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

Подождите немного. Документ загружается.

9 Invariant Subspaces for Discrete Operators 453

has the solutionsU(x , t) = C

(t)+C

(t) f (x ) on W

with the coefﬁcientssatisfying

a DS, which is similar to (9.95),





= 

, C



= 

, C

(9.107)

By approximation, 

and 

converge as h → 0tothecoefﬁcients of the DS

(9.95).

Temporarily assuming that the mesh is ﬁxed and uniform, with a given h > 0, we

introduce the natural approximations of the spatial derivatives



i+1

−U

i−1



, u



i+1

+ U

i−1

− 2U



, ... . (9.108)

Substituting those into operators F

– F

yieldsthe correspondingdiscrete operators

– F

11h

. As was showninthe previoussection, the polynomialinvariant subspaces

then remain the same for discrete operators. Concerning trigonometricand exponen-

tial subspaces, operator F

in (9.97) admits

= L{1, cos(λx )}, with

sin

(λh) = 1, (9.109)

and F

in (9.99) admits the same subspace as in (9.109), where λ satisﬁes

sin

(

λh

)



1 +

cos

(

λh

)



= σ. (9.110)

The exponential subspace

L{1, cosh(λx )}, with λ = λ(h), (9.111)

exists for the fourth-order operator F

in (9.100), etc.

Example 9.31 Consider the discrete equation (9.102). On W

given by (9.109),

it reduces to the DS (9.54), with a = b = 0, for the coefﬁcients {C

, C

} of the

expansion (9.103). For h & 1, the parameter λ(h) of the basis function satisﬁes

(9.55), so that λ(h) →

√

σ +1

as h → 0

, and, in the limit, the discrete DS (9.54)

coincides with the continuous system (9.104).

Forstandard approximationsof derivatives(9.108),subspaces(9.109),(9.111) can

be invariant on uniform ﬁxed meshes only. In order to preserve this invariance on

non-uniform MMs, different approximations have to be used.

Invariance of moving mesh operators. We are going to show that invariant sub-

spaces can be preserved by MM operators. Consider a general MMM for (9.90).

Assume that, given initial data u

, we approximate the evolution orbit {u(t), t > 0}

by the approximate solution U (t) ={U

(t), i ∈ I }, I ={1, 2, ..., K },deﬁned on

the ﬁnite movingmesh (U) ={X

(t), i ∈ I ; t > 0}. Here, U satisﬁes the moving

mesh equation (MME)



[U]



= U



− [D



− F

[U] = 0, (9.112)

where B

is the moving mesh operator (MMO) deﬁned at the internal points i ∈

={2, ..., K − 1}⊂I . Assume that suitable initial data are prescribed at t = 0,

and boundary conditions are given on the lateral boundary (U ) of (U ), i ∈ I

(if

S = IR ), so that the solution U is well-deﬁned locally in time on (U).Foraﬁxed

454 Exact Solutions and Invariant Subspaces

τ ≥ 0, we denote 

(U) = (U) ∩{t = τ}.By∂(U) we denote the parabolic

boundary of (U ), i.e., that part of the boundary where initial and boundary data

are prescribed.

We will study the invariant properties of the MMO B

. We do not take into ac-

count the mechanism of generation of the moving mesh (U), which is crucial for

construction of MMMs, [293, 292, 99]; see also applications to blow-up problems

in [88, 87]. We construct MMEs with an arbitrary MMs (U ), which possess so-

lutions on the subspaces of the discrete nonlinear operators F

. A special choice of

MMs can improve invariant properties of the MMMs. There are examples, for which

the requirement of optimal approximations of invariant subspaces (forming a stable

attractor of the givenﬂow) plays an important role in the correct description of evolu-

tion properties of PDEs. Moreover, this may be the decisive principle of construction

of the corresponding MMOs.

We restrict ourattention to the problem of invariantsubspacesofthe MMO (9.112).

We will deal with 2D subspaces introduced above.

