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

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3 Thin Film, Kuramoto-Sivashinsky, and Magma Models 143

0 20 40 60 80 100 120

−2

−1.5

−1

−0.5

0.5

1.5

φ(s)

Figure 3.7 Formation of a heteroclinic connection ϕ

→ ϕ

−

for (3.166) as n → n

−

0 2 4 6 8 10

−0.01

−0.008

−0.006

−0.004

−0.002

0.002

0.004

0.006

0.008

0.01

φ(s)

(a) stabilization to ϕ

0 1 2 3 4 5

−4

−3

−2

−1

x 10

−5

φ(s)

(b) periodic behavior, enlarged

Figure 3.8 Solutions of (3.155) for m = 3andn = 1: stability of constant equilib-

ria ϕ

(a) and the unstable periodic behavior in between (b), ϕ(0) = 0.04, ϕ



(0) =

−0.0176896526588... .

ﬁrst critical exponent n

until the corresponding heteroclinic bifurcation at

= 1.215053... (m = 4); (3.167)

see Figure 3.10. For n > n

, the behavior becomes more and more unstable. Smaller

values of n are discussed below.

For m = 5, the unstable periodic behavior is shown in Figure 3.11. Cauchy data

for the periodic motion are: ϕ(0) = 10

−7

, ϕ



(0) = 4.838491256 × 10

−7

for n = 1

in Figure (a). In Figure (b), we have ϕ



(0) = 1.942533642 × 10

−6

for n =

.The

rest of the derivatives in both Figures (a) and (b) are equal to zero, ϕ



(0) = ϕ



(0) =

... = ϕ

(8)

(0) = 0.

According to these results checked numerically for several even and odd m,letus

state the following:

144 Exact Solutions and Invariant Subspaces

0 5 10 15 20

−8

−6

−4

−2

x 10

−5

φ(s)

(a) n = 1

0 10 20 30 40 50

−6

−4

−2

x 10

−3

φ(s)

(b) n = n

= 1.1666...

Figure 3.9 Stable periodic behavior in the ODE (3.155), m = 4, for n = 1 (a) and n =

(b).

0 50 100 150

−0.03

−0.02

−0.01

0.01

0.02

0.03

φ(s)

(a) n = 1.215

0 50 100 150

−0.03

−0.02

−0.01

0.01

0.02

0.03

φ(s)

(b) n → n

−

= 1.2150534...

Figure 3.10 (a) Periodic structures of (3.155), m = 4, for n close to n

−

in (3.167); (b) de-

scribes formation of a heteroclinic connection (the wave moves to the right as n increases).

0 2 4 6 8 10 12 14 16 18

−3

−2

−1

x 10

−7

φ(s)

(a) n = 1

0 2 4 6 8 10 12 14 16 18

−1

−0.8

−0.6

−0.4

−0.2

0.2

0.4

0.6

0.8

x 10

−5

φ(s)

(b) n = n

Figure 3.11 Unstable periodic behavior in (3.155), m = 5, for n = 1 (a) and n =

(b).

3 Thin Film, Kuramoto-Sivashinsky, and Magma Models 145

Conjecture 3.1 (“On periodic solutions”) For any n ∈ (0, 1], the ODE (3.155)

admits a stable periodic solution for even m = 2, 4, ... and an unstable one for for

odd m = 3, 5, ... .

Thus, for even m, stable periodic solutions, existence of which for small n > 0

can be associated with continuity in n, persist for all n ∈ (0, 1]. Moreover, based on

numerical evidence, we also claim that they still exist for

n ∈ (1, n

), with some n

∈



2m−1

2m−2

2m−1

2m−3



= (n

, n

where n = n

is a heteroclinic bifurcation point for the corresponding (2m-1)th-

order ODE, as explained above. Such a non-oscillatory character of interfaces for

n > n

suggests that, somewhere in this parameter range, sign-preserving proper-

ties of special classes of solutions may exist leading to nonnegative solutions of the

Cauchy problem for sufﬁciently large n. This needs extra investigation and is an

OPEN PROBLEM.

