Quasi-periodic Solutions Of Nonlinear Wave Equations With Quasi-periodic Forcing

The problem of the existence of finite-dimensional tori for infinite-dimensional hamiltonian systems, such as nonlinear PDEs, has been extensively studied in the liter-ature. So far there are two main approaches to deal with the periodic and quasi-periodic solutions of nonlinear PDEs. The first one is the Craig-Wayne-Bourgain method(CWB method for simplicity). It is a generalization of the Liapunov-Schmidt reduction and the Newtonian method. The second approach is the infinite-dimensional KAM theory which is the extension of classical KAM theory. The classical KAM theory which is constructed by three famous mathematicians Kolmogorov [35], Arnold [1] and Moser[49] in the last century is the landmark of the development of Hamiltonian systems. It made the stabil-ity of solar system got resonable explanation and brought a new method to the study of Hamiltonian systems. The classical KAM theory which constructed on 2n-dimensional smooth manifold asserts that under the Kolmogorov non-degenerate condition, the ma-jority of the non-reasont tori of integrable system are persistent under small pertubation. In the later 1990's, the celebrated KAM theory was successfully generalized to infinite dimensional setting by Wayne[64] and Kuksin[36], where the Hamiltonian systems are those with infinite many normal frequencies. Later, Poschel[57] restated the result. The two techniques are somehow complementary. We point out the advantage of CWB method is its weaker dependence on the second Melnikov conditions, so that it is more convenient to solve the case of multiple normal frequencies. The disadvantage of CWB method is that one knows nothing on the dynamics around constructed quasi-periodic solution. Comparing with Lyapunov-Schmidt reduction method, the KAM approach has its own advantages. Besides obtaining the existence of quasi-periodic solutions it allows one to construct a local normal form in a neighborhood of the obtained solu-tions, and provides more information of the dynamics, for instance on the stability of the solutions and Lyapunov exponent be 0. The disadvantage of this approach is its stonger dependence on the second Melnikov conditions, such that it is difficult to treat the case of multiple normal frequencie.In this paper, we will are concerned with existence of quasi-periodic solutions for quasi-periodically forced nonlinear wave equation utt=uxx-μu-μφ(t)h(u),μ> 0 (0.1) on the finite x-interval [0,Ï€] with Dirichlet boundary conditions u(t,0)= 0= u(t,Ï€),-∞< t<∞whereεis a small positive parameter,φ(t) is a real analytic quasi-periodic function in t with frequency vectorω=(ω1,ω2...,ωm) (?) [ρ,2ρ]m for some constantρ> 0, and the nonlinearity h is a real analytic odd function of the form It is shown that under a suitable hypothesis onφ(t) and h, there are many quasi-periodic solutions for the above equation via KAM theory.The main step is to reduce the equation to a setting where KAM theory for PDE can be applied. This needs to reduce the linear part of Hamiltonian system to constant coefficients by a linear quasi-periodic change of variables with the same basic frequen-cies 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 fact, the question of reducibility of infinite-dimensional linear quasi-periodic systems is also interesting itself and remains open in the general case. There are D. Bambusi and S. Graffi[5], Kuksin and Elias-son[31] in this line. However, it would seem that their results cannot be directly applied to our problems because of the difference in the orders of corresponding morphism of the Hilbert scales between Schrodinger equations and wave equations.The paper is organized as follows:In Chapter 1 we introduce the existence of finite-dimensional tori for nonlinear PDEs obtained in the literature and our main work in this paper.In Chapter 2 firstly, we give the KAM theory of infinite-dimensional Hamiltonian systems in Kuksin [36] which will applied to the reducibility of wave equations in§3.2. Secondly, we give the infinite-dimensional KAM theorem for partial differential equa-tions in P6schel[57] which will applied to prove our main result Theorem3.1.1.In Chapter 3 we discuss the main result of this paper. Without losing generality, we assume that h(u)=u+u3. In particular, in§3.2 we discuss the Hamiltonian setting and reducibility of wave equations.Firstly, let us rewrite the wave equation(0.1) as follows H=H+εG4, where andFor convenience we introduce another coordinates(…,ω₂,ω-1,ω1,w2,…)by setting zj=ωj,zj=ω_j. There is a real analytic canonical transformation∑∞0 changes Hamiltonian H into where And Hamiltonian G4 is changed into This implies the Hamiltonian H=H+εG4 is changed by the transformation∑∞0 into H=H0+εG4.In§3.3 we transform the Hamiltonian obtained in§3.2 into some partial Birkhoff normal form of order four for using the KAM theorem in [57]. There exists a real analytic,symplectic change of coordinates XF1 in some neighborhood of the origin on the now complex Hilbert space la,s that takes the hamiltonian H=H0+εG4 into H o XF1=Ho+εG+εG+ε2K, where XG,XG,XK∈A(la,s,la,a,s), with uniquely determined coefficients whereω(ω,ε)depends smoothly onεandωand there is an absolute constant C such that‖ωij(ω,ε)‖Ω*≤Cερforεsmall enough,and we have |G|=O(‖z‖a4,s),|K|=O(‖z‖a6,s), uniformly for |ImÏ…|<σo/3,ω∈Ω,Z=(zn+1,zn+2,...). We introduce the action-angle variable by setting the normal form becomes with I=(I1,…,In),A=(Gij)1≤i,j≤n,B=(Gij)1∈Ωarbitrarily.Forω∈Ω:={ω∈Ω||ω-ω_|≤ε},we can introduce new parameterωby the followingω=ω₊εω,ω∈[0,1]m. Hence,the Hamiltonian becomes whereω(ξ)=ω(?)ωwithω=α+ε2 Aξ,Ω(ξ)=β+ε2Bξ,andξ=ω(?)ξ, y=J(?)ρ,α=(μ1,…,μn),β=(μn+1,μn+2,…)We apply the infinite dimensional KAM theorem in [57] to the above Hamiltonian. We have that the nonlinear wave equation(0.1) possess a solution of the form where fi(Ï…,ω,ε)=λ(?)ερfj*(Ï…,ω,ε)is of period 2Ï€in each component ofÏ…and for j∈(?),υ∈Θ(σ0/2),ω∈Ω,we have |fj*(Ï…,ω,ε)|≤C.