Theorem 9.32 Assume that, given moving mesh (U ), the discrete operator F

(U ) admits the 2D subspace W

= L{1, f (x )}, with a smooth function f . Let the

discrete gradient operator satisfy

f (X )]

= f



) on (U). (9.113)

Then B

: W

→ W

and the MME (9.112) on (U) is equivalent to a nonlinear

2D dynamical system that coincides with (9.107) for the corresponding ﬁxed mesh.

Forinstance,wecanuse the followingdiscretegradientoperatorsatisfying(9.113):

i+1

−U

i−1

f (X

i+1

)− f (X

i−1

)



). (9.114)

On the mesh satisfying |X

i+1

− X

i−1

|=O(h), it approximates

, with at least

order O(h).

Proof of Theorem 9.32. We are looking for a solution of (9.112) in the form of

= U(X

(t), t) = C

(t) + C

(t) f (X

(t)) ∈ W

. (9.115)

Plugging (9.115) into (9.112) yields



+ C



f (X

) + C





) − [D

f (X )]





= F

+ C

f (X ) ∈ W

(9.116)

In view of (9.113), the term depending on X



on the left-hand side of (9.116) van-

ishes, and, due to (9.106), the DS (9.107) is obtained.

We consider evolution PDEs with the above nonlinear operators, F

– F

,and

show how to construct suitable discretizations F

of the operators to get solutions of

the MMEs on the invariant subspaces.

9.4.3 Invariant subspace L{1, x

}

We have presented a number of such subspaces with m = 2, 3, or 4.

9 Invariant Subspaces for Discrete Operators 455

Semilinear operator F

. Consider the PDE (9.40) with the operator (9.96). The

corresponding MME (9.112) has the form



− [D



= F

[U] ≡ U

+ (U

)

on (U), (9.117)

with the approximation F

to be determined later. In view of (9.114) with the func-

tion f (x) = x

, the gradient operator is

i+1

−U

i−1

i+1

−X

i−1

, (9.118)

so [D

]

= 2X

by (9.113). Consider the discrete operator F

admitting W

the discrete derivatives are given by

= D

U and U

= d

U, (9.119)

where d

denotes the standard symmetric gradient approximation,

U =

i+1

−U

i−1

i+1

−X

i−1

, so that d

= 2. (9.120)

Finally, given the mesh function U = C

(t) + C

(t)X

∈ W

, the following holds:

[U] = C

+ 2C

+ C

(1 + 4C

∈ W

, (9.121)

so that equation (9.117) on W

is the DS (9.42). Hence, the discrete evolution on W

coincides with the differential one induced by the PDE (9.40) on W

On conditionally invariant MMOs. Observe that, in general, the natural approx-

imation U

= D

U does not satisfy the second condition in (9.120). One can

calculate that





i+1

i−1

and, therefore, taking the condition D

= 2 as one of the generating conditionsof

the MMs, we arrive at X

i+1

i−1

), i.e., the MM hasto be uniform. Hence,

on any uniform MM (U), the MME (9.117) with U

= D

U admits solutions

on W

. This MME is conditionally invariant, i.e., not for every MM. On the other

hand, the invariance of the standard approximation U

= d

U implies that





≡

i+2

−X

i−2

i+1

−X

i−1

= 2,

which is true for uniform MMs (U). Finally, the invariant MMO in (9.117) leaves

the extended subspace W

= L{1, x, x

} invariant on any uniform mesh, since

the discrete gradient operator correctly differentiates the linear function x on such

meshes, [D

i+1

−X

i−1

= 1.

Porous medium operators F

, F

,andF

. Consider the PDE

= uu

)

+ αu + β on W

= L{1, x

}. (9.122)

Using (9.118) and (9.119), we introduce the MME on (V )



− [D



= F

[U] ≡ U

U +

+ αU

+ β,

which admits solutions U = C

(t) + C

(t)X

. The expansion coefﬁcients {C

, C

}

456 Exact Solutions and Invariant Subspaces

solve the same DS as in the differential case,





= 2C

+ αC

+ β,



2(σ +2)

+ αC

The invarianceresultis truefor the extension W

on any uniformmesh. The subspace

remains invariant if we add any linear differential operator L[u] = γ u

+δu

→ W

to the right-hand side of (9.122).

