The Attractor And The Approximate Inertial Manifold For The Fractional Long-short Wave Equations

The dynamic behavior of fractional long-short wave equations is discussed in the paper.Firstly,we can explore the generalized long-short wave equations with fractional derivative.we verify that there is a global solution and it is unique.Secondly,we show that stochastic fractional long-short wave equations have random attractors.Thirdly,considering the(2+1)-dimensional fractional long-short wave equations in the case of non-autonomous,it includes the exploration of the attractor and the AIM.The paper carries on the elaboration from the following 4 chapters.In Part 1,a number of definitions and sources associated with fractional calculus,the setting and relevant basic knowledge for the fractional long-short wave equations and stochastic differential equation are stated.We review the results of the research and explain the work of this paper.In Part 2,we mainly discuss the(2+1)-dimensional generalized fractional long-short wave equations.The first step is to make a priori estimation through some basic inequalities.Then we can use the standard Gal?rkin method to obtain the existence of the solution and verity that it still satisfies the uniqueness.In Part 3,we discuss the dynamic behavior of the solution for stochastic fractional long-short wave equations.We prove that the system has random attractors.In Part 4,we discuss the(2+1)-dimensional fractional long-short wave equations in the case of non-autonomous.The first step is to obtain a priori estimation and can be verified by the Gal?rkin method for the existence of a unique solution.The second step,the existing results of non-autonomous dynamical systems are applied to obtain the existence of uniform attractor.The third step,the AIM of this system is obtained by expanding the phase plane and the operator projection method.