Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2011, Article ID 452087, 13 pages
doi:10.1155/2011/452087
Research Article
Random Attractors for the Stochastic Discrete
Long Wave-Short Wave Resonance Equations
Jie Xin and Hong Lu
School of Mathematics and Information, Ludong University, Shandong,
Yantai 264025, China
Correspondence should be addressed to Jie Xin, fdxinjie@sina.com
Received 21 June 2011; Accepted 15 August 2011
Academic Editor: F. Marcellán
Copyright q 2011 J. Xin and H. Lu. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We prove the existence of the random attractor for the stochastic discrete long wave-short wave
resonance equations in an infinite lattice. We prove the asymptotic compactness of the random
dynamical system and obtain the random attractor.
1. Introduction
There has been considerable progress in the study of infinite-dimensinal dynamical systems
in the past few decades see 1–5. Recently, the dynamics of infinite lattice systems has
attracted a great deal of attention from mathematicians and physicists; see 6–11 and the
references therein. Various properties of solutions for lattice dynamical systems LDSs have
been extensively investigated. For example, the long-time behavior of LDSs was studied in
5, 10. Lattice dynamical systems play an important role in their potential application such as
biology, chemical reaction, pattern recognition and image processing, electrical engineering,
and laser systems. However, a system in reality is usually affected by external perturbations
within many cases that are of great uncertainty or random influence. These random effects are
introduced not only to compensate for the defects in some deterministic models but also to
explain the intrinsic phenomena. Therefore, there is much work concerning stochastic lattice
dynamical systems. The study of random attractors gained considerable attention during
the past decades; see 12 for a comprehensive survey. Bates et al. 13 first investigated the
existence of global random attractor for a kind of first-order dynamic systems driven by white
noise on lattice Z; then, Lv and Sun 14 extended the results of Bates to the dimensional
lattices. Stochastic complex Ginzburg-Landau equations, FitzHugh-Nagumo equation, and
2
Journal of Applied Mathematics
KGS equations in an infinite lattice are studied by Lv and Sun 15, Huang 16, and Yan et
al. 17, respectively.
The long wave-short wave LS resonance system is an important model in nonlinear
science. Long wave-short wave resonance equations arise in the study of the interaction of
surface waves with both gravity and capillary modes present and also in the analysis of
internal waves as well as Rossby waves as in 18. In the plasma physics they describe the
resonance of the high-frequency electron plasma oscillation and the associated low-frequency
ion density perturbation in 19.
Due to their rich physical and mathematical properties the long wave-short wave
resonance equations have drawn much attention of many physicists and mathematicians.
The LS system is as follows:
iut uxx − uv iαu fx, t,
vt γ|u|2x βv gx, t,
x ∈ R, t ≥ 0,
1.1
where u denotes a complex-valued vector and v represents a real-valued function; fx, t
and gx, t are given complex- and real-valued functions, respectively. The constants α, β are
positive, and γ ∈ R \ {0} is real. For the LS equations, Boling Guo obtained the global solution
in 20. The existence of global attractor was studied in 21–23.
Throughout this paper, we set
L 2
u un n∈Z , un ∈ C :
2
|un | < ∞ ,
v vn n∈Z , vn ∈ R :
2
n∈Z
n∈Z
vn2
<∞ .
1.2
For brevity, we use H to denote Hilbert space L2 or 2 and equip H with the inner product
and norm as
u, v un v n ,
u2 u, u n∈Z
|u|2 ,
∀u, v ∈ H,
n∈Z
1.3
where vn denotes the conjugate of vn .
In this paper, we consider the following stochastic discrete LS equations
i
dwn1
dun
− Aun − vn un iαun fn t an un
, n ∈ Z, t > 0,
dt
dt
dwn2
dvn
γ B |u|2
, n ∈ Z, t > 0,
βvn gn t bn
n
dt
dt
1.4
1.5
with the initial conditions
un 0 un0 ,
vn 0 vn0 ,
n ∈ Z,
1.6
where |u|2 |un |2 n∈Z , un t ∈ C, vn t ∈ R C, R are the sets of complex and real numbers,
resp., a an n∈Z ∈ 2 and b bn n∈Z ∈ 2 , ft fn tn∈Z , gt gn tn∈Z ∈ CB R, 2 ,
the space of bounded continuous functions from R into 2 . {wn1 t : n ∈ Z} and {wn2 t : n ∈ Z}
Journal of Applied Mathematics
3
are two independent two-side real-valued standard Wiener process. Z is the integer set, i is
the unit of the imaginary numbers such that i2 −1, and A, B are linear operators defined,
respectively, by
Aun un1 un−1 − 2un ,
Bun un1 − un ,
n ∈ Z, ∀u un n∈Z ,
n ∈ Z, ∀u un n∈Z .
