Random Attractor For Zakharov Lattice Dynamical Systems With White Noises

Random dynamical system is the generalization of dynamicalsystem, due to considering uncertain and random factor duringresearching process, thus playing an important role in practicalapplication. Moreover, Lattice dynamical systems apply to manypractical fields, such as, the theory of chemical reaction,pattern-recognition etc. And that global random attractor is one ofmain content in studying the the long term asymptotic behavior andunitary character of stochastic lattice dynamical system, which isimportant to discuss the existence of random attractor. The paper isdevoted to prove the existence of the random attractor to thestochastic dynamical system generated by the stochastic Zakharovlattice equations with white noises.Firstly, I introduce the development survey and research result ofdynamical system, stochastic dynamical system, lattice system,Zakharov system, random attractor and the research work of thispaper.Then, I represent the preliminary conceptions, theorem andfrequently used inequalities, such as, random attractor, the theorem forexistence of attractor, Gronwalls inequality.Finally, I discuss the stochastic Zakharov lattice equations withtwo kinds of white noise possess random attractor. First, usingOrnstein-Uhlenbeck transformation, the stochastic Zakharov latticeeystem is turned into a random differential equation with randomcoefficients, testify the existence and uniqueness of solutions, thereforethe solution of the system determine a continuous random dynamicalsystem with tempered random absorbing set, in which the randomdynamical system is asymptotically compactness, then obtain randomattractor and absorbing all tempered random sets.