equation from thin ﬁlm theory (see

more on such degenerate parabolic models in Section 3.1),

= F[u] ≡−uu

xxxx

+ αu

xxx

+ β(u

)

+ γ uu

. (1.131)

Let  ={0, 1, 2, 3}, so the PDE is restricted to the subspace

= L{1, e

, e

}, (1.132)

and let us look for a solution in the form of expansion

u(x , t) = C

(t) + C

(t)e

+ C

(t)e

+ C

(t)e

. (1.133)

Since 1 + 2 = 3, the structure of this solution corresponds to the 2-solitons. The

polynomial of operator F in (1.131) is

P =−



+ Y





+ X



+ β X



+ Y



42 Exact Solutions and Invariant Subspaces





={4, 5, 6}, so, by Proposition 1.47, subspace (1.132) is invariant under F iff

P(1, 3) = P(2, 3) = P(3, 3) = 0.

This yields a linear system for parameters,



15α + 9β + 5γ = 41,

78α + 72β + 13γ = 97,

9α +9β + γ = 9.

This has a unique solution, so that there exists a single thin ﬁlm operator

F[u] =−uu

xxxx

+ 10 u

xxx

−

)

− 7uu

preserving (1.132). The PDE (1.131) on W

then reduces to the fourth-order DS













= 0,



= 2P(0, 1)C



= 2P(1, 1)C

+ 2P(0, 2)C



= 2P(0, 3)C

+ 2P(1, 2)C

that can be solved explicitly. We may take C

= 1 from the ﬁrst ODE, and, from

the second, C

(t) = A

2P(0,1)t

, with the constant A

= C

(0). The rest of the

ODEs can be explicitly solved step by step. The resulting exact solution (1.133)

is not always composed of elementary solitonic exponential terms, such as e

px+ωt

For instance, linear (and quadratic) polynomials in t appear in the resonance case

P(0, 2) = 2P(0, 1). For solutions not having the 2-soliton structure, e.g.,

u(x , t) = C

(t) + C

(t)e

+ C

(t)e

+ C

(t)e

where 



={3, 5, 6, 8}, we need four conditions of invariance of the corresponding

subspace

P(1, 2) = P(1, 4) = P(2, 4) = P(4, 4) = 0.

The maximal number of such invariance conditions of general exponential spaces

is six for the case where no correlation between products of exponential terms in

(1.126) is available.

Example 1.51 (5D subspaces) Consider a general second-order PDE

= F[u] =



(i, j )

i, j

on W

u(x , t) = C

(t) + C

(t)e

+ C

(t)e

−x

+ C

(t)e

+ C

(t)e

Then  =



−1, 0,

, 1, 2



and 





−2, −

, 3, 4



, with seven conditions

P(−1, −1) = P(−1,

) = P(

) = P(1,

) = P(

, 2) = P(1, 2) = P(2, 2) =

0. The DS is













= P(0, 0)C

+ 2P(1, −1)C



= 2P(0, 1)C

+ 2P(−1, 2)C



= 2P(0, −1)C



= 2P(0,



= 2P(0, 2)C

+ P(1, 1)C

with the expansion (cf. the canonical 5D subspace (1.122))