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

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5 Quasilinear Wave and Boussinesq Models. Systems 253

Choosing the positive function C

(t) = t

−2

from the second ODE, and integrating

the ﬁrst equation yields the following solution:

u(x , t) =

−

lnt +

. (5.56)

It is interesting ﬁrst to analyze the initial singularity as t → 0

,where

u(x , t) =

+ η

+ ... , with the rescaled variable η =

This shows that initial functions for such solutions are entirely singular:

u(x , 0) ≡+∞ and u

(x , 0) ≡−∞ in IR .

Nevertheless, such initial data give rise to a local positive analytic solution with the

given quadratic growth as x →∞.

Consider next the quenching phenomenon at the moment t = T > 1 such that

ln T.

Then, for all t < T , (5.56) is a smooth strictly positive solution of (5.55), and, as

t → T

−

, the solution exhibits the following asymptotics:

u(x , t) = a

(T − t) + a

+ ... ≡ (T − t)



+ a



+ ... , (5.57)

where a

and a

are positive constants, depending on T .Hereξ is the quenching

rescaled spatial variable

ξ =

√

T −t

. (5.58)

Asymptotics (5.57) of the solutions showsa regularapproach to the singularitypoint.

In rescaled variables, there exists the limit as t → T

−

of the rescaled function,

v(ξ,τ) ≡ (T − t)

−1

u(x , t) → g(ξ ) ≡ a

+ a

. (5.59)

A general asymptotic stability problem is

OPEN. Let us present some comments. By

differentiating of u = e

−τ

v(ξ,τ), u

= v

− v +

ξ, (uu

)

= e

−τ

(vv

)

,etc.,

we derive a singular perturbed PDE,

ττ

+ v

τξ

ξ −v

ξξ

−

ξ = e

−2τ

(vv

)

− e

−τ

, (5.60)

where the second-orderquadratic operator gives, on the right-hand side, an exponen-

tially small perturbation as τ →∞. The proﬁle g given in (5.59) satisﬁes the limit

(i.e., at τ =+∞) stationary ODE

C[g] ≡



−



ξ = 0.

On linear stability. The problem of the passage to the limit τ →∞in (5.60) re-

mains

OPEN.Webrieﬂy discuss the linear stability setting using similarities with that

in Example 5.1. Linearization in (5.60) about the stationary proﬁle by setting

v(ξ,τ) = g(ξ ) + Y (ξ, τ )

leads to the following equation:

ττ

+ Y

τξ

ξ −Y

ξξ

−

= e

−2τ



(gY

)

+ (Yg



)



− e

−τ

+ e

−2τ

(YY

)

(5.61)

254 Exact Solutions and Invariant Subspaces

In the ﬁrst step of linearized analysis, we neglect both linear and nonlinear exponen-

tially small perturbations on the right-hand side. Then, using the standard method of

separation of variables for linear homogeneousPDEs and looking for

Y (ξ, τ ) = e

λτ

ψ(ξ) (5.62)

yields the following eigenvalue problem for the quadratic pencil of linear operators:



I + λB + C



[ψ] ≡ λ

ψ + λ(ψ



ξ −ψ) +



−



ξ = 0. (5.63)

Spectral theory of linear polynomial pencils is an important and classical area of

differential operators theory; see Markus [412].

In general, for pencils of non-self-adjoint operators, even in the ordinary differ-

ential setting, the problem of discreteness of the spectrum, as well as completeness

and closure of the eigenfunction sets, are often difﬁcult. In the present case, we use

a special advantage associated with the blow-upcharacter of scaling variables. Since

the coefﬁcients of operators in (5.63) are unbounded as ξ →∞, the functional set-

ting plays a key role. For such operators, a right setting is available in the weighted

(IR ) space with the exponential weight ρ(ξ) = e

−ξ

. Since the operators are

not self-adjoint, we are not obliged to be very determined in choosing a particular

weight. As often happens in blow-up rescaled problems, our pencil generates some

polynomial eigenfunctions.