The fourth-order operator F

. For the quadratic operator (9.101), the invariant

approximation of the gradient operator is

i+1

−U

i−1

i+1

−X

i−1

Then the discrete DS is the same as for the PDE u

= F

[u]onW

. Exact solutions

of both the continuous and discrete DS are u = C

+ C

and U = C

+ C

where





=−24C

+ γ C

+ δ,



= (−24 + 96α + 144β)C

+ γ C

Any operator F

+ L, for any linear higher-order operator L with constant coefﬁ-

cients, admits the same subspace. This also applies to other higher-order operators

on polynomial subspaces.

9.4.4 Invariant subspace L{1, cos x}

Unlike the polynomials, the trigonometric (and exponential) functions need special

discrete approximations.

Semilinear operator F

. The MME, corresponding to the evolution PDE

= u

+ (u

)

+ u

on W

= L{1, cos x }, (9.123)

has the form



− [D



= F

[U] ≡ U

+ (U

)

on (U ), (9.124)

where, by (9.114),

i+1

−U

i−1

cos(X

i+1

)−cos(X

i−1

)

(−sin(X

)). (9.125)

Then [D

cos X]

=−sin(X

) by (9.113), and we choose U

= D

U.Inorderto

keep invariant properties of the second derivative, we take the approximation

= D

U, with [D

i+1

−U

i−1

sin(X

i+1

)−sin(X

i−1

)

cos(X

). (9.126)

Therefore, [

cos X]

=−cos(X

). Finally, the invariant MMO (9.124) pre-

serves the continuous evolution on W

. For the solutions

U = C

(t) + C

(t) cos X ∈ W

, (9.127)

the DS is the same as for PDE (9.123),





= C

+ C



=−C

+ 2C

9 Invariant Subspaces for Discrete Operators 457

The DS is not explicitly solved, but is easily studied on the phase-plane.The equation

(9.123) describes regional blow-up, where the blow-up sets of bell-shaped solutions

have measure 2π; see [245, Ch. 9]. On any uniform MM,

sin X]

= cos X

and [D

cos X]

=−sin X

so that the operator F

with U

= D

U and U

= D

U admits the extended

subspace W

= L{1, cos x , sin x}.

Quasilinear operators F

and F

. As the next example, consider the equation

(9.90), (9.99) in the case where F

admits W

= L{1, cos x },sothat

λ =

√

σ +1

= 1, i.e., σ = σ

(1 ±

√

(note that −σ

−

is the Golden Mean). The MME on (U ) is



− [D



= F

[U] ≡ U

U +

+ σ U

where the derivativesare givenin (9.125) and (9.126).This MME exactlyreproduces

the continuous solutions (9.127) and the corresponding DS is





= σ C



= (2σ − 1)C

On uniform MMs, the MMO approximates the corresponding extended subspace

= L{1, cos x, sin x}. A similar analysis applies to the Kuramoto–Sivashinsky

equation, u

= F

[u], and to many other PDEs in Section 3.1 on trigonometric

subspaces.

9.4.5 Higher-order evolution equations

We describe invariant properties of the equation (9.92). Setting

u = v

for j = 1, ..., m − 1,

reduces it to the system of PDEs



= v

)

= v

j +1

, j = 1, ..., m − 2,

m−1

)

= F



u,v

, ..., v

m−1

, u

, ..., D



Introducing the variable W = (u,v

, ..., v

m−1

)

∈ IR

yields the DS

F[W ], t > 0.

The corresponding MME for W takes the standard form in IR

similar to (9.112),



− [D

W ]



[W ].

By checking the invariant property of the gradient operator, we see that Theorem

9.32 is true, and the gradient approximation (9.114) preserves invariance of W

Example 9.33 (Boussinesq-type PDE) Consider the PDE with operator F

= uu

)

+ αu + β.