1.7
In addition, simple computation shows that, for u un n∈Z ∈ H, there holds
Bu2 |un1 − un |2 ≤ 2
n∈Z
|un1 |2 |un |2 4u2 .
n∈Z
1.8
This paper is organized as follows. In the next section, we recall some basic concepts
and already know results to random dynamical system and random attractors. In Section 3,
we prove the existence of the global random attractor for stochastic LS lattice dynamical
systems 1.4–1.6.
2. Preliminaries
In this section, we first introduce the definitions of the random dynamical systems and
random attractor, which are taken from 13. Let H, · H be a Hilbert space and Ω, F, P a
probability space.
Definition 2.1. Ω, F, P, θt t∈R are called metric dynamical systems; if θ : R × Ω → Ω is
BR × F, F-measurable, θ0 I, θts θt ◦ θs for all t, s ∈ R, and θt P P for all t ∈ R.
Definition 2.2. A stochastic process φt, ω is called a continuous random dynamical system
RDS over Ω, F, P, θt t∈R if φ is BR × F × BH, BH-measurable, and for all ω ∈ Ω
i the mapping φ : R × Ω × H → H is continuous;
ii φ0, ω I on H;
iii φt s, ω, χ φt, θs ω, φs, ω, χ for all t, s ≥ 0 and χ ∈ H cocycle property.
Definition 2.3. A random bounded set Bω ⊂ H is called tempered with respect to θt t∈R if
for a.e. ω ∈ Ω and all > 0
lim e−
t dBθ−t ω 0,
t→∞
2.1
where dB supχ∈B χH .
Consider a continuous random dynamical system φt, w over Ω, F, P, θt t∈R , and
let D be the collection of all tempered random sets of H.
Definition 2.4. A random set Kω is called an absorbing set in D if for all B ∈ D and a.e. ω ∈ Ω
there exist tB ω > 0 such that
φt, θ−t ω, Bθ−t ω ⊂ Kω,
t ≥ tB ω.
2.2
4
Journal of Applied Mathematics
Definition 2.5. A random set Aω is a random D-attractor for RDS φ if
i Aω is a random compact set, that is, ω → dχ, Aω is measurable for every
χ ∈ H and Aω is compact for a.e. ω ∈ Ω;
ii Aω is strictly invariant, that is, φt, ω, Aω Aθt ω, for all t ≥ 0 and for a.e.
ω ∈ Ω;
iii Aω attracts all sets in D, that is, for all B ∈ D and a.e. ω ∈ Ω we have
lim d φt, θ−t ω, Bθ−t ωAω 0,
2.3
t→∞
where dX, Y supχ∈X infy∈Y χ − yH , X, Y ⊂ H.
The collection D is called the domain of attraction of A.
Definition 2.6. Let φ be an RDS on Hilbert space H. φ is called asymptotically compact if, for
any bounded sequence {χn } ⊂ H and tn → ∞, the set {φtn , θ−tn ω, χn } is precompact in H,
for any ω ∈ Ω.
From 13, we have the following result.
Proposition 2.7. Let K ∈ D be an absorbing set for an asymptotically compact continuous RDS φ.
Then φ has a unique global random D-attractor
Aω φt, θ−t ω, Kθ−t ω
2.4
κ≥tKω t≥κ
which is compact in H.
Let W 1 t n∈Z an wn1 ten and W 2 t n∈Z bn wn2 ten , where an n∈Z , bn n∈Z ∈ 2 .
Here {en } denotes the standard complete orthonormal system in 2 , which means that the
nth component of en is 1 and all other elements are 0. Then W 1 · and W 2 · are 2 -valued
Q-Wiener processes. It is obvious that EW 1 t EW 2 t 0. For details we refer to 24.
