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 133

Recall that, for second-order operators in Section 1.4, similar different trigono-

metric and exponential subspaces occurred for the unique remarkable operator

[u] = uu

− (u

)



= F

rem

[u]



. (3.127)

Its natural fourth-order analogy corresponds to β = 1 in (3.125),

[u] =−uu

xxxx

+ (u

)

. (3.128)

Using exponential functions. Continuing to detect other invariant properties, us-

ing Proposition 3.19, consider the following differential operator for functions u =

u(x , t) of two independent variables:

F[u] = u

− F

[u].

We choose solutions in the form of

u(x , t) = C

(t) + C

(t)x +C

(t)e

γ(t )x

, (3.129)

with a function γ(t) to be speciﬁed. Our analysis here is similar to that shown in

Example 1.37, so for functions (3.129) we ﬁnd

F[u] = C



+ C



x +





+ γ

− (1 − β)γ



γ x





+ γ



γ x

Then the ODE γ



=−γ

is a characterization of an invariant set M in the linear

subspace (a partially invariant module) W

,sothat

F : M → W

Example 3.20 The fourth-order PDE

=−uu

xxxx

+ β(u

)

+ (1 − β)u

xxx

+ µu + ν

admits solutions (3.129), where the coefﬁcients satisfy the DS













= µC

+ ν,



= µC



=−γ

+ (1 − β)γ

+ µC



=−γ

The second and the fourth ODEs give all possible functions γ(t).

Using trigonometric functions. According to Proposition 3.19, we now deal with

operator (3.128). Then, the ﬁfth-order operator

F[u] =

[u] ≡−uu

xxxxx

− u

xxxx

+ 2u

xxx

(3.130)

also preserves 3D trigonometric subspaces in (3.126) for arbitrary γ = 0. The extra

differentiation in (3.130) is necessary to create a single ODE for γ(t). Recall that,

similarly, in Example 1.36 we used the quadraticoperator of the RPJ equation as the

derivative of (3.127). Such invariant properties of the corresponding operator

F[u] = u

−

F[u]

are listed in the following example.

134 Exact Solutions and Invariant Subspaces

Example 3.21 The sixth-order semilinear parabolic equation

= εu

xxxxxx

− uu

xxxxx

− u

xxxx

+ 2u

xxx

+ µu + ν

possesses exact solutions

u(x , t) = C

(t) + C

(t)x +C

(t) cos(γ (t)x ) + C

(t) sin(γ (t)x),













= µC

+ ν,



= µC



=−εγ

− γ

+ µC



=−εγ

+ γ

− γ

+ µC



=−γ

3.6 2mth-order thin ﬁlm operators and equations

We describe basic subspaces and extensions for 2mth-order TFEs, such as (3.16).

3.6.1 Basic polynomial subspaces

Consider the standard higher-order quadratic thin ﬁlm operator

F[u] = (−1)

m+1



2m−1



Since M(l) = 2l − 2m, i.e., F[x

] ∼ x

2l−2m

, equating 2l − 2m = l yields the

following basic subspace:

2m+1

= L{1, x, x

, ..., x

}. (3.131)

This makes it possible to study various singularity formation phenomena, such as

ﬁnite time extinction, quenching, turning points, interface dynamics, etc., for the

corresponding TFE with absorption

= (−1)

m+1



2m−1



− 1.

The corresponding exact solutions are given by

u(x , t) =



(t) + C

(t)x +C

(t)x

+ ... + C

2m+1

(t)x



where {C

(t)} solve a DS. These nonnegative functions are weak solutions of FBPs

with Stefan–Florin free-boundary conditions, which can be detected in a manner

similar to above lower-order models (cf. a simpler derivation below).

Example 3.22 (Cubic TFE) In the cubic equation

= F[u] ≡ (−1)

m+1



+ bu

+ cu



the operator admits the basic subspace W

m+1

= L{1, x, x

, ..., x

}. Free-boundary

and asymptotic analysis of the extinction and quenching behavior in the correspond-

ing absorption model

= F[u] −1onW

m+1

are similar to the fourth-order models in Section 3.2.