458 Exact Solutions and Invariant Subspaces

It is equivalent to the system



= v,

= F

[u] + αu + β,

and the corresponding MME is





− [D



= V



− [D

V ]



= F

[U] +αU

+ β.

We deﬁne the gradient operator by (9.118), and then, by Theorem 9.32, the MMO

admits the subspace W

= L{1, x

9.4.6 Positivity and comparison

In order to use solutions on invariant subspaces, we ﬁrst need to establish the Max-

imum Principle aspects of the MMMs for the second-order parabolic PDEs. Let us

begin with the positivity property.

Weak Maximum Principle. We return to the MMO (9.112) and consider second-

order equations of parabolic type, assuming a natural three-point approximation of

the nonlinear operator F,so

[U]]

= F

i−1

, U

i+1

, i, X).

Theorem 9.34 Let U(t) ≥ 0 on ∂(U ). Then the inequality





+ F

[U]





≥ 0 (9.128)

for all i ∈ I

={2, ..., K − 1} and U

≥ 0 for i = j, t > 0, is necessary and

sufﬁcient for the positivity of U(t) with arbitrary nonnegative data on ∂(U).

Proof. It follows from (9.112) that, at any instant, when U

= 0 at an internal point

of (U), we have to have U



≥ 0, which gives the desired result.

This applies to other homogeneous boundary conditions, including Neumann’s,

= 0 on the lateral boundary of (U ).

The criterion in the theorem can easily be checked for invariant or standard MMOs.

Example 9.35 (Semilinear parabolic PDEs) Consider

= u

+ g(x , t, u, u

where g satisﬁes g(x, t, 0, u

) ≡ 0. We take the standard MME on the MM (U ),



− [D



= U

+ g(X

, t, U

, U

The difference operators are

≡ [D

i+1

−U

i−1

and U

i−1



i+1

−U

−

−U

i−1



where h

= X

i+1

− X

> 0. Then the condition (9.128) reads

i+1

−U

i−1



i−1



i+1

i−1



≥ 0, or

i+1







i−1



i−1

− X





≥ 0

9 Discrete Operators 459

for any U

i±1

≥ 0. This gives the criterion of the (internal) positivity of the solution

−

≤ X



≤

i−1

for i ∈ I

, t > 0.

Both inequalities are valid if



(t)|≤2min



i−1



for i ∈ I

, t > 0, (9.129)

i.e., if the MM does not move very fast. Thus any MM moving sufﬁciently slow

preserves positivity of the solution just as the stationary mesh does. This is a nat-

ural continuity property of discrete parabolic operators that describe a transitional

behavior from ﬁxed meshes to the MMs.

Comparison principle. This is a consequence of the positivity result.

Theorem 9.36 Let U and W be two solutions of the MME with (U ) = (W ).Let

U ≥ Won∂. The inequality



(U − W)]



+ F

[U] − F

[W ]





≥ 0 (9.130)

for all i ∈ I

and U

≥ W

for i = j, t > 0, is necessary and sufﬁcient for the

comparison U ≥ Win for arbitrary ordered data on ∂.

Proof is straightforward, and the result is equivalent to the positivity of the differ-

ence U − W ≥ 0in satisfying a linearized MME. Since the proof is essentially

of the interior nature, comparison is true for the homogeneous Neumann conditions

or monotone nonlinear Neumann conditions preserving ordering of solutions on the

lateral boundary.

Example 9.37 Consider the semilinear heat equation

= u

+ g(x , t, u).

Taking the standard MME as given in Example 9.35, one obtains that (9.129) guar-

antees comparison of solutions of this MME on any such coinciding MMs.

On asymptotic properties of MMEs. The standard comparison of different solu-

tions U and W assumes that the corresponding MMs, (U) and (W), coincide.

Since the MMs depend on the solutions, this makes the comparison a delicate mat-

ter. Using invariant MMOs makes such a comparison not only possible, but also is

an effective way to study asymptotic properties of discrete equations. By invariant

MMOs, we actually compare general solutions with exact solutions taken on the

same MMs. The main conclusion directly follows from the invariance Theorem 9.32

and the comparison Theorem 9.36.