Proposition 5.15

The quadratic pencil

(5.63)

(IR )

has the discrete spectrum,

consisting of two series of real eigenvalues

=−

(k − 2)

and

−

=−

for

k = 0, 1, 2, ... , (5.64)

with eigenfunctions

(ξ)

being

th-order polynomials.

The eigenvaluesλ

in (5.64) are obtainedby pluggingψ

(ξ) = ξ

+... into (5.63).

We thus observe from (5.64) that, in the symmetric setting, there exist just two bad

modes:

(i) k = 0 with λ

= 1, corresponding to the instability via perturbations of the

blow-up time, and

(ii) k = 1 with λ

, corresponding to the shifting in x of the quenching point.

Both modes are excluded by ﬁxing the time T and the point x = 0 of quenching in

rescaled variables (5.58), (5.59). Since the polynomials are complete in any suitable

weighted L

-spaces, [352, p. 431], this gives certain evidence about the linear stabil-

ity of the stationary proﬁle g; though we should take into account that the constants

and a

in (5.59) are arbitrary and depend on the initial data (this corresponds to

the centre subspace behavior with λ

= 0fork = 2 and 0). The extension of the

linear stability to the nonlinear one is a difﬁcult

OPEN PROBLEM.

On related blow-up problems on multiple zeros. It is curious that polynomial

eigenfunctions of such pencils describe various pattern formation of multiple zeros

in x of solution u(x, t) of the linear wave equation

= u

in IR × IR

−

. (5.65)

Namely, if a multiple zero occurs at the point (0, 0), we perform the blow-up scaling

5 Quasilinear Wave and Boussinesq Models. Systems 255

near this point (see [231])

u(x , t) = Y (ξ, τ ), with ξ =

(−t)

,τ=−ln(−t),

ττ

+ Y

+ 2Y

ξτ

ξ = A[Y ] ≡ (1 − ξ

ξξ

− 2Y

ξ.

Looking for solutions in separate variables (5.62) yields the eigenvalue problem for

a quadratic pencil,

{(λ

+ λ)I + 2λξ D

− A}[ψ] = 0, with eigenvalues

=−k and λ

−

=−k − 1fork ≥ 0,

and eigenfunctions ψ

(ξ) being kth-order polynomials. Again, in view of complete-

ness and closure of the eigenfunction set  ={ψ

} in weighted L

spaces, these

eigenfunctions give a full countable set of different patterns of multiple zeros at

(0, 0) for the wave equation (5.65),

(x , t) = (−t)

−λ



(−t)



< 0



Therefore, each multiple zero of kth order is formed as t → 0

−

by k zero curves

focusing at x = 0 with the behavior

(t) = ξ

(−t), where ψ

(ξ

) = 0.

For the linear parabolic heat equation

= u

a similar classiﬁcation of multiple zeros with the heat kernel rescaled variable ξ =

√

−t leadsto the classic eigenvalueproblem,where  ={ψ

}consists of Hermite

polynomials. These computations were ﬁrst performed by Sturm (1836), [538], and

initiated deep mathematical theory in relation to both PDEs and ODEs; see survey

and references in [226, Ch. 1].

For hyperbolicequations, zero set theory is less developed.Similar quadratic pen-

cils with the same spectrum occur [231] in the study of a different blow-up phe-

nomenon for the semilinear wave equation with a nonlinear force term,

= u +|u|

p−1

u ( p > 1).

Interface propagation. For t > T , solution(5.56)takesnegativevalues.To continue

to deal with nonnegativesolutions,one needs another FBP frameworkby introducing

free-boundary conditions at the interfaces. Take now a simpler solution

u(x , t) =−

lnt +

, (5.66)

with the positive interface at

s(t) =

√

lnt for t > T = 1. (5.67)

The dynamic interface equation is derived by using TWs u(x, t) = f (y), with y =

x − λt, satisfying the ODE λ



= ( ff



)



− 1, so that, for regular solutions,



)

= λ



[( f



)

− 1] ≡

[(u

)

− 1] at x = s(t). (5.68)

to get the PDE (cf. (5.61))

256 Exact Solutions and Invariant Subspaces

Noticing that (5.68) holds for any regular solution (so it is a manifestation of regu-

larity), we derive the actual dynamic equation by calculating u

from (5.66),

hence, t =



. Differentiating (5.67) yields



√





1/2





−1/2



This interface equation seems to be sufﬁcient for the second-order PDE (5.55) to

admit a local unique solution via the von Mises transformation (the analysis can be

affected by possible formations of shock waves nearby). This is an

OPEN PROBLEM.