3 Thin Film, Kuramoto-Sivashinsky, and Magma Models 135

3.6.2 Solutions with zero contact angle conditions

In order to obtain solutions v(x, t) satisfying m + 1 free-boundary conditions at the

interface, including m − 1 generalized zero contact angle conditions and a zero-ﬂux

condition, i.e.,

v = v

= ... = D

m−1

v = v D

2m−1

v = 0atx = s(t), (3.132)

we need to introduce models with speciﬁc algebraic and invariant properties. As

above, we consider two types of such equations.

Example 3.23 (Strong absorption)Theﬁrst model is the 2mth-order TFE with

strong absorption

= (−1)

m+1



v D

2m−1



− v

m−1

. (3.133)

Setting v = u

yields the operator F

m+1

of the algebraic homogenuity m + 1,

= F

m+1

[u] −

≡ (−1)

m+1



+ u

2m−1



−

. (3.134)

m+1

admits W

= L{1, x , x

}, and hence, the FBP (3.133), (3.132) has solutions

v(x , t) = u

(x , t) =



(t) + C

(t)x



, (3.135)





= (−1)

m+1

(2m)!

−



= (−1)

m+1

(2m+1)!

m+1

The dynamic interface equation is readily derived from (3.134),



=−

= S[u] ≡ (−1)

2m−1

at x = s(t).

This equation is the main regularity condition for solvability of the degenerate PDE,

written in terms of the vonMises variable X = X (u, t) near the interface. Themanip-

ulations become more technical for large m than those performed before for lower-

order equations. Exact solutions (3.135) describe singular phenomenaof quenching,

extinction, and others.

On the Cauchy problem. For solutions of changing sign that exhibit the maximal

regularity, we consider the TFE with absorption

= (−1)

m+1



|v|D

2m−1



−|v|

−

v in IR × IR

. (3.136)

As in Example 3.10, we use the TWs v(x, t) = f (x − λt) satisfying the ODE

−λf



= (−1)

m+1

(| f |f

(2m−1)

)



−|f |

−

or, neglecting the non-stationary, λ-dependent term for small f ,

(−1)

m+1

(| f |f

(2m−1)

)



−|f |

−

f = 0fory > 0, f (0) = 0.

This gives the following oscillatory structure of solutions:

f (y) = y

ϕ(s), s = ln y, where γ =

m+1

, (3.137)

which applies to many higher-order ODEs. Here the oscillatory component ϕ(s)

136 Exact Solutions and Invariant Subspaces

solves a 2mth-order autonomous ODE. Such ODEs will be shown to admit oscil-

latory, periodic solutions that describe different types of multiple zeros of solutions

of the PDE (3.136); see Figure 3.5(a) as an illustration. The exponent γ in the enve-

lope in (3.137) is such that

γ>2m − 2, so that f (y) = o(y

2m−2

) as y → 0.

Hence, maximal regularity solutions satisfy on the interface the conditions



(0) = ... = f

(2m−2)

(0) = 0,

and are smoother than those of the FBP, which are O(y

), as (3.132) suggests.

Example 3.24 (Backward diffusion perturbation) The second model includes a

special divergent second-order operator generating an extra instability feature of the

thin ﬁlm ﬂow,

= (−1)

m+1



v D

2m−1



−





, or

= (−1)

m+1



+ u

2m−1



−



+ m(u

)



for v = u

. For solutions (3.135), the following DS is obtained:





= (−1)

m+1

(2m)!

− 2C



= (−1)

m+1

(2m+1)!

m+1

− 2(2m + 1)C

It follows from the second ODE that, given an initial value C

(0)<0, the following

holds:C

(t) →−a =−[(2m −1)!]

−1/(m−1)

as t →∞. This means the exponential

stabilization to unstable stationary solutions,

u(x , t) = (B − ax

)

+ O(e

−γ

), with γ

= (m − 1)(2m + 1)a,

where B > 0 is a constant, depending on the initial data.

3.6.3 Extensions of polynomial subspaces

Consider a generalization of the thin ﬁlm operator,

F[u] = (−1)

m+1



u + βu

2m−1



(β ∈ IR ). (3.138)

Here we keep two monomial operators of the same total differential order, where

i + j = k, with k = 2m; cf. the general quadratic operator (1.86). The next result is

proved directly.