Extrascalingasn → 0. The above ﬁgures show that the oscillatory behavior is not

directly detectable for small n > 0 and is numericallyinvisibleif n <

. To revealing

the limit oscillatory behavior as n → 0, where solutions of changing sign are of the

order

max|ϕ(s)|∼A(n) ≡



2m−1



2m−1

for small n > 0,

an extra rescaling in equation (3.155) is necessary,

ϕ(s) = Aψ(η), η =

, where a(n) = A

2m−1

. (3.168)

Then, for small n > 0, function ψ(η) solves the ODE with Euler’s differential oper-

ator, which can be written in the form of

−η

(ψe

)

(2m−1)

− c

|ψ|

= 0 (3.169)

(as usual, all higher-order perturbations have been omitted). This is an easier ODE

than the original one (3.155), and contains the binomial coefﬁcients. It turns out that

(3.169) possesses similar periodic orbits.

Numerical analysis gives solid evidence of the existence of stable periodic motion

for m = 2 in Figure 3.12, and for m = 4 in Figure 3.13, where the ODEs have the

form

m = 2: ψ



+ 3ψ



+ 3ψ



+ ψ +

|ψ|

= 0, and

(7)

+ 7ψ

(6)

+ 21ψ

(5)

+ 35ψ

(4)

+ 35ψ



+ 21ψ



+ 7ψ



+ ψ +

|ψ|

= 0

for m = 4(wetakec

=−1). Already for n = 0.1 in Figure 3.13(a), oscillations are

extremely small, since, by (3.168),

max|ϕ|∼10

−130

(n = 0.1, m = 4).

For n = 0.03 in (b), max|ϕ|∼10

−460

For m = 3, the corresponding rescaled ODE

m = 3: ψ

(5)

+ 5ψ

(4)

+ 10ψ



+ 10ψ



+ 5ψ



+ ψ −

|ψ|

= 0 (3.170)

admits an unstable periodic solution for small n > 0; see Figure 3.14(a) for n =

146 Exact Solutions and Invariant Subspaces

0 5 10 15 20 25 30

−0.02

−0.015

−0.01

−0.005

0.005

0.01

0.015

0.02

ψ(η)

(a) n =

0 10 20 30 40 50 60 70 80

−3

−2

−1

x 10

−6

ψ(η)

(b) n = 0.16

Figure 3.12 Stable periodic behavior of (3.169), m = 2, for n =

(a) and n = 0.16 (b).

0 50 100 150 200

−1

−0.8

−0.6

−0.4

−0.2

0.2

0.4

0.6

0.8

x 10

−3

ψ(η)

(a) n = 0.1

0 50 100 150 200 250 300 350 400

−2

−1

x 10

−11

ψ(η)

(b) n = 0.03

Figure 3.13 Oscillatory solutions of (3.169), m = 4, for n = 0.1 (a) and n = 0.03 (b).

(the Cauchy data to reveal the periodic motion are ϕ(0) = 2 × 10

−5

, ϕ



(0) =

−9.9850972244 × 10

−5

, other derivatives are equal to zero). Figure 3.14(b) shows

the case n = 0.2, where ϕ(0) = 5 × 10

−10

, ϕ



(0) =−1.4296767 × 10

−9

,andthe

periodic behavior is still not clearly seen. More details on such solutions and passage

to the limit n → 0

can be found in [175].

0 2 4 6 8 10 12

−3

−2

−1

x 10

−5

ψ(η)

(a) n =

0 2 4 6 8 10

−3

−2

−1

x 10

−10

ψ(η)

(b) n = 0.2

Figure 3.14 Unstable periodic behavior for (3.170) for n =

(a) and n = 0.2(b).

3 Thin Film, Kuramoto-Sivashinsky, and Magma Models 147

0 1 2 3 4 5 6 7 8

0.2

0.4

0.6

0.8

g(ξ)

FBP: g

CP: n=0

CP: n=1

FBP: g

Figure 3.15 Similarity proﬁles g(ξ) satisfying (3.151), m = 2, for n = 1andn = 0. The

dashed lines denote the ﬁrst FBP proﬁles, g

(ζ ) and g

(ζ ).

On existence of the fundamental similarity solution: dimension of linear bun-

dles. For m = 2, existence and uniqueness of the fundamental proﬁle g satisfying

(3.151) are proved for n ∈ (0, 1),[174,Sect.9].Form ≥ 3, such results are un-

known. For using a shooting strategy from the interface point ξ = ξ

in (3.154)

(where, e.g., ξ

= 1isﬁxed by scaling invariance of the ODE) to the origin to match

m − 1 symmetry conditions, one needs to have an (m-1)-parametric stability bundle

about the periodic solution ϕ(s). According to (3.165) for n = 0, by continuity in

n, this bundle is expected to be precisely (m-1)-dimensionalfor small n > 0. Recall

that ξ

= 1, i.e., one parameter is ﬁxed. This shows existence of g(ξ),inviewof

extremely good matching topology of the oscillatory bundle. Uniqueness is

OPEN.