Theorem 9.38 Let the hypotheses of Theorem 9.32 hold. Fix a solution U on (U )

and an exact invariant solution W (t) ∈ W

for t ≥ 0 such that

U ≥ W (U ≤ W ) on ∂(U ).

Assume that the hypothesis (9.130) is true for solutions that are deﬁned on the same

MM  = (U). Then,

U ≥ W (U ≤ W ) in (U).

460 Exact Solutions and Invariant Subspaces

Example 9.39 (Comparison and asymptotic behavior) Consider the semilinear

equation (9.40). Let U be a discrete solution of the MME on an MM (U ),



− [D



= U

+ (U

)

+ U

with the gradient approximation D

given in (9.118), and invariant approximations

(9.119)of the derivativesU

and U

. The quadratic operator F

on the right-hand

side admits the same subspace as in (9.96), with the basis function f (X) = X

. Let

W (t) ∈ W

be an exact solution. By Theorem 9.32, this solution satisﬁes the same

MME on the MM (U). The comparison condition then reduces to

i+1

− W

i+1

)







+ W

)



+(U

i−1

− W

i−1

)



−



i−1

−

+ W

)



≥ 0

for all U

i±1

≥ W

i±1

,where





)

f (X

i+1

)−f (X

i−1

)

= X

i+1

− X

,andH

+ h

i−1

). This gives the following condition of

comparison:





(t)|≤min



i−1



−

sup



(|U

+ W

|),

i.e., if, for sufﬁciently regular solutions with bounded derivatives U

and W

,the

MM does not move faster than O





(similar to the positivity (9.129)).

The above comparison is true for any solution on W

. Choose the solution

W (x, t) = C

(t) + C

(t)x

∈ W

where the coefﬁcients {C

, C

} satisfy the DS (9.42). Assume that U ≥ W on the

lateral boundary of (U),sothat

U(X, 0) ≥ W (X, 0) ≡ C

(0) + C

(0)X

in 

(U).

Then, by comparison,

U(X, t) ≥ W (X, t) ≡ C

(t) + C

(t)X

in (U),

provided that the MM satisﬁes the above comparison condition. A similar estimate

from belowis also proved.Finally, solving the DS (9.42), we obtain the exactasymp-

totic behavior as t →∞of a wide class of solutions of this invariant MME. In this

case, as t →∞, the discrete solutions converge to the principle separable solution,

as in the easy continuous case of the parabolic PDE, [509, p. 93].

Moreexampleson the comparisonwith solutions on invariantsubspacesand asymp-

totic behavior can be constructed for parabolic PDEs and invariant approximations

of other quadratic operators given above.

9.5 Applications to anharmonic lattices

In this section, we present examples of exact solutions of nonlinear lattices. The

dynamics of such discrete systems have been extensively studied since the 1950s,

9 Invariant Subspaces for Discrete Operators 461

starting with the celebrated work of Fermi–Pasta–Ulam [183]. In fact, the soliton

concept appeared for the ﬁrst time in the context of nonlinear lattices, before becom-

ing widely used in many areas of physics and mechanics; see a history review in

Remoissenet [487]. Another classical example is the Toda lattice



= e

n−1

−u

− e

−u

n+1

It was introduced by M. Toda in 1967 and had an earlier geometric origin detected by

Darboux(1887)via iterationsof, in modernterminology,Laplace–Darbouxtransfor-

mations, [141]. The Toda lattice was used to describe anharmonic oscillations in a

1D crystal lattice. This is known to be an integrable Hamiltonian system admitting a

Lax pair, N-soliton solutions given by Hirota’s expression, etc.; see Toda [556]. The

related discrete sine-Gordon (FK) model was discussed in Example 5.5.