We will give further details of such a construction in the next model dealing with

zero contact angle free-boundary conditions.

5.3.2 Zero contact angle and oscillatory solutions

Example 5.16 (Zero contact angle solutions) In order to create exact solutions

with the zero contact angle at the interfaces,another modiﬁcation of quasilinear wave

equations is necessary. Consider the PDE with special left and right-hand sides



√



= (vv

)

+ αv + βv

in IR × IR

(5.69)

for nonnegativesolutionsv ≥ 0. This equation looks awkward, but by setting v = u

we obtain a simpler quadratic equation,

= (u

)

+ 4(u

)

+ α + βu

that possesses solutions u = C

+ C

x + C

on the subspace L{1, x, x

}.Obvi-

ously, then v(x, t) = u

(x , t) satisﬁes the zero contact angle condition

v = v

= 0 at interfaces x = s(t). (5.70)

To illustrate the behavior of such interfaces, set α = 0andβ = 1, to get the PDE

= (u

)

+ 4(u

)

+ u

possessing solutions

u(x , t) = C

(t) + C

(t)x

, (5.71)





= 4C

+ C



= 28C

+ C

Choosing the equilibrium C

=−

from the second ODE and substituting into the

ﬁrst equation yields C



, which gives, e.g., C

(t) = cosh



t. The exact

solution of the original equation (5.69) with α = 0andβ = 1 takes the form

v(x , t) =



cosh



t −



, (5.72)

which has exponentially expanding interfaces with the zero contact angle

(t) =±R(t) ≡±

28cosh



t for t > 0. (5.73)

5 Quasilinear Wave and Boussinesq Models. Systems 257

The regularity criterion at interfaces is obtained from the identity

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

yielding



+ 2u



+ u



)

+ u

= 0atx = s(t). (5.74)

The interface equation for the particular solutions (5.71), (5.72) is of stationary,

Florin type,

at x = s(t).

In view of two already available free-boundary conditions (5.70), other dynamic

interface equations look like regularity conditions. Introducing the von Mises trans-

formation near the interface

X = X (u, t),

and assuming that it applies locally, by computations similar to those in Example

3.10, we arrive at the following second-order hyperbolic equation:

−

)



− X



= 2u

)

−

− uX

. (5.75)

We next impose the transversality assumption at the interface, X

= 0atu = 0, and

then the local solvability and uniqueness of a sufﬁciently smooth solution depend

on the spectral properties of the linear operator B = u

. It is self-adjoint in the

weighted space L

(0, 1), ρ =

, has a compactresolvent, a discrete spectrum, and a

complete, closed set of eigenfunctions. We refer to Naimark’s monograph [432]. For

the solvability of (5.75) by the semigroup approach and existence of a local regular

solution, one needs to impose the conditions that this PDE is valid at u = 0, i.e.,

)

− (2X

− X

) =−6X

which is the identity (5.74), written in terms of the von Mises variable X .Asa

consequence and an

OPEN PROBLEM, it is believed that the compactly supported

function (5.72) is a unique solution of the CP for the original equation (5.69).

Principlesof formationof quenchingpatternsremain similarfor higher-orderwave

equations with extra linear operators, e.g., for the Boussinesq-type equation

=−u

xxxx

+ (uu

)

− 1. (5.76)

Questions of interface propagation becomes more delicate, as is usual for higher-

order equations; cf. examples in Section 3.2. The fourth-order linear operator u

xxxx

vanishes on W

, and hence, will not affect the asymptotics of the singular extinction

patterns.