Proposition 3.25

Operator

(3.138)

preserves the following extensions of the sub-

space

(3.131)

2m+2

= L{1, x , ..., x

, x

2m+1

}

iff

β = β

=−

2m+1

; (3.139)

2m+3

= L{1, x , ..., x

, x

2m+1

, x

2m+2

}

iff

β = β

=−

2m+2

. (3.140)

Further extensions of polynomial subspaces are not possible for operator (3.138),

but can be obtained for other similar operators containing extra monomials of the

3 Thin Film, Kuramoto-Sivashinsky, and Magma Models 137

same exponent M(l) with more free parameters β, γ , etc. (see (3.142) below). Con-

structing the corresponding exact solutions leads to higher-order DSs. In the case of

(3.140), the DS is simpliﬁed if we take only even polynomials by introducing

even

m+2

= L{1, x

, ..., x

, x

2m+2

} (3.141)

that is also invariant under operator (3.138) for β = β

. Such subspacesare sufﬁcient

for studying most singularity,free boundary,and asymptotic phenomena for the TFE

with symmetric initial data. Using subspaces of even functions, we formulate the

following pretty general observation that is convenient for future computations and

was observed before in particular cases.

Proposition 3.26

If subspace

(3.141)

is invariant under a polynomial quadratic op-

erator with constant coefﬁcients

(

again, in each monomial,

i + j = k = 2m)

F[u] = (−1)

m+1



u + βu

2m−1

u + γ u

2m−2

u + ...



(3.142)

for some

β =

depending on

and other parameters. Then subspace in

(3.140)

also invariant under

Proof. Indeed, taking

u = C

+ C

+ ... + C

2m+1

+ C

2m+3

2m+2

(3.143)

and controlling the higher-degree term only yields

F[u] = (β −

2m+3

)

2m+4

+ ... , (3.144)

where A

= 0 is a constant. The second (omitted) term contains x

2m+2

∈ W

even

m+2

Hence, always F[u] ∈ W

even

m+2

iff β =

.Takingnowu from subspace (3.140),

u = C

2m+3

2m+2

+ C

2m+2

2m+1

+ ... , where C

2m+3

= 0

2m+3

= 0 is easy), and performing the translation x → x − a gives

u = C

2m+3

2m+2



2m+2

− (2m + 2)aC

2m+3



2m+1

+ [·]x

+ ... .

Choose now a such that C

2m+2

− (2m + 2)aC

2m+3

= 0tovanishthesecondco-

efﬁcient. Then we get that the last two terms contain x

2m+2

and x

as in (3.143).

Hence, the argument for even degree polynomials applies, so the term with x

2m+3

does not appear in F[u].

Similarly, a more general result holds, which applies to arbitrary quadratic opera-

tors with monomials of the same total order such as (1.86).

Lemma 3.27

If operator

(3.142)

admits

(1)

2m+2

= L{1, x , ..., x

2m−1

, x

2m+2

}

(

a subspace with “1-gap”

)

, then it admits

2m+3

= L{1, x, ..., x

, x

2m+1

, x

2m+2

}

(

the full subspace

)

Remark 3.28 (Open problem: gap completing) The result is true for the “2-gap”

subspace W

(2)

2m+1

= L{1, x , ..., x

2m−3

, x

2m−2

, x

2m+2

}.AsanOPEN PROBLEM,

taking the “(m+1)-gap” subspace of even polynomials (3.141), we conjecture that if

F : W

even

m+2

→ W

even

m+2

,thenF : W

2m+3

→ W

2m+3

. (For cubic operators, this is not

true; see Proposition3.5.) Invariantsubspaces with more general distribution of gaps

are also unclear.

138 Exact Solutions and Invariant Subspaces

3.6.4 Subspaces of irrational functions: dipole-type solutions

The origin of such subspaces and exact solutions is the quadratic PME

= F[u] ≡ (u

)

in IR × IR

, (3.145)

possessing the self-similar dipole Barenblatt–Zel’dovich solution [28]

u(x , t) = t

−



A − t

−



for x ≥ 0 (A > 0), (3.146)

which is extended for x < 0 as an odd function. The solution has the ﬁxed point at

x = 0, where u(0, t) ≡ 0, and the graph of u(x , t) has a typical dipole form in x.The

corresponding initial function at t = 0 is proportional to δ



(x ), the weak derivative

of Dirac’s delta. Actually, this is a solution on the subspace

= L



, x



that is invariant under the quadratic operator F in (3.145). At the same time, it is a

standard similarity solution induced by a group of scaling transformations.