For n ∈ (0, n

), we need to check the spectrum (the number of eigenvalues with

positive real parts) of the singular, non self-adjoint linearized operator with periodic

coefﬁcients in equation (3.155),

2m−1

[ϕ] −(1 − n)c

|ϕ|

−n

This is an

OPEN PROBLEM.

In Figure 3.15, we present similarity proﬁles of the CP given by the ODE (3.151),

m = 2, for n = 1 and, for comparison, the fundamental rescaled kernel for n = 0.

CorrespondingCauchy data are g(0) = 1andg



(0) =−0.3696375... for n = 1, and



(0) =−0.337989123... for n = 0. Here, g

1,2

(ζ ) are the ﬁrst similarity proﬁles of

the zero contact angle, zero-ﬂux FBP.

On negative exponents n. By continuity in n → 0, it can be expected that oscilla-

tory behavior of solutions of the Cauchy problem for TFEs persists for n < 0. The

representation (3.159) suits this case with the interface at y =+∞, where the oscil-

latory component ϕ solves the same ODE (3.155). Unlike in the case where n > 0,

for n < 0, the changing sign character of such solutions is difﬁcult to detect, since

the ODEs involved exhibit extra blow-up singularities in ﬁnite y or s.Thisiseasily

148 Exact Solutions and Invariant Subspaces

seen from (3.158), written in the form of

(2m−1)

= c

| f |

|n|



= λ(−1)



where the right-hand side is now superlinear for f  1, and, hence, there exists

monotone or oscillatory (always if c

< 0) blow-up in ﬁnite y. Moreover, typi-

cally an oscillatory behavior is not available, so that solutions of changing sign are

constructed by asymptotic expansion as n → 0

−

and extension in n;suchWKBJ

asymptotics are explained in [174, Sect. 7.6] for the TFEs with n → 0

3.8 Invariant subspaces in Kuramoto-Sivashinsky type models

The Kuramoto–Sivashinsky (KS) equation

=−u

xxxx

− u

+ (u

)

(3.171)

is a semilinear fourth-order parabolic PDE with a quadratic Hamilton–Jacobi term

)

. It was originally introduced as a model that describes ﬂame front propaga-

tion in turbulent ﬂows of gaseous combustible mixtures [526], and later it found

other applications in many areas of physics, including 2D turbulence; see Remarks.

The modiﬁed Kuramoto–Sivashinsky (mKS) equation contains an extra second-order

quadratic operator,

= F[u] ≡−u

xxxx

− u

+ (1 − λ)(u

)

+ λ(u

)

, (3.172)

where λ ∈ [0, 1] is a constant (λ = 0 leads to (3.171)), and describes dynamical

properties of hyper-cooled melt.

Example 3.33 (Instability) Let us begin with simple instability phenomena for the

mKS equation, which occur on the invariant subspace of periodic functions.

Proposition 3.34

For any

λ ∈ (0, 1)

, the quadratic operator in

(3.172)

admits sub-

space

= L{1, cosγ x, sinγ x}

,where

γ =



1−λ

Consider solutions in the subspace of cos-functions,

u(x , t) = C

(t) + C

(t) cosγ x,





(1 − λ)



(1 − λ)(2λ − 1)C

It follows from the second ODE that λ>

leads to the exponentialinstability, while

for λ =

it follows that C

(t) = O(t) for t  1. On the other hand, for λ<

(t) is exponentially small as t →∞,andC

(t) is bounded, thus describing the

stability of the origin.

Let us introduce two models, exhibiting ﬁnite-time singularities.

Example 3.35 (Blow-up) Consider the KS equation with source

=−u

xxxx

+ (u

)

+ u

, (3.173)

3 Thin Film, Kuramoto-Sivashinsky, and Magma Models 149

where the reaction term u

is added, and the linear second-order term −u

(not im-

portant for blow-up solutions) is ignored. Invariant subspaces of ﬁrst-order quadratic

operators, such as (u

)

+ u

in (3.173), have been described in Section 1.4. Take

= L{1, cos x, sin x} on which the solutions u = C

+ C

cosx + C

sin x are

driven by the DS





= C

+ C



= 2C

− C



= 2C

− C

(3.174)

This quadratic system is similar to those studied for the TFEs in Example 3.17.