Lattice dynamical systems occur in a variety of applications, where the spatial

structure has a discrete character. These apply in chemical reaction theory, cellular

neural networks, image processing, and pattern recognition, in different areas of ma-

terial science, in electrical engineering in laser physics, etc. Discrete breathers, i.e.,

time-periodic patterns of lattice dynamics with spatially localized oscillations, are

supported by many nonlinear lattices, and occur in several physical models, such as

dynamical properties of crystals, Josephson junction arrays, DNA denaturation, vi-

bration dynamics of ionic crystals, etc. See [304] for further references and a survey

on mathematical results in this area of lattice theory.

Example 9.40 (Parabolic quadratic equation)Considera quadraticoperator,which

was studied in Example 5.6 in the differential case,

[u] = 2|u

+ u

. (9.131)

In computations,we use the followingsymmetricexpressionsfor discrete derivatives

(later on, we often set h = 1):



n+1

− u

n−1



, u

= (u

)

≡



n−1

− 2u

+ u

n+1



xxx

= (u

)

≡



−u

n−2

+ 2u

n−1

− 2u

n+1

+ u

n+2



xxxx

= (u

xxx

)

, u

xxxxx

= (u

xxxx

)

, etc.

(9.132)

The results can easily be recalculated for other derivative approximations. Some ap-

proaches are extended to uniform lattices in IR

, e.g., for the operator

[u] = 2|∇u|u + u

with natural deﬁnitions of the discrete gradient and the Laplacian operators.

The discrete operator F

in (9.131) is associated with the 2D subspace

= L{1, f (x)}, with f

= f (n), (9.133)

where f solves the equation

[ f ] − f

= 0. (9.134)

Consider the corresponding discrete parabolic equation



= F

[u]. (9.135)

462 Exact Solutions and Invariant Subspaces

Plugging the expression

(t) = C

(t) + C

(t) f

∈ W

, with C

≥ 0 (9.136)

(in fact, the cone, K

={C

1,2

≥ 0}⊂W

, is invariant under F

) into (9.135) yields

the same DS as in the differential case,





= C



= C

+ 2C

Setting C

(t) ≡ 0 yields C



= C

and the separate variable solution

(t) =

T −t

where T > 0 is the blow-up time. Such separation of variables in ﬁnite-difference

equations is well known in blow-up theory; see [509, p. 491] and references therein.

Choosing a nontrivial C

(t) from the ﬁrst ODE, C

(t) =−

, we obtain more inter-

esting blow-up solutions on such a uniform grid,

(t) =−

(T −t)t

On inﬁnite propagation in the lattice. It is easy to see that implicit schemes such

as (9.135), with operator (9.131), cannot describe processes with ﬁnite propagation.

For the stationary (“elliptic”) equation (9.134), this means that f

= f (n) is not

compactly supported, unlike in the differential case in Example 5.6. The asymptotic

decay of the monotone fast decreasing solution {f

> 0} for large n is easy to obtain

from the following asymptotic representation of (9.134):

= F

[ f ] = f

n−1

(1 + O( f

)) for n  1,

where we have used sharp estimates f

( f

n+1

− f

n−1

) ≈−

n−1

and f

n−1

− 2 f

+ f

n+1

≈ f

n−1

. Hence, for some constants c

∗

∈ (0, 1) and C > 0,

∼ C(c

∗

)

as n →∞.

We thus observe a fast double-exponentialdecay of the function f (n).

Concerning the discrete parabolic equation with a gradient dependent diffusion

(we omit the quadratic term u

that is negligible on such decay asymptotics)



=|u

n+1

− u

n−1

|(u

n−1

+ u

n+1

− 2u

), (9.137)

assuming that the initial data {u

} are compactly supported, we ﬁnd that, for small

t > 0asn →∞, the sharp asymptotic form of the equation is given by



= u

n−1

(1 + O(u

)). (9.138)

Let n

> 0 be the interface point of u

,sothatu

= 0andc

= u

−1

> 0. In

this case, (9.137) at n = n

yields that

(t) = c

t (1 +o(1)) as t → 0. (9.139)

Further iterations of the equation (9.138) leads to the following asymptotic behavior

as k = n − n

→∞:

(t) =





i=0

i+2

− 1)

−2

k−i



+ ... . (9.140)