On maximal regularity of oscillatory solutions. For solutions of changing sign in

the CP, we consider the signed PDE with absorption

=−u

xxxx

− signu in IR × IR ,

where we omit the quadratic term that is negligible near interfaces. The TWs satisfy



=−f

(4)

− sign f, (5.77)

or, neglecting the left-hand side, which is smaller as f → 0, we have a simple ODE

(4)

+ sign f = 0fory > 0, f (0) = 0, so

258 Exact Solutions and Invariant Subspaces

f (y) = y

ϕ(s), with s = ln y, (5.78)

where the oscillatory component ϕ solves

(4)

+ 10ϕ



+ 35ϕ



+ 50ϕ



+ 24ϕ + sign ϕ = 0.

This is precisely the ODE (4.96) for a KS-type equation in Example 3.36, and for

a ﬁfth-order nonlinear dispersion equation in Example 4.10. This indicates a uni-

versality feature of formation of oscillatory patterns for nonlinear PDEs of different

types. The stable periodic solution ϕ(s) (the bold line) is shown in Figure 3.16 (see

also Figure 4.13(b)). Therefore, the TWs (5.78) are oscillatory near interfaces. The

λ-dependent right-hand side in (5.77) determines the next expansion term in (5.78),

which states the dynamic interface equations for the CP (cf. Example 4.29).

5.3.3 Compactons in higher-order nonlinear dispersive Boussinesq equations

We nowdescribe the dynamics of 2π -periodic solutions of the sixth-order dispersive

Boussinesq equation B(m, n, k)

− u

+ α(u

)

+ β(u

)

xxxx

+ γ(u

)

xxxxxx

= 0. (5.79)

Taking m = n = k = 2 yields

− u

+ α(u

)

+ β(u

)

xxxx

+ γ(u

)

xxxxxx

= 0, (5.80)

where the quadratic operator admits W

= L{1, cos x , sin x} iff −α +4β −16γ = 0.

Hence, (5.80) restricted to W

possesses solutions

u(x , t) = C

(t) + C

(t) cos x + C

(t) sin x ,





= 0,



=−(µC

+ 1)C



=−(µC

+ 1)C

where µ = 2(β − α − γ). Compactons are obtained for constant C

(t) ≡ A,so

denoting λ

= µA + 1 > 0, we obtain u(x, t) = A + B cos(x − λt). Setting A = B

yields the compacton

(x , t) =

2(λ

−1)

2(β−α−γ)

cos

[

(x −λt)], (5.81)

which is localized on {|x − λt| <π} and moves with constant speed λ>0, as in

Figure 4.1. The dynamic equation of the interfaces at s(t) = λt ± π is derived from

the PDE, as above, showing that compactons need an FBP setting.

Including the term δu

into (5.80) leads to blow-up behavior on W

.IfC

(t) is

not constant, the dynamics of such periodic solutions are more complicated.

Oscillatory solutions of maximal regularity. The actual maximal regularity of so-

lutions differs from the C

presented by (5.81). Consider the corresponding family

of hyperbolic PDEs of the type (5.80), keeping only the main term

− u



|u|



xxxxxx

(n > 0).

5 Quasilinear Wave and Boussinesq Models. Systems 259

The TWs satisfy, after integrating twice,

(λ

− 1) f = (| f |

f )

(4)

Therefore, for λ = ±1, the behavior near the interface at y = 0isgivenby

f (y) = y

ϕ(s), with s = ln y, (5.82)

where F =|ϕ|

ϕ satisﬁes an ODE of the form (4.93) from nonlineardispersionKdV

analysis (Example 4.10). Hence, there exists a periodic orbit ϕ(s) if |λ| < 1(TWs

are oscillatory) and no such ϕ for |λ| > 1 (non-oscillatory TW interfaces). The same

is true for n = 0.