Now, we abstract 1.4–1.6 as stochastic ordinary differential equations with respect
to time t in E L2 × 2 . Let a an n∈Z , b bn n∈Z , f fn n∈Z , and g gn n∈Z . Then
1.4–1.6 can be written as the following integral equations:
ut u0 t
−iusvs − αus − iAus − ifs ds − i
0
vt v0 t
t
usdW 1 ,
2.5
0
−βvs − γB |u|2 gs ds W 2 .
2.6
0
Remark 2.8. The special form of multiplicative noise in 2.5 is more suitable than the white
noise “adW” and the additive noise “ nj1 aj dW j ”, because it is more approximative to the
perturbations of the short wave for this model.
Journal of Applied Mathematics
5
For our purpose we introduce the probability space as
Ω ω ∈ C R, 2 : ω0 0
2.7
endowed with the compact open topology 12. P is the corresponding Wiener measure, and
F is the P-completion of the Borel σ-algebra on Ω.
Let θt ω· ω· t − ωt, t ∈ R. Then Ω, F, P, θt t∈R is a metric dynamical system
with the filtration Ft : ∨s≤t Fst , t ∈ R, where Fst σ{Wt2 − Wt1 : s ≤ t1 ≤ t2 ≤ t} is
the smallest σ-algebra generated by the random variable Wt2 − Wt1 for all t1 , t2 such that
s ≤ t1 ≤ t2 ≤ t; see 12 for more details.
3. The Existence of a Random Attractor
In this section, we study the dynamics of solutions for the stochastic LS 1.4–1.6. Then we
apply Proposition 2.7 to prove the existence of a global random attractor for stochastic lattice
LS equations.
Before proving the existence of global solution for 2.5-2.6, we need the following a
priori estimate.
Lemma 3.1. Suppose that ft fn tn∈Z ∈ CB R, 2 . Then, the solution of 1.4–1.6 satisfies
ut, ω; u0 2 ≤ e−αt u0 2 1
f 2 ,
α
t ≥ 0,
3.1
for all ω ∈ Ω and f supt∈R |ft|2 .
Proof. We write 1.4 in the form of vector as
iut − Au − vu iαu f uWt1 ,
t > 0.
3.2
Taking the imaginary part of the inner product of 3.2 with u, we obtain
α
1 1 d
f 2 .
u2 αu2 − Im f, u ≤ u2 2 dt
2
2α
3.3
d
1 2
ut, ω; u0 2 αut, ω; u0 2 ≤ f .
dt
α
3.4
So we have
By the Gronwall inequality, we get
ut, ω; u0 2 ≤ e−αt u0 2 Thus, we derive 3.1.
1
f 2 .
α
3.5
6
Journal of Applied Mathematics
By Lemma 3.1, we know that u2 is bounded in any bounded subset of 0, ∞, that is,
u ≤ e−αt u0 2 1/αf2 , 0 ≤ t ≤ T , for any fixed constant T > 0.
In order to show the existence of global solutions of 2.5-2.6, we first change 2.52.6 into deterministic equations. First, due to special linear multiplicative noise, 2.5 can be
reduced to an equation with random coefficients by a suitable change of variable. Consider
1
the process zt eiW t , which satisfies the stochastic differential equation
2
1
dzt − ztdt iztdW 1 .
2
3.6
The process u
ztut follows the random differential equation,
i
1
d
u
− A
ui α
u
− vu
− ftzt 0.
dt
2
3.7
We denote v vt − W 2 t, then 2.5-2.6 can be changed into the following
equations:
t
1
u
t u
0 −i
usvs − α u
s − iA
us − ifszs ds,
2
0
t
−βv s − γB |u|2 gs − βW 2 s ds.
v t v0 3.8
0
Remark 3.2. For the general multiplicative noise, we can also choose a suitable process and a
change of variable to convert the stochastic equations into deterministic equations.
For each fixed ω ∈ Ω, 3.8 are deterministic equations, and we have the following
result.
Theorem 3.3. For any T > 0, 2.5-2.6 are well posed and admit a unique solution ut, vt ∈
L2 Ω; C0, T ; E. Moreover, the solution of 2.5-2.6 depends continuously on the initial data
u0 , v0 .