Let us show that such subspaces and solutions exist for higher-order parabolic

PDEs. Consider the 2mth-order quadratic operator

F[u] = (−1)

m+1





. (3.147)

Proposition 3.29

Operator

(3.147)

preserves the

(m+1)

-dimensional subspace

m+1

= L



, x

, ..., x

2m−3

, x

2m−1

, x



Proof. It follows that, for any u ∈ W

m+1

∈ L



x, x

, ..., x

2m−1

, x

2m+

, ..., x

2m+

2m−1

, x



which yields D

) ∈ W

m+1

Example 3.30 The fourth-order equation

=−(u

)

xxxx

is parabolic in the positivity domain {u > 0}. The corresponding solutions are

u(x , t) = C

(t)x

+ C

(t)x

+ C

(t)x

∈ W

for t ≥ 0,













=−

945



=−

3465



=−1680C

Solving the DS yields the explicit formula

u(x , t) =



−

128

+ Bt

−

128

1680t



. (3.148)

For different values of A and B, (3.148) describes a dipole-like singularity at t = 0,

extinction/quenching phenomena, and the interface propagation. A proper setting of

the corresponding Stefan–Florin FBP with positive solutions can be also revealed.

3 Thin Film, Kuramoto-Sivashinsky, and Magma Models 139

3.6.5 Trigonometric subspace

A typical example is the following TFE with source possessing blow-up solutions:

= F[u] ≡ (−1)

m+1



2m−1



+ 2u

. (3.149)

Proposition 3.31

Operator

(3.149)

admits

= L{1, cos x , sin x}

Equation (3.149) on W

is a third-order DS that describes blow-up of 2π-periodic

solutions. As shown in Example 3.17, the 2D restrictions L{1, cos x } and L{1, sin x}

are also invariant and generate simpler DSs on which blow-up similarity solutions

are asymptotically stable.

3.7 Oscillatory, changing sign behavior in the Cauchy problem

For a moment we digress from invariant subspaces for TFEs and discuss some as-

pects concerning general solutions of changing sign of the Cauchy problem.

Source-type solutions of the TFE. Consider the signed 2mth-order TFE with the

higher-order term only,

= (−1)

m+1



|u|

2m−1



in IR × IR

, (3.150)

where n > 0 is a parameter. The fundamental source-type solution has the similarity

form

b(x, t) = t

−β

g(ξ), ξ =

, where β =

n+2m

and, on integration once, g solves the (2m − 1)th-order ODE

(−1)

m+1

|g|

(2m−1)

+ βgξ = 0inIR . (3.151)

The unit mass condition is also imposed



g(ξ) dξ = 1 (3.152)

and can always be achieved by scaling. Another usual normalization condition is

g(0) = 1 that will be used sometimes later on. For the Cauchy problem, we are

looking for solutions of the maximal regularity, which can be admitted by the ODE

(3.151). We pose m − 1 symmetry boundary conditions at the origin,



(0) = g



(0) = ... = g

(2m−3)

(0) = 0 (g(0)>0). (3.153)

For m = 2, rigorous mathematical results on existence and uniqueness for the prob-

lem (3.151), (3.152) are known for n ∈ (0, 1]; see [174] where Bernis–McLeod

approach [47] was used. For other n and m ≥ 3, we rely on analytic-numerical

evidence to be presented.