For n = 1 in (5.82), we have the behavior f (y) = O(y

) as y → 0, so the one-

hump compacton (5.81), exhibiting less regularity ∼ O(y

), is not a solution of the

Cauchy problem in IR × IR . The formula (5.82) gives a ﬁrst approximation of mul-

tiple zeros behavior for such quasilinear hyperbolic equations and can be treated as

a nonlinear counterpart of the linear operator pencil properties detected in Example

5.14. A classiﬁcation of multiple zeros assumes solving a difﬁcult nonlinear eigen-

value

OPEN PROBLEM, which was studied for the second-order quasilinear parabolic

equations and linear 2mth-order PDEs, [226, pp. 29–34].

5.3.4 On the modiﬁed Zabolotskaya–Khokhlov equation

The dissipative Zabolotskaya–Khokhlov (ZK) equation

=−u

xxx

− (uu

)

(5.83)

arises in the theory of acoustic signals propagating through stratiﬁed media and has

other applications, [587, 588, 363]. Actually, (5.83) is a stationary version of the

following full PDE which is also known as the 3D Burgers equation:

+ (uu

)

+ νu

xxx

+ 

⊥

u = 0, (5.84)

where ν is a constant and 

⊥

u denotes u

. Therefore, in (5.83), u = u(x , y) with

y replaced by t.Forν = 0, (5.84) gives the dispersionless Kadomtsev–Petviashvili

equation

+ (uu

)

+ u

= 0.

The ZK equation (5.83) contains a single quadratic operator of the PME type,

with known polynomialor other subspacesthat do not produceinterestingsingularity

phenomena. Consider a modiﬁcation of the ZK equation by adding an extra lower-

order term

=−u

xxx

− (uu

)

− 2u

Thequadraticoperator F[u] = (uu

)

+2u

admitsthe basic trigonometricsubspace

= L{1, cos, sin x}, on which the solutions can be written as

u(x , t) = C

(t) + C

(t) cos(x + γ(t)), (5.85)









=−2C

− C





(γ



)

− 3C





=−



− 1.

(5.86)

260 Exact Solutions and Invariant Subspaces

According to representation (5.85), the speed of propagation of such 2π-periodic

waves is not constant(as in Example 4.3 for some extendedKdV equations), and de-

pends on the solutions under consideration. If solutions blow-up, the speed remains

bounded,which follows from the last ODE in (5.86).Finite interfaces and oscillatory

solutions can occur for PDEs with singular lower-order terms such as

=−u

xxx

− (|u|

p−1

)

, or u

=−u

xxx

−|u|

p−1

where p < 1. These give

OPEN PROBLEMS on existence, uniqueness and regularity.

5.4 Invariant subspaces in systems of nonlinear evolution equations

5.4.1 Main Theorem on invariant subspaces

Here we extendmain results on linear subspaces(Section 2.1) to systemsof evolution

PDEs with vector-valued solutions U = (u

, ..., u

)

. In this case, we deal with

symmetries of systems of linear ODEs

= P(x )y, (5.87)

where x ∈ IR is the independent variable, y = (y

, ... , y

)

is a vector function,

and P(x) is a given n × n square matrix. Denote

F(x , y) =



(x , y), ... , F

(x , y)



and

∂

∂y



∂

∂y

, ... ,

∂

∂y



All symmetries of system (5.87) are point and given by the operators

X = (F(x , y ))

∂

∂y

≡ F

(x , y

, ... , y

)

∂

∂y

+ ... + F

(x , y

, ... , y

)

∂

∂y

(5.88)

with coefﬁcients that are obtained from the invariance criterion

DF(x, y) = P(x )F(x , y).

Here, D is the operator of differentiation via system (5.87), i.e.,

D ≡



(5.87)

∂

∂x

+ (P(x )y)

∂

∂y

≡

∂

∂x

+ (P

(x )y)

∂

∂y

+ ... + (P

(x )y)

∂

∂y

where P

(x ), ... , P

(x ) denote rows of the matrix P(x).

