Journal of Applied MathematicsVolume 2011 (2011), Article ID 452087, 13 pagesdoi: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

Received 21 June 2011; Accepted 15 August 2011

Academic Editor: F. Marcellán

Copyright © 2011 Jie Xin and Hong 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.

Abstract.

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 ; then, Lv and Sun [14] extended the results of Bates to the dimensional lattices. Stochastic complex Ginzburg-Landau equations, FitzHugh-Nagumo equation, and 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: where denotes a complex-valued vector and represents a real-valued function; and are given complex- and real-valued functions, respectively. The constants are positive, and 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 For brevity, we use to denote Hilbert space or and equip with the inner product and norm as where denotes the conjugate of .

In this paper, we consider the following stochastic discrete LS equations with the initial conditions where , , ( are the sets of complex and real numbers, resp.), and , , , the space of bounded continuous functions from into . and are two independent two-side real-valued standard Wiener process. is the integer set, is the unit of the imaginary numbers such that , and are linear operators defined, respectively, by In addition, simple computation shows that, for , there holds

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 be a Hilbert space and a probability space.

Definition 2.1. are called metric dynamical systems; if is measurable, , for all , and for all .

Definition 2.2. A stochastic process is called a continuous random dynamical system (RDS) over if is measurable, and for all (i)the mapping is continuous;(ii); (iii) for all , and (cocycle property).

Definition 2.3. A random bounded set is called tempered with respect to if for a.e. and all where .

Consider a continuous random dynamical system over , and let be the collection of all tempered random sets of .

Definition 2.4. A random set is called an absorbing set in if for all and a.e. there exist such that

Definition 2.5. A random set is a random -attractor for RDS if (i) is a random compact set, that is, is measurable for every and is compact for a.e. ;(ii) is strictly invariant, that is, and for a.e. ;(iii) attracts all sets in , that is, for all and a.e. we have where .

The collection is called the domain of attraction of .

Definition 2.6. Let be an RDS on Hilbert space . is called asymptotically compact if, for any bounded sequence and , the set is precompact in , for any .

From [13], we have the following result.

Proposition 2.7. Let be an absorbing set for an asymptotically compact continuous RDS . Then has a unique global random attractor which is compact in .

Let and , where . Here denotes the standard complete orthonormal system in , which means that the th component of is 1 and all other elements are 0. Then and are valued Wiener processes. It is obvious that . For details we refer to [24].

Now, we abstract (1.4)–(1.6) as stochastic ordinary differential equations with respect to time in . Let , and . Then (1.4)–(1.6) can be written as the following integral equations:

Remark 2.8. The special form of multiplicative noise in (2.5) is more suitable than the white noise “” and the additive noise “”, because it is more approximative to the perturbations of the short wave for this model.

For our purpose we introduce the probability space as endowed with the compact open topology [12]. is the corresponding Wiener measure, and is the completion of the Borel algebra on .

Let . Then is a metric dynamical system with the filtration , where is the smallest algebra generated by the random variable for all such that ; 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 . Then, the solution of (1.4)–(1.6) satisfies for all and .

Proof. We write (1.4) in the form of vector as Taking the imaginary part of the inner product of (3.2) with , we obtain So we have By the Gronwall inequality, we get Thus, we derive (3.1).

By Lemma 3.1, we know that is bounded in any bounded subset of , that is, , for any fixed constant .

In order to show the existence of global solutions of (2.5)-(2.6), we first change (2.5)-(2.6) 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 the process , which satisfies the stochastic differential equation The process follows the random differential equation,

We denote , then (2.5)-(2.6) can be changed into the following equations:

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 , (3.8) are deterministic equations, and we have the following result.

Theorem 3.3. For any , (2.5)-(2.6) are well posed and admit a unique solution . Moreover, the solution of (2.5)-(2.6) depends continuously on the initial data .

Proof. By standard existence theorem for ODEs, it follows that (3.8) possess a local solution , where is the maximal interval of existence of the solution of (3.8). Now, we prove that this local solution is a global solution. Let ; from (3.8) it follows that By the Young inequality and (1.8), direct computation shows that Combining the above inequalities with Lemma 3.1, we obtain where are constants depending on , and is a sufficiently small positive number. By the Gaussian property of and , (3.11) implied that (3.8) admit a global solution . The proof of the lemma is completed.

From the definition , we know and combining the above theorem we have the following result.

Theorem 3.4. System (2.5)-(2.6) generates a continuous random dynamical system over .

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 possesses a bounded absorbing set . We introduce an Ornstein-Uhlenbeck process in on the metric dynamical system given by Wiener process: where and are positive. The above integral exists in the sense of for any path with a subexponential growth and solve the following Itô equations:

In fact, the mapping , is the process. Furthermore, there exists a invariant set of full measure such that (1)the mapping , is continuous for each ,(2)the random variables , are tempered.

Lemma 3.5. There exists a invariant set of full measure and an absorbing random set , for the random dynamical system .

Proof. We use the estimates in Theorem 3.3. By (3.11), we have where .By the Gronwall inequality, we have Replace by in the above inequality to construct the radius of the absorbing set and define Define Then, is a tempered ball by the property of , and, for any . Here, denotes the collection of all tempered random sets of Hilbert space . The proof of the lemma is completed.

Lemma 3.6. Let , the absorbing set given in Lemma 3.5. Then, for every and a.e. , there exist and such that the solution of system (2.5)-(2.6) satisfies

Proof. Let be a cut-off function satisfying and (a positive constant).Taking the inner product of (3.8) with and , respectively, we get We also use the estimates in Theorem 3.3. Similar to (3.11), it follows that Replace by in (3.22). Then, we estimate each term on the right-hand of (3.22); it follows that Since , are tempered and , are continuous in , there is a tempered function such that Combining (3.23) with (3.24), there is a such that Next, we estimate Let and be fixed positive constants. Then, for and , by the Lebesgue theorem, we have Since and , there exists such that for Therefore, let Then, for and , we obtain Direct computation shows that Therefore, we obtain The proof of the lemma is completed.

Lemma 3.7. The random dynamical systems are asymptotically compact.

Proof. We use the method of [25]. Let . Consider a sequence with as . Since is a bounded absorbing set, for large , , where . Then, there exist and a sequence (denoted by itself) such that Next, we show that the above weak convergence is actually strong convergence in .From Lemma 3.6, for any , there exist positive constants and such that, for , Since , there exists such that Let , then, from (3.33), there exists such that, for , By (3.34)–(3.36), we obtain that, for , 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 possess a global random attractor in .

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.

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