Though the last two ODEs contain extra linear terms, the proof of Proposition 3.18

remains unchanged. This gives a typical stability result on the subspace W

,where

all the blow-up solutions converge to the similarity pattern belonging to the 1D sub-

spaces W

. It is curious that the whole linear operator −u

xxxx

− u

from (3.171)

vanishes identically on W

, the resulting DS does not contain any linear terms on

the right-hand side, simplifying stability analysis. The precise asymptotic behavior

as t → T for the DS (3.174) gives important properties of blow-up of 2π-periodic

solutions of (3.173); see references in the Remarks.

Example 3.36 (Extinction) In order to exhibit extinction phenomena, let us choose

the KS equation with absorption

=−u

xxxx

+ (u

)

− 1. (3.175)

We restrict our attention to a lower-dimensional subspace W

= L{1, x

, x

} (which

is extended to W

by adding extra basic functions x and x

), and consider the evolu-

tion of the solutions u(x, t) = C

(t) + C

(t)x

+ C

(t)x

with the DS









= 4C

− 24C

− 1,



= 48C



= 144C

This system can be integrated and studied in a manner similar to that considered

in Section 3.2. The positive solutions satisfy the corresponding FBP which can be

studied by the von Mises transformation (see Example 3.10). The related singularity

formation phenomena for the uniformly parabolic PDE (3.175)are not as interesting

as those for the degenerate TFEs. On the other hand, the presence in the PDE of

the non-Lipschitz absorption term −1, which is actually |u|

p−1

u for p = 0(the

Heaviside function), makes the problem rather consistent, especially in the Cauchy

problem to be discussed next.

Finite propagation and oscillatory patterns in the Cauchy problem. Considering

(3.175) in IR × IR

, we face the question of the maximal regularity of solutions at

the interfaces. We present the signed version of this PDE for solutions of changing

sign,

=−u

xxxx

+ (u

)

− signu.

As for TFEs with absorption, using the TWs u(x , t) = f (y), y = x − λt, yields

−λf



=−f

(4)

+ ( f



)

− sign f for y > 0, f (0) = 0.

150 Exact Solutions and Invariant Subspaces

0 5 10 15 20

−0.015

−0.01

−0.005

0.005

0.01

0.015

φ(s)

Figure 3.16 The stable periodic behavior in (3.177).

Using our TFE experience, we keep two leading terms,

(4)

+ sign f = 0,

so the oscillatory component ϕ in the solution representation

f (y) = y

ϕ(s), where s = ln y, (3.176)

satisﬁes the autonomous fourth-order ODE

(4)

+ 10ϕ



+ 35ϕ



+ 50ϕ



+ 24ϕ + sign ϕ = 0. (3.177)

Existence and uniqueness of periodic solutions are

OPEN.

Conjecture 3.2 The ODE (3.177) has a unique nontrivial periodic solution ϕ(s)

which is asymptotically stable as s →+∞.

Figure 3.16 showsthis stable periodicmotion obtainedfor differentinitial data. We

expect that, according to (3.176), such TW patterns describe the generic behavior of

solutions near interfaces in the CP.

Example 3.37 (Blow-up and interfaces in a quasilinear KS-type equation)Fi-

nally, let us present a simple quasilinear KS-type equation, which allows us to de-

scribe some crucial nonlinear phenomena by using simple mathematical tools. Con-

sider the following quasilinear fourth-order parabolic equation:

= F[u] ≡ u(−u

xxxx

+ u), (3.178)

which is initially formulated for nonnegative solutions. Clearly, the quadratic opera-

tor F preserves

= L{1, cos x , sin x, coshx , sinh x},

with simple blow-up dynamics. We restrict our attention to blow-up solutions in

separate variables

u(x , t) =

T −t

θ(x), (3.179)

3 Thin Film, Kuramoto-Sivashinsky, and Magma Models 151

where θ solves the linear ODE

−θ

(4)

+ θ = 1. (3.180)

Zero contact angle FBP. Let us posefor (3.178)the FBP with the zero contactangle,

u = u

= u

= 0 at the interface at x = s(t), (3.181)

and bounded compactly supported initial data u

(x ) ≥ 0. The well-posednessof this

by exploiting a regularization in the degenerate term −uu

xxxx

, e.g., by replacing

u → ε +|u| (ε > 0),

in the PDE (3.178) and studying the convergence of the sequence {u

(x , t)} of so-

lutions of such uniformly parabolic equations. (It is likely that more “singular” as

