KAM-Type Theorem For Infinite-dimensional Hamiltonian Systems

There two parts in my paper. The first part is the persistance of random Hamiltonian systems. The other is the KAM-tye theorem for indimensional Hamiltonian with special spectrum Asymptotics. Throughout the paper, we use KAM iteration to proof the persistance of nealy integrable Hamiltonian.The classical KAM theorem, constructed by Kolmogorov [37], Arnold [1], Moser[53], interpreted dynamics behaviour of solar system successfully.KAM-type theorems is an important tool to study the existence of quai-solutions for PDEs with Hamiltonian structure. So, we can summarize the development of the theorems according to the structure of the Strum Liouville operator A=-(d2)/(dx2)+V(x,ξ).Letλn is an eigenvalue of A. For Dirichlet boundary problems, the eigenvalues can be denoted asλ1,λ2,…, and the componens are unequal to each other. This kind problems can be sovled by Poschel (see[57]). However, for the periodic boundary problems, the mul-tiplicity of the eigenvalues is finite. This problem solved by Chierchia (see [14]). When the multiplicity goes to infinite, one can find the results in [30].In section 3, we consider the infinite dimensional Hamiltonian system with small per-turbation where (x,y,z,z) lies in the complex neighborhood of Tn×{0}×{0}×{0},Tn is an n-torus, and a≥0, p≥0. The frequenciesωandΩdepend onξ∈∏(?)Rn,∏is a closed bounded region. With respect to the symplectic form the associated unperturbed motion of(0.0.12)can be described by Hence,for eachξ∈П,there is a family of invariant n-dimensional torus Tξ=П"×{0}×{0)×{0}with fixed frequencyω(ξ).Our aim is to prove that most of the invariant tori persists under small perturbation P. To formulate out theorem,we make the following assumptions:A1)Nondegeneracy.The mapξ→ω(ξ)is a homemorphism betweenΠand its image, and Lipschtiz continuous in both directions.Moreover,for l=(…,li,…)i∈Z1p,▕l▕=∑i∈Z1p,il≤2,let△_denote those l with two non-zero components of opposite sign,and△+denote the others.Then for▕l▕=0,▕k▕f≠0,or l∈△+,or l∈△_,▕k▕+║i▕-▕j║≠where i,jare the site of the nonzero components of