Quasi-periodic Solutions Of Nonlinear Beam Equations With Quintic Quasi-periodic Nonlinearities

The integrable Hamiltonian system is the most simple one, it’s phase space is 2n dimension and hierarchical to a family of n-dimension invariant torus, all the motion starting from the torus is quasi-periodic, and Hamiltonian is only dependent of the action variables H=H(I), with motion equation the frequency ω(I) defines a map from action space to frequency space.Many scientists spent their whole life to find a canonical transformation to change a general Hamilton system into integrable one. But it is shown that the integrable Hamilton system is less and less. Since 1960’s, scientists have been focusing on nearly-integrable system. For infinite-dimensional nearly-integrable Hamilton systems, we not only concern about periodic solutions, but also their quasi-periodic solutions and their almost-periodic solutions. Up till now there are two main methods to deal with the periodic and quasi-periodic solutions of infinite-dimensional systems. One is the Craig-Wayne-Bourgain method [20], [21], [45], [46], it is a generalization of the Lyapunov-Schmidt reduction and the Newtonian method. The other is based on superconvergent (Newton’s) method-s, as the KAM theory or the Nash-Moser Implicit Function Theorem, allow to extend well known methods and results from the finite-dimensional case and are the natural ways to deal with the lack of regularity due to the "small divisors" problem. The classical KAM theory which is constructed by three famous math-ematicians Kolmogorov, Arnold and Moser in the last century, is the landmark of the development of hamiltonian systems. It gived reasonable explanation of the stability of solar system. In the later 1990’s, the KAM theory was successful-ly generalized to the infinite-dimensional setting by Wayne [4] and Kuksin [38]. Later, Poschel [18] restated the result. Such techniques are somewhat comple-mentary to the variational ones allowing us to obtain periods. However, unlike the variational methods, they are local, perturbed in nature and, therefore, re-stricted to equations with weak nonlinearities or, equivalently, to solutions of small amplitudes.The infinite dimensional KAM theory and results are mainly applied to the research of the existence of quasi-periodic solutions of partial equation,typical ex-amples,nonlinear wave equations, nonlinear Schrodinger equations,and nonlinear beam equation etc. Recently, many progresses have been done concerning the dy-namic behavior for nonlinear beam equations by making use of different methods, for instance, [1],[5],[7],[8],[30],[36],[39], In these works, little is considered about quasi-periodic solutions of this kind of equations. In the last years, there are also many significant results have been obtained with respect to quasi-periodic solu-tions of autonomous beam equations by KAM theory, see [10]-[12]. In particular, Liang and Geng [53] considered the existence of the quasi-periodic solutions of completely resonant beam equations with hinged boundary conditions where the case B=1.All works mentioned above do not conclude the case of non-autonomous. More recently, Wang [49] obtained the existence of quasi-periodic solutions for non-autonomous beam equations under hinged boundary conditions In this paper, we will consider the existence of quasi-periodic solutions for non-linear beam equations with quintic quasi-periodic nonlinearities subject to periodic boundary conditions where B is a positive constant, ε is a small parameter and φ(t) is real analytic quasi-periodic function in t with frequency vector ω=(ω1,ω2...,ωm). The e-quation can be regarded as a perturbation (with perturbation term εφ(t)u5) of completely resonant nonlinear beam equation We first consider the existence quasi-periodic solutions of a ordinary differential equation with respect to the unknown function x(t) then we obtain the following nonlinear beam equationWe consider the quasi-periodic solutions of above beam equation by letting u= u0(t)+εv(x,t), here u0(t) is a nonzero quasi-periodic solution of (0.0.6) and Vk (k=1,2,...,4) defined as in Chapter 2 section 3/construct the invariant tori or quasi-periodic solutions of (0.0.7) by means of KAM theory. Finally, we will obtain that (0.0.5) under the periodic conditions have many quasi-periodic solutions in the neighborhood of a quasi-periodic solutions to (0.0.6).As our previous works [42],[15], the method in this paper is based on infinite-dimensional KAM theory as developed by Kuksin [37] and Pochel [17]. Thus the main step is to reduce the equation to a setting where KAM theory for PDE can be applied. We note that equation (0.0.7) is a nonlinear beam equation with quasi-periodic potential and quasi-periodic nonlinearities. This needs to reduce the linearized system of (0.0.7) to constant coefficients by a linear quasi-periodic change of variables with the same basic frequencies as the initial system. However, we cannot guarantee in general such reducibility. A large part of the present paper will be devoted to the proof of reducibility of an infinite-dimensional linear quasi-periodic systems.In the recent years, the reducibility problems of infinite-dimensional linear quasi-periodic systems, by KAM techniques, has become an active field of re-search. In this directions, the first result was obtained by Bambusi and Graffi [9], later, Eliasson and Kuksin [31], Yuan [47], Liu and Yuan [16] and B.Grebert and L.Thomann [3]. However, in general, the question of reducibility of infinite-dimensional linear quasi-periodic systems remains open and very attracting.