Proof. By standard existence theorem for ODEs, it follows that 3.8 possess a local solution
ut, v t ∈ C0, T ; E, where 0, Tmax is the maximal interval of existence of the solution
of 3.8. Now, we prove that this local solution is a global solution. Let ω ∈ Ω; from 3.8 it
follows that
vt2
ut2 t
t
t
1
u0 2 v0 2 − 2 α s ds − 2β us2 ds − 2 Re ifz, u
vs2 ds
3.9
2
0
0
0
t
t t
2
B |u| , v s ds 2
W 2 s, v s ds.
− 2γ
gs, v s ds − 2β
0
0
0
Journal of Applied Mathematics
7
By the Young inequality and 1.8, direct computation shows that
1
2 2
α
1 α
4
f 4 −2 Re ifz, u
≤ f z2 u2 ≤
u 2 ,
W α
2
α
2
4γ 2
2
2 v −2γ B |u| , v ≤ 2 γ B |u| v ≤ γ ε
u4 ,
ε
1 2
v2 g ,
2 g, v ≤ γ ε
γ ε
3.10
β 2
v2 W 2 .
−2β W 2 , v ≤ 2βW 2 v ≤ γ ε
γ ε
Combining the above inequalities with Lemma 3.1, we obtain
2
2
2
2
vt ≤ u0 v0 − C0
ut t
vs2 ds
us2 0
t
2 4 2 2 f g C1
W 2 W 1 ds
0
2
2
v0 C1
≤ u0 t
3.11
2 4 2 2 2
f g W W 1 ds,
0
where C0 , C1 are constants depending on α, β, γ, and ε is a sufficiently small positive number.
By the Gaussian property of W 1 and W 2 , 3.11 implied that 3.8 admit a global solution
ut, v t ∈ L2 Ω; C0, T ; E. The proof of the lemma is completed.
From the definition θt t∈R , we know
Wt h, ω Wt, θh ω Wh, ω,
∀t, h ∈ R,
3.12
and combining the above theorem we have the following result.
Theorem 3.4. System 2.5-2.6 generates a continuous random dynamical system φt, θ−t ωt≥0
over Ω, F, P, θt t∈R .
The proof is similar to that of Theorem 3.2 in 13, so we omit it.
Now, we prove the existence of a random attractor for system 2.5-2.6. By
Proposition 2.7, we first prove that RDS φ possesses a bounded absorbing set Kω. We
introduce an Ornstein-Uhlenbeck process in 2 on the metric dynamical system Ω, F, P, θt given by Wiener process:
z θt ω −ν
1
0
−∞
νh
e θt ωhdh,
t ∈ R,
z θt ω −μ
2
0
−∞
eμh θt ωhdh,
t ∈ R, 3.13
8
Journal of Applied Mathematics
where ν and μ are positive. The above integral exists in the sense of for any path ω with a
o equations:
subexponential growth and z1 , z2 solve the following It
dz1 νz1 dt dW 1 t,
dz2 μz2 dt dW 2 t.
3.14
In fact, the mapping t → zi θt ω, i 1, 2, is the O − U process. Furthermore, there
exists a θt invariant set Ω ⊂ Ω of full P measure such that
1 the mapping t → zi θt ω, i 1, 2, is continuous for each ω ∈ Ω ,
2 the random variables zi ω, i 1, 2, are tempered.
Lemma 3.5. There exists a θt invariant set Ω ⊂ Ω of full P measure and an absorbing random set
Kω, ω ∈ Ω , for the random dynamical system φt, θ−t ωt≥0 .
Proof. We use the estimates in Theorem 3.3. By 3.11, we have
d vt2
ut2 dt
≤ −C0 vt2 C1 ρθt ω,
ut2 4
3.15
2
where ρθt ω f2 g2 W 1 W 2 .
By the Gronwall inequality, we have
t
vt2 ≤ v0 2 e−C0 t e−C0 t−s ρθs ωds.
ut2 u0 2 3.16
0
Replace ω by θ−t ω in the above inequality to construct the radius of the absorbing set and
define
2 ω 4 lim
t→∞
t
e−C0 t−s ρθs−t ωds 4 lim
0
0
t→∞
−t
e−C0 s ρθs ωds.
