Invariant Manifolds For Two Classes Of Partial Differential Equations

In this paper we consider invariant manifolds,i.e.,center manifold,stable manifolds and unstable manifolds,of partial differential equations(PDEs).The form of the PDEs is where is a partial differential operator which is defined in a domain inside a Banach space X.We assume A is linear and N is a nonlinear operator.Moreover,N will be of lower order with respect to A and the spectrum of A is of trichoto-my.In the theory of dynamical systems,the scholars are very interested in the existences and stability of the special solutions:the equilibrium,periodic solution,quasi-periodic solutions and so on.The behavior of other solutions of the systems can also be known by studying these special solutions.The KAM(Kolmogorov-Arnold-Moser)theory is a very powerful tool to study the peri-odic and quasi-periodic solutions which mainly contains normal form quadratic approach and Newton-Nash-Moser implicit function theorem.One obtains the linear stability of the solution by the normal form quadratic approach.In 2003,Xavier Carbe,Ernest Fontich and Rafael de la Llave introduced a new method,named parameterization method,to prove existence and regularity of invari-ant manifolds for dynamical systems.This method mainly take the advantage of the trichotomy of the(discrete)spectrum of linear operator(?)to reduce the whole system into a finite dimensional subspace and get the quasi-periodic solution of the original system by solving the homological equations in the fi-nite dimensional subspace.However,to use the parameterization method one needs the approximated solution K(t),which can be obtained by,for example,the numerical simulation,i.e.where ||e||Y is small enough.Two concrete models considered in this paper are Boussinesq equation and complex Ginzburg-Landau equationThere are plenty of results about these two equations.Particularly,in 2009,Rafael de la Llave constructed the center manifold of(0.5).In 2015,J.Xu proved that(0.5)admits quasi-periodic solutions under some conditions.In 2016 Rafael de la Llave and Yannick Sire constructed the finite dimension-al quasi-periodic solutions by the parameterization method.In 2008,K.W.Chung and X.Yuan proved that(0.6)possesses periodic and 2-dimensional quasi-periodic solutions when d = 1.In 2009,H.Cong,J.Liu and X.Yuan proved the existence of 2-dimensional quasi-periodic solutions of(0.6)when d>1.In 2011 H.Cong and M.Gao discussed the existence of 2-dimensional quasi-periodic solutions for the generalized Ginzburg-Landau equation with derivatives in the nonlinearity.In 2013,H.Cheng and J.Si constructed the(m + 2)-dimension quasi-periodic solutions of complex Ginzburg-Landau e-quation with quasi-periodic forced perturbations and the forcing frequency w =(w1,...,wm)satisfying the Diophantine condition.The following is the organization of this paper.Chapter 1 is divided into five sections.In the first section we introduce the background of our problems,we mainly introduce the background of the two concrete models.In second section we give the basic definitions,inequalities,lemmas and propositions.In the third section we give two classical KAM theorems for the Hamilton system,one is for the finite dimensional system and the other is for the infinite dimensional system.In the fourth section we give one KAM theorem for Non-Hamilton system with quasi-periodic forcing,where the frequency subjects to Diophantine condition.In the fifth section we talk about the parameterization method and give a brief introduction to how the authors,Rafael de la Llave and Yannick Sire,use this method to construct the quasi-periodic solutions.In Chapter 2 we construct the quasi-periodic solutions of the complex Ginzburg-Landau equation with the quasi-periodic forcing whose frequency satisfies the Diophantine condition.More details,we assume the solution we want is the linear sum of the basis with the factor being the functions of the time t.Once we submit this sum into the main equation we can get the lattice equation,then we make a change of variables to kill the resonant terms.By action-angle variables changes and some basic calculations we get a system which is a integral system with a small perturbation.Applying the Theorem1.5 we get the result we want.In Chapter 3 we construct the stable manifolds to the bounded solutions of the Boussinesq equation and the complex Ginzburg-Landau equation,what we want to say is that the bounded solutions can be just forward bounded solutions obtained by any methods.Concretely,we assume u(t)= K(t)is the solution of(0.4)and we want to get another function ?(t)such that u(t)= K(t)??(t)is also the solution of(0.4).Submit u(t)= K(t)and u(t)= K(t)??(t)to(0.4)and with some easy calculations we get the evolution equation of ?(t)(the variational equation).Take the quasi-periodic solution K(?+wt)for example,its stable manifold we get is the set WWe call W as the graph of function w,where w satisfiesWe use the contraction mapping theorem to get the function w.Chapter 4 is the appendix,we give the proof of some lemmas in this part.