ε → 0

approximation is necessary to catch this FBP; such hard questions are not

discussed here and we refer to [44], where the dependence of solutions of TFEs on

approximations of the degenerate coefﬁcient was studied ﬁrst.) Further references

are given in the Remarks. The correspondence of the regularization to prescribed

free-boundary conditions is a difﬁcult asymptotic problem that, in general, is

OPEN

for most degenerate parabolic PDEs, including the present one. The analytic regular-

ization

u →

√

+ u

is expected to induce, as ε → 0, the solutions of the Cauchy problem; see below.

Thus, looking for even solutions of the ODE (3.180),

θ(x) = 1 + C

cos x + C

coshx ∈ W

and assuming that x

> 0 is the corresponding interface point, where (3.181) are

valid, we obtain the algebraic system



1 + C

cos x

+ C

coshx

= 0,

−C

sin x

+ C

sinh x

= 0,

−C

cosx

+ C

coshx

= 0.

(3.182)

The last two equations give a single transcendental equation for x

tan x

= tanh x

that possesses a countable set of roots {x

} such that

= π



k +



+ ... for k  1.

Solving the rest of equations in (3.182) yields the sequence of solutions

(x ) = 1 −

2cosx

cos x −

2coshx

coshx for |x| < x

These functions are strictly positive on (−x

, x

), which is easily seen for k  1,

and the oscillatory patterns satisfy

(x ) → θ

∞

(x ) ≡ 1 +

(−1)

√

cosx > 0ask →∞ (3.183)

uniformly on compact subsets (θ

corresponds to even k and θ

−

to odd). There

FBP can be checked locally by using the von Mises variables (see Example 3.10), or

152 Exact Solutions and Invariant Subspaces

exists a countable set {u

(x , t)} of standing wave blow-up solutions (3.179) of the

FBP localized in bounded intervals (−x

, x

On solutions of the Cauchy problem. Consider now the signed PDE

=|u|(−u

xxxx

+ u) in IR × IR

, (3.184)

with boundedcompactly supportedinitial data u

(x ) in IR . Thentheseparate-variable

solutions (3.179) yield the ODE

−θ

(4)

+ θ = signθ. (3.185)

Evidently, there exist strictly positive solutions θ

∞

(x ) given in (3.183), but we are

interested in compactly supported patterns.

It follows from (3.185) that, in the positivity and negativity domains respectively,

the proﬁles are

(x ) =±1 +C

cosx + C

sin x +C

coshx + C

sinh x .

Despite such a simple form of proﬁles, the matching assumes four conditions of

continuity (here [·] denote the jumps of functions at zeros)

[θ] = [θ



] = [θ



] = [θ



] = 0,

so that the construction of changing sign solutions of the FBP leads to algebraic

problems for the eight expansion coefﬁcients {C

} at each zero point x = x

.This

is an

OPEN PROBLEM that can be tackled numerically.

The existence of changing sign solutions of the ODE (3.185) for both the FBP

and the CP problems is connected with general oscillatory properties of solutions of

the PDE (3.184). It is convenientto describe the oscillatory character of solutions by

TWs u(x, t) = f (x − λt),so

−λf



=−|f |f

(4)

in IR ,

whereweomit| f |f that is smaller near interfaces. Integrating once yields



= λ sign f ln| f | for y > 0, f (0) = 0.

We next introduce the oscillatory componentϕ,

f (y) = y

ϕ(ln y) ⇒ ϕ



+6ϕ



+11ϕ



+6ϕ = 3λs + λ signϕ ln |ϕ|. (3.186)

For λ<0, there exists the positive solution ϕ(s) =

λs + ... for s &−1. By

transformation in (3.186), this gives non-oscillatory behavior at the interface,

f (y) =

λy

ln y + ... > 0asy → 0, (3.187)

which corresponds well to the free-boundaryconditions (3.181). Oscillatory patterns

in the CP need extra study.

Oscillatory component. As usual, we introduce the family of PDEs

=−|u|

xxxx

, with parameter n ∈ (0, 1). (3.188)

Dividing by |u|

and setting v =|u|

−n

u yields a divergent quasilinear equation,

1−n

= [v] ≡−(|v|

xxxx

, with σ =

1−n

> 0. (3.189)