3.17
Define
R2 ω 2 ω 1
4
4 2
f 2 W W 2 .
α
3.18
Then, Kω K0, Rω is a tempered ball by the property of W 1 , W 2 , and, for any B ∈
D, ω ∈ Ω. Here, D denotes the collection of all tempered random sets of Hilbert space H. The
proof of the lemma is completed.
Lemma 3.6. Let u0 , v0 ∈ Kω, the absorbing set given in Lemma 3.5. Then, for every ε > 0
and P-a.e. ω ∈ Ω, there exist T ε, ω > 0 and Nε, ω > 0 such that the solution u, v of system
2.5-2.6 satisfies
|un t, θ−t ω|2 |vn t, θ−t ω|2 ≤ ε,
|n|>Nε,ω
∀t ≥ T ε, ω.
3.19
Journal of Applied Mathematics
9
Proof. Let ηx ∈ CR , 0, 1 be a cut-off function satisfying
ηx ⎧
⎨1,
∀x ∈ 2, ∞,
⎩0,
∀x ∈ 0, 1,
3.20
and |η x| ≤ η0 a positive constant.
vn n∈Z ,
Taking the inner product of 3.8 with η|n|/M
un n∈Z and η|n|/M
respectively, we get
|n| d |n|
1 |n|
un |2 −2 α un |2 − 2 Im η
fn zn , u
η
η
n ,
|
|
dt n∈Z M
2 n∈Z M
M
n∈Z
|n| |n|
|n| 2 d |n|
2
2
vn | −2β η
vn | − 2γ η
B |u| , vn 2 η
gn , vn
η
|
|
n
dt n∈Z M
M
M
M
n∈Z
n∈Z
n∈Z
|n| Wn2 , vn .
− 2β η
M
n∈Z
3.21
We also use the estimates in Theorem 3.3. Similar to 3.11, it follows that
|n| η
vn |2
un |2 |
|
M
n∈Z
|n| |n| 2
2
−C0 t−Tk η
η
≤e
un Tk , ω| vn Tk , ω|
|
|
M
M
n∈Z
n∈Z
C1
t
eC0 s−t
Tk
3.22
2 2 2
2
z1n θs ω z2n θs ω fn gn ds.
|n|≥M
Replace ω by θ−t ω in 3.22. Then, we estimate each term on the right-hand of 3.22;
it follows that
e
−C0 t−Tk |n| η
vn Tk , θ−t ω|2
un Tk , θ−t ω|2 |
|
M
n∈Z
≤ e−C0 t−Tk |u0 θ−t ω|2 |v0 θ−t ω|2 e−C0 Tk
Tk
e
−C0 Tk −s
4 2 1
z θs−t ω z2 θs−t ω ds
0
e
−C0 t−Tk Tk
e
0
−C0 Tk −s
!
f 2 g 2 ds .
3.23
10
Journal of Applied Mathematics
Since zi ω, i 1, 2, are tempered and zi θt ω, i 1, 2, are continuous in t, there is a
tempered function rω > 0 such that
4 2
1
z θt ω z2 θt ω ≤ rθt ω.
3.24
Combining 3.23 with 3.24, there is a T1 ε, ω > Tk such that
e−C0 t−Tk ε
|n| η
vn Tk , θ−t ω|2 ≤ .
un Tk , θ−t ω|2 |
|
M
3
n∈Z
3.25
Next, we estimate
t
C1
e
C0 s−t
Tk
2 4 2
1
zn θs−t ω zn θs−t ω ds.
|n|≥M
3.26
Let T ∗ ≥ 1/C0 ln6C1 rω/C0 ε and N1 ε, ω be fixed positive constants. Then, for t >
T ∗ Tk and M > N1 ε, ω, by the Lebesgue theorem, we have
t
e
C1
C0 s−t
Tk
|n|≥M
C1
0
−T ∗
C1
≤ C1
≤
4 2 2
1
zn θs−t ω zn θs−t ω ds
|n|≥M
−T ∗
Tk −t
0
−T ∗
4 2
z1n θξ ω z2n θξ ω ds
eC0 ξ
e
C0 ξ
4 2 2
1
zn θξ ω zn θξ ω ds
3.27
|n|≥M
eC0 ξ
4 2
C1
∗
rωeC0 T
z1n θξ ω z2n θξ ω ds C
0
|n|≥M
ε ε ε
.
