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9 Invariant Subspaces for Discrete Operators 463

Therefore, the formula

(t) ∼ e

lnt

→ 0

gives the sharp asymptotic rate of convergenceof the solution as t → 0tocompactly

supportedinitial data u

. This has an easy relation to asymptotics (1.61)of the delay

ODE (1.60).

Example 9.41 (Oscillatory behavior near “interfaces”) Considerthe higher-order

parabolic lattice (9.135) with the operator

[u] =−(|u

)

+ u

which uses the central differences as in (9.132). This model is associated with the

differential analogy in Example 5.7 (for n = 1), where we have detected a speciﬁc

oscillatory behavior of weak solutions near interfaces. Apparently, there occur the

same subspace and a DS on W

,where f solves (9.134).

Studying the decay behavior for n  1, we have, instead of (9.138),the following

asymptotic equation for small solutions:



=−|u

n−2

. (9.141)

Posing, similar to (9.139), the positive conditions at two “boundary” points

(t) = c

t, u

(t) = d

t as t → 0 (0 < d

< c

we obtain from (9.141) the following asymptotic behavior as t → 0:

(t) =−

< 0, u

(t) =−

< 0,

(t) =





> 0, u

(t) =





> 0,

(t) =−





2·2





< 0, u

(t) =−





2·2





< 0,

etc. This shows a clear oscillatory structure of solutions, and, writing the general for-

mula as in (9.140) (with 2

→ 2

k/2

), we will observe a rescaled 2-periodic behavior

as n →∞. This is a discrete analogy of solutions of changing sign in the differential

counterpart in Example 5.7.

Example 9.42 (Quadratic hyperbolic equation) Consider the hyperbolic lattice

with the second-order operator (9.131),



= F

[u] − u

Using the subspace (9.133), we obtain solutions (9.136) with the DS





= C

− C



= C

+ 2C

− C

≥ 0). The analysis of inﬁnite propagation with compactly supported initial data

, u

} is performed in a similar fashion, where the governing equation is



= u

n−1

(1 + O(u

)) for n  1,

that admits asymptotics as in (9.140).

464 Exact Solutions and Invariant Subspaces

Example 9.43 (Cubic hyperbolic equation) The next lattice is associated with the

PDE (5.52) possessing exact solutions on the invariant subspace

= L{1, cos(λx ), sin(λx )}, with λ>0.

We now take the corresponding lattice in the form of



= F

[u] ≡ u

n−1

+ u

n+1

− 2u

) − α u



n+1

−u

n−1



+ βu

. (9.142)

Proposition 9.44

The discrete operator

(9.142)

preserves

α =

1+cosλ

and

β = 1 − cosλ(λ= π(2k + 1)).

Therefore, the dynamics on W

is as follows:

(t) = C

(t) + C

(t) cos(λn) + C

(t) sin(λn),



= (1 −cos λ)C

− C

), i = 1, 2, 3.

For convenience, we present the results of calculations for the general discrete

cubic operator in Proposition 1.17. Here λ = 1.

Proposition 9.45

The discrete operator

(1.57)

admits

= L{1, cos x, sin x }



+ b

− b



, b



−b

+ 3b

+ 2b

− 3b





−



where

sinh (= sin 1

for

h = 1)

and

sin

(

)(= 4sin

(

))

Example 9.46 (5th-order PDE from compacton theory) Consider the nonlinear

dispersion lattice from compacton theory in Section 4.3 (equation (4.97), α = 1)

= F

[u] + δu + ε ≡ (u

)

xxxxx

+ β(u

)

xxx

+ γ(u

)

+ δu + ε. (9.143)

We will ﬁrst study it on the same subspace, W

, as in Proposition 9.45. For con-

venience of passing to the limit, we introduce the mesh parameter h > 0inthe

deﬁnition of discrete operators in (9.132). Then the following results hold:

(i) W

is invariant if

− Q

β + γ = 0, where now Q

sin

(

λh

). (9.144)

Therefore, Q

→ 4ash → 0, so, in the limit, the invariance condition (9.144)

coincides with the differential one (4.12) with α = 1.

(ii) TheDS for the lattice (9.143)on W

is (4.98) with the coefﬁcientsµ = 2R

−

)(β − Q

− Q

) (we again use the step h), where R

sin(λh). Hence, µ →

6(β − 5) as h → 0 as in (4.98). The exact solutions are the same as in (4.99) and

(4.100), so these are not presented here.