Let  be an n × n fundamental matrix of solutions of system (5.87), i.e., the

matrix, whose n columns are linearly independent solutions of this system. Let  be

the fundamental matrix of the adjoint system

=−(P(x))

where z = (z

, ... , z

)

. According to ODE theory [132, Ch. 3], the product 



is a constant matrix and the components I

, ... , I

of the column

I = 

y (5.89)

give a system of independent integrals of system (5.87). Similarly to the study of

single equations in Section 2.1, the following result is obtained.

5 Quasilinear Wave and Boussinesq Models. Systems 261

Theorem 5.17 (“Main Theorem”)

The fullalgebraof symmetriesof

(5.87)

is given

by Lie-B¨acklund operators

(5.88)

with coefﬁcients

F(x , y) =  A(I),

where

A(I) = (A

, ..., I

), ..., A

, ..., I

))

is a column of

arbitrary smooth

functions.

The fundamental matrix of the adjoint equation can be chosen as follows:

 = (

−1

)

Then the product ()

 = E is the identity matrix and integrals (5.89) are

I = 

−1

y. (5.90)

5.4.2 Examples: applications of the Main Theorem

We begin with simple examples illustrating Theorem 5.17.

Example 5.18 Consider the following system of ODEs:

= y

=−y

. (5.91)

Its fundamental matrix takes the form

 =

cosx sin x

−sin x cos x

⇒ 

−1

= 

cosx −sin x

sin x cos x

and we ﬁnd the integrals by formula (5.90),

= y

cosx − y

sin x and I

= y

sin x + y

cos x. (5.92)

According to Theorem 5.17, the full algebra of symmetries of system (5.91)is given

by operators

X = F

(x , y

, y

)

∂

∂y

+ F

(x , y

, y

)

∂

∂y

, where

= A

, I

) cos x + A

, I

) sin x,

=−A

, I

) sin x + A

, I

) cos x ,

(5.93)

and A

(1)

and A

(2)

are arbitrary smooth functions of integrals (5.92).

Let us determine the symmetries that are independent of x, i.e., satisfying

∂ F

∂x

∂ F

∂x

= 0.

This yields the system



− I

= A

− I

=−A

which determines A

and A

. Plugging these into (5.93), we obtain

(I ) +

−y

(I ),

where I = (y

)

+ (y

)

and B

(I ), B

(I ) are arbitrary smooth functions.

262 Exact Solutions and Invariant Subspaces

Example 5.19 (Reaction-diffusion system) Consider the system



= u

+ v B(u

+ v

= v

− uB(u

+ v

(5.94)

where B is an arbitrary function. Using the result of the previous example yields that

the operator on the right-hand side admits W

= L{f

, f

}, where

cosx

−sin x

and f

sin x

cosx

This makes it possible to look for solutions

= C

(t)

cos x

−sin x

+ C

(t)

sin x

cosx

. (5.95)

Substituting (5.95) into (5.94) yields



+ C



=−C

− C

+ (−C

+ C

)B(C

+ C

), or





=−C

+ C



+ C





=−C

− C



+ C



Solving this DS and using (5.95) gives

= D

−t

cos(x +

t + D

)

−sin(x +

t + D

)

, where

t =



B(D

−2s

) ds,

where D

and D

are arbitrary constants.This solutionis invariantunder the operator

X =

∂

∂x

− u

∂

∂v

+ v

∂

∂u

Example 5.20 (Nonlinear Schr

odinger equation)Thefollowingsystemof second-

order PDEs:



=−v

− νv(u

+ v

= u

+ ν u(u

+ v

is the real form of the cubic nonlinear Schr

odinger (NLS) equation

∗

+ z

+ ν |z|z = 0, (5.96)

where z = u + iv and u, v are real-valued functions. It is straightforward that the

operator on the right-hand side admits W

= L{f

, f

}, where

cos x

sin x

and f

−sin x

cos x

Therefore, we look for solutions

= C

(t)

cosx

sin x

+ C

(t)

−sin x

cos x

. (5.97)

∗

Its derivation goes back to Da Rios, 1906.