6 6 3
Since ft ∈ CB R, 2 and gt ∈ CB R, 2 , there exists N2 ε, ω such that for M >
N2 ε, ω
t
C1
Tk
eC0 s−t
2 2 fn gn ds ≤ ε .
3
|n|≥M
3.28
Therefore, let
T ε, ω max{T1 ε, ω, T ∗ ε, ω},
" ω max{N1 ε, ω, N2 ε, ω}.
Nε,
3.29
Journal of Applied Mathematics
11
" ω, we obtain
Then, for t > T ε, ω and M > Nε,
vn t, θt ω|2 ≤ ε.
un t, θt ω|2 |
|
|n|>M
3.30
Direct computation shows that
2
v2 4z2 θt ω .
u2 u2 v2 ≤ 2 3.31
Therefore, we obtain
|un t, θt ω|2 |vn t, θt ω|2 ≤ ε.
|n|>M
3.32
The proof of the lemma is completed.
Lemma 3.7. The random dynamical systems φt, θ−t ωt≥0 are asymptotically compact.
Proof. We use the method of 25. Let ω ∈ Ω. Consider a sequence tn n∈N with tn → ∞ as
n → ∞. Since Kω is a bounded absorbing set, for large n, un , vn φtn , θ−tn ωu0 , v0 ∈
Kω, where u0 , v0 ∈ Kω. Then, there exist u, v ∈ E and a sequence un , vn denoted by
itself such that
un , vn u, v weakly in E.
3.33
Next, we show that the above weak convergence is actually strong convergence in E.
"1 such
From Lemma 3.6, for any ε > 0, there exist positive constants N3 ε, ω and M
"1 ,
that, for n > M
i>N3
ε
|uin tn , θtn ω|2 |vin tn , θtn ω|2 ≤ .
6
3.34
Since u, v ∈ E, there exists N4 ε, ω > 0 such that
|i|≥N4
ε
|ui |2 |vi |2 ≤ .
6
3.35
" ω max{N3 ε, ω, N4 ε, ω}, then, from 3.33, there exists M
"2 > 0 such
Let Nε,
"2 ,
that, for n > M
"
|i|≤N
ε
|uin − u|2 |vin − v|2 ≤ .
3
3.36
12
Journal of Applied Mathematics
"2 },
" max{M
"1 , M
By 3.34–3.36, we obtain that, for n > M
un tn , θ−tn ω − u2 vn tn , θ−tn ω − v2
|uin tn , θ−tn ω − ui |2 |vin tn , θ−tn ω − vi |2
"
|i|≤N
|uin tn , θ−tn ω − ui |2 |vin tn , θ−tn ω − vi |2
"
|i|>N
≤
|uin tn , θ−tn ω − ui |2 |vin tn , θ−tn ω − vi |2
3.37
"
|i|≤N
2
2
|ui | |vi |2
uin tn , θ−tn ω2 |vin tn , θ−tn ω|2 2
"
|i|>N
≤
"
|i|>N
ε 2ε 2ε
≤ ε.
3
6
6
The proof of the lemma is completed.
Now, combining Lemmas 3.5, 3.7 with Proposition 2.7, we can easily obtain the
following result.
Theorem 3.8. The random dynamical systems φt, θ−t ωt≥0 possess a global random attractor in
E.
Acknowledgments
The authors are grateful to the anonymous referees for their careful reading of the paper
and precious comments. The authors also thank the editors for their kind help. They were
supported by the NSF of China nos. 11071162, 11001116, the NSF of Shandong Province,
the Project of Shandong Province Higher Educational Science and Technology Program
nos. J10LA09, J11LA03, and the Project of Discipline Construction Foundation in Ludong
University.
References
1 V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, vol. 49 of AMS
Colloquium Publications, AMS, Providence, RI, USA, 2002.
2 J. K. Hale, Asymptotic Behavior of Dissipative Systems, vol. 25 of Mathematical Surveys and Monographs,
AMS, Providence, RI, USA, 1988.