Concerning 5D subspaces, let us mention that, as in Example 4.13, the discrete

equation (9.143) possesses solutions

(t) = C

(t) + C

(t) cosn + C

(t) sin n + C

(t) cos2n + C

(t) sin 2n,

9 Invariant Subspaces for Discrete Operators 465

provided that γ = Q

and β = Q

(here Q

→ 9andQ

→ 16 as h → 0),

and the DS for δ = ε = 0is













= 0,



= µ(C

− C

+ 2C



=−µ(C

+ C

+ 2C



= 2ν(2C

+ C



= ν



− C

− 4C



The coefﬁcients µ and ν satisfy µ = (Q

− Q

)(Q

− Q

) → 120 and ν =

− Q

)(Q

− Q

) → 60 as h → 0, so in the limit h = 0, we obtain the

DS (4.104) for the continuous PDE. The dynamical evolution properties of such

solutions on W

are quite similar in both continuous and discrete cases, so one can

observe the discrete TW (4.105).

Example 9.47 (4th-order PDE from thin ﬁlm theory) In this last example, we

consider the lattice counterpart of the thin ﬁlm PDE in Example 3.17,

=−uu

xxxx

− u

xxx

+ βu

Then L{1, cos x , sin x} is invariant if β = Q

+ R

) → 2ash → 0, and for

solutions

(t) = C

(t) + C

(t) cosn + C

(t) sin n,

we obtain the DS (cf. (3.117) for h = 0)





= Q



+ R



+ R



= µC

, C



= µC

(9.145)

where µ = Q

+ 2R

) → 3. For C

= C

and C

= 0, we have the ODE



= µC

and the following localized blow-up pattern on this lattice:

(t) =

µ(T −t )

cos

(

Remark 9.48 1. (Other approximations) Using another discrete derivative u

,in-

stead of (9.132), slightly changes the coefﬁcients of the resulting DSs according to

the easy rule



n+1

− u



⇒ (sinλx)

= R

cos x −

sin x .

2. (Exponential functions) For subspaces composed of exponential or hyperbolic

functions, the calculations yield

λx

)

= Q

λx

, where Q



λh

− e

−λh



3. (Polynomial subspaces) These were especially important for a number of PDE

examples in the previous chapters. All the computations are easily translated to the

discrete case using the rule

)



(x +h)

− (x −h)





k=1



1 + (−1)



n−k

k−1

Of course, this affects the coefﬁcients of the DSs.

466 Exact Solutions and Invariant Subspaces

Remarks and comments on the literature

There are many classical texts on difference equations; see references in [367, p. 34, p. 72].

Concerning applications of Lie group theory to difference equations, see lists of references in

[151, 152, 490].

§ 9.1, 9.2. The main results are taken from [242].

§9.3.Some of the results were published in [221]. There exists a group-invariant ﬁnite-

difference scheme for (9.40), which admits explicit solutions, and then the corresponding

space grid depends on the time variable (and on the solution itself); see [23]. Some solutions

in this section can be justiﬁed by generalized conditional symmetries, [119].

It has been known for a long time that, for a number of fully integrable PDEs, there exist

techniques of their discretization without destroying integrability, where the discrete equations

admit N-soliton solutions that are obtained by similar methods over linear spaces of exponen-

tial functions. The idea of such discretization techniques was proposed by Hirota in 1973; see

[286], where the extra references can be found. Algebro-geometric and elliptic solutions of the

fully discretized KP and 2D Toda equations were constructed in [357] by using Baker func-

tions. One can expect that, for non-integrable PDEs possessing one and two-soliton solutions

similar to those in Section 8.5 and in [100, 329, 595], there exist discretizations preserving

such exact solutions constructed by similar manipulations with exponential functions.

§9.4.The main results are available in [222]. The basics of MMM theory are given in [88,

293, 292, 99, 98], some applications to blow-up solutions of higher-order parabolic PDEs can

be found in [87].

§9.5.Discrete “almost compact” breathers of latticesassociated with cubic PDEs were studied

in [500]. More recent references on exact TW, and other solutions of lattice differential equa-

tions can be found in [291]. References to main results and applications of lattice dynamical

systems in chemical reaction theory, image processing, pattern recognition, material sciences,

and biology are given in [124]. The most well-studiedexact solutions of lattice parabolic PDEs

are TWs; see a more recent survey [408] and also [404], where non-local models were treated.

Concerning integrable cases, complexiton-type exact solutions of the Toda lattice were con-

structed in [406]. Notice a relation of embedded solitons in dynamical lattices [258] to typical

invariant subspaces and sets.

For higher-order linear and nonlinear PDEs, using various approximating discrete dynam-

ical systems via Saint-Venant’s principle (originally formulated by A.-J.-C. Barr´edeSaint-

Venant in linear elasticity theory in 1855 [508]) is an effective tool for existence, uniqueness,

and asymptotic analysis. This discrete approach leads to concepts of energy solutions, and can

be applied in general singular nonlinear cases. We refer to the fundamental Oleinik–Radkevich

paper [443]; see also a survey on more recent results and applications in [240].

Open problems

• The general forward Problem I of invariant subspaces (and several other related

aspects): given a nonlinear ﬁnite-order difference operator, determine all the linear

subspaces which it admits; see Section 9.2.

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