Let g be supportedon the interval [−ξ

,ξ

]. We can also usenormalization ξ

= 1

instead of (3.152). At the right-hand interface, for ξ ≈ ξ

−

, we then introduce the

oscillatory component ϕ by

g(ξ) = (ξ

− ξ)

ϕ(s), s = ln(ξ

− ξ), where γ =

2m−1

. (3.154)

Setting

ξ = ξ

− e

for s &−1

140 Exact Solutions and Invariant Subspaces

and omitting the exponentially small non-autonomous perturbation yields that ϕ(s)

satisﬁes

2m−1

[ϕ] = c

|ϕ|

−n

ϕ in IR , where c

= βξ

(−1)

m+1

. (3.155)

Here, {P

[ϕ], k ≥ 0} denote operators that are constructed by the iteration

k+1

[ϕ] = (P

[ϕ])



+ (γ − k)P

[ϕ]fork = 0, 1, ... , P

[ϕ] = ϕ. (3.156)

For instance, for m = 2



γ =



and m = 3



γ =



, respectively,

[ϕ] = ϕ



+ 3(γ − 1)ϕ



+ (3γ

− 6γ + 2)ϕ



+γ(γ − 1)(γ − 2)ϕ and

[ϕ] = ϕ

(5)

+ 5(γ − 2)ϕ

(4)

+ 5(2γ

− 8γ + 7)ϕ



+5(γ − 2)(2γ

− 8γ + 5)ϕ



+ (5γ

− 40γ

+ 105γ

−100γ + 24)ϕ



+ γ(γ − 1)(γ − 2)(γ − 3)(γ − 4)ϕ.

(3.157)

Traveling waves. For TWs u(x , t) = f (y), with y = x − λt, the ODE is easier

(−1)

m+1

| f |

(2m−1)

+ λ f = 0fory > 0, f (0) = 0, (3.158)

where the left-hand interface is at y = 0. The oscillatory component is given by

f (y) = y

ϕ(s), s = ln y, where γ =

2m−1

, (3.159)

where ϕ solves the ODE (3.155) with c

= (−1)

λ.

As before, we are mainly interested in periodic solutions ϕ(s) of ODEs (3.155),

which, according to (3.154), determine typical (and sometimes generic) oscillatory

behavior of solutions near interfaces, as shown in Figure 3.5(a). Existence and mul-

tiplicity of periodic solutionsof dynamical systems in IR

is a classical area of appli-

cations of topologicaland geometric methods of nonlinear analysis, such as rotations

of vector ﬁelds and index-degreetheory; see Krasnosel’skii–Zabreiko[355, Sect. 13,

14]. Another mathematical direction is classical branching theory; see Vainberg–

Trenogin [565, Ch. 6]. In our case, an n-branching approach may be effective, since

for n = 0, the unique solution of the problem (3.151), (3.152) is indeed the rescaled

kernel of the fundamental solution. A mathematical justiﬁcation of such a branching

is a hard

OPEN PROBLEM. We also refer to [569, 399, 336] as a guide to modern

theory of periodic solutions of higher-order nonlinear ODEs. In general, for large

m, equations (3.155) are difﬁcult to study, and especially as the main concern is the

number of their different periodic solutions, as well as the identiﬁcation of the most

stable solution that describes a general structure and complexity of multiple zeros of

solutions near interfaces. We will also rely on careful numerical evidence on exis-

tence, uniqueness and stability of periodic solutions. It is natural to begin recalling

the properties of the linear PDE (3.150) with n = 0, which explain the oscillatory

patterns in the quasilinear case for small n > 0.

Oscillations in the linear equations. For n = 0 in (3.150), the linear polyharmonic

PDE is obtained,

= (−1)

m+1

u in IR × IR

3 Thin Film, Kuramoto-Sivashinsky, and Magma Models 141

whose oscillatory properties are described by its fundamental solution

b(x, t) = t

−

g(ξ), ξ =

1/2m

, (3.160)

where g is the unique exponentially decaying solution of the ODE

(−1)

m+1

(2m)



ξ +

g = 0inIR ,



g = 1. (3.161)

See Eidelman’s classic book [164, Ch. 1] for the existence and sharp estimates of

fundamental solutions of higher-order linear parabolic operators. By substituting, it

is not hard to check that the behavior as ξ →+∞of solutions of (3.161) is given by

g(ξ) ∼ ξ

−µ

aξ

, with α =

2m−1

and µ =

m−1

2m−1

, (3.162)

where a is the root with the maximal Rea < 0 of the characteristic equation

(−1)

(αa)

2m−1

. (3.163)

Estimating the number of complex conjugate pairs of roots, a, and their real parts

yields the following useful conclusion.

Proposition 3.32 (i)

The asymptotic representation

(3.162)

of the rescaled kernel

g(ξ)

of the fundamental solution

(3.160)

can be represented in terms of a quasi-

periodic function containing not more than

([·]

denotes the integer part

)





fundamental frequencies.

(3.164)

(ii)

There exists a constant

D > 0

such that

|g(ξ)|≤De

−d|ξ|

IR ,

where

d =

2m−1

(2m)



cos



mπ

2m−1





It follows from (3.164) that the total asymptotic linear bundle of exponentially

decaying solutions (3.162) satisﬁes

the bundle is m-dimensional. (3.165)

For m odd, this includes the 1D non-oscillatory bundle, corresponding to the real

negative root of (3.163)

=−

2m−1

(2m)

Oscillatory periodic patterns for m = 2 and a heteroclinic bifurcation. Let us

begin with the simplest case, m = 2, where the third-order ODE (3.155) takes the

form (here c

=−1andγ =

)

[ϕ] ≡ ϕ



+ 3(γ − 1)ϕ



+ (3γ

− 6γ + 2)ϕ



+γ(γ − 1)(γ − 2)ϕ =−

|ϕ|

(3.166)

Numerical experiments (Figure 3.6) show that (3.166) admits a unique stable peri-

odic solution for all n ∈ (0, n

),where

= 1.758665... (m = 2)

is a critical heteroclinic bifurcation exponent.

The exponent n

plays an important role and shows the precise parameter range

142 Exact Solutions and Invariant Subspaces

0 2 4 6 8 10

−0.025

−0.02

−0.015

−0.01

−0.005

0.005

0.01

0.015

0.02

0.025

φ(s)

(a) n = 1

0 5 10 15 20 25 30 35 40

−0.5

−0.4

−0.3

−0.2

−0.1

0.1

0.2

0.3

0.4

0.5

φ(s)

(b) n =

Figure 3.6 Stable periodic behavior of (3.166), m = 2, for n = 1 (a) and n =

(b).

of n for which all the ODE proﬁles near interfaces are oscillatory (this may also be

key for the corresponding PDE). Numerical results reveal a non-local heteroclinic

bifurcation at n = n

associated with two unstable equilibria of (3.166)

=±



−

γ(γ−1)(γ −2)



for n ∈



, 3



Note that (3.159) with ϕ(s) ≡ ϕ

yields a non-oscillatory behavior; see the dashed

line in Figure 3.5(a). Figure 3.7 shows a typical “heteroclinic” deformation of peri-

odic patterns as n → n

−

, where, in order to reveal the widest periodic pattern (the

bold line), we need to take n = 1.758664976837300 (where not all the 15 decimals

are reliable). This is a standard scenario of homoclinic-heteroclinic bifurcations of

periodic solutions in ODEs; see Perko [460, Ch. 4]. A rigorous justiﬁcation of such

bifurcations is difﬁcult and is an

OPEN PROBLEM, especially for higher-order equa-

tions with m = 4, 6, ... to be considered below.

For n > n

, the behavior of solutions of the ODE (3.166) becomes exponentially

unstable and oscillatory or changing sign patterns are not observed. It is likely that

precisely above n = n

, the ODE (and, to some extent, the correspondingPDE) loses

its natural similarities with the linear equation for n = 0, i.e., some local properties

of solutions dramatically change at n = n

m ≥ 3: unstable periodic behavior for odd m and stable for even. In numerical

experiments, we have observed a single stable oscillatory behavior for even m ≥ 2

and an unstableorbitfor odd m. Figure 3.8(a) showssolutionsof the ODE (3.155)for

m = 3 and stability of equilibria ϕ

. In this case, the unstable periodic solution lies

in between those two stable ﬂows of orbits tending to ϕ

and ϕ

−

as (b) explains. It

turns out that such an unstable periodic solution exists until the “heteroclinic” value

ˆn

= 1.909... (m = 3).

In view of its unstability, the corresponding analytical and numerical techniques be-

come more involved; see [175] for extra details.

For m = 4, i.e., for the eighth-order TFE, the solutions ϕ(s) approach a stable

periodic pattern; see Figure 3.9. The periodic solution is easily detected above the