3 O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press,
Cambridge, UK, 1991.
4 G. R. Sell and Y. You, Dynamics of Evolution Equations, Springer, New York, NY, USA, 2002.
5 R. Temam, Infinite Dimensional Dynamical System in Mechanics and Physics, Springer, New York, NY,
USA, 1995.
6 P. W. Bates, K. Lu, and B. Wang, “Attractors for lattice dynamical systems,” International Journal of
Bifurcation and Chaos, vol. 11, no. 1, pp. 143–153, 2001.
Journal of Applied Mathematics
13
7 W. J. Beyn and S. Y. Pilyugin, “Attractors of reaction diffusion systems on infinite lattice,” Journal of
Dynamics and Differential Equations, vol. 15, pp. 485–515, 2003.
8 S. N. Chow, Lattice Dynamical Systems, vol. 1822 of Lecture Notes in Mathematics, Springer, Berlin,
Germany, 2003.
9 N. I. Karachalios and A. N. Yannacopoulos, “Global existence and compact attractors for the discrete
nonlinear Schrödinger equation,” Journal of Differential Equations, vol. 217, no. 1, pp. 88–123, 2005.
10 B. Wang, “Dynamics of systems on infinite lattices,” Journal of Differential Equations, vol. 221, no. 1, pp.
224–245, 2006.
11 S. Zhou and W. Shi, “Attractors and dimension of dissipative lattice systems,” Journal of Differential
Equations, vol. 224, no. 1, pp. 172–204, 2006.
12 L. Arnold, Random Dynamical Systems, Springer, Berlin, Germany, 1998.
13 P. W. Bates, H. Lisei, and K. Lu, “Attractors for stochastic lattice dynamical systems,” Stochastics and
Dynamics, vol. 6, no. 1, pp. 1–21, 2006.
14 Y. Lv and J. H. Sun, “Dynamical behavior for stochastic lattice systems,” Chaos, Solitons and Fractals,
vol. 27, no. 4, pp. 1080–1090, 2006.
15 Y. Lv and J. H. Sun, “Asymptotic behavior of stochastic discrete complex Ginzburg-Landau
equations,” Physica D, vol. 221, no. 2, pp. 157–169, 2006.
16 J. Huang, “The random attractor of stochastic FitzHugh-Nagumo equations in an infinite lattice with
white noises,” Physica D, vol. 233, no. 2, pp. 83–94, 2007.
17 W. Yan, S. Ji, and Y. Li, “Random attractors for stochastic discrete Klein-Gordon-Schrödinger
equations,” Physics Letters A, vol. 373, no. 14, pp. 1268–1275, 2009.
18 R. H. J. Grimshaw, “The modulation of an internal gravity-wave packet and the resonance with the
mean motion,” Studies in Applied Mathematics, vol. 56, no. 3, pp. 241–266, 1977.
19 D. R. Nicholson and M. V. Goldman, “Damped nonlinear Schrödinger equation,” Physics of Fluids,
vol. 19, no. 10, pp. 1621–1625, 1976.
20 B. Guo, “The global solution for one class of the system of LS non-linear wave resonance equations,”
Journal of Mathematical Research and Exposition, vol. 1, pp. 69–76, 1987.
21 W. Du and B. Guo, “The global attractor for LS type equation in R1 ,” Acta Mathematicae Applicatae
Sinica, vol. 28, pp. 723–734, 2005.
22 Y. Li, “Long time behavior for the weakly damped driven long-wave-short-wave resonance
equations,” Journal of Differential Equations, vol. 223, no. 2, pp. 261–289, 2006.
23 R. Zhang, “Existence of global attractor for LS type equation,” Journal of Mathematical Research and
Exposition, vol. 26, pp. 708–714, 2006.
24 G. D. Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, vol. 229 of London Mathematical
Society Lecture Note Series, Cambridge University Press, Cambridge, UK, 1996.
25 P. W. Bates, X. Chen, and A. J. J. Chmaj, “Traveling waves of bistable dynamics on a lattice,” SIAM
Journal on Mathematical Analysis, vol. 35, no. 2, pp. 520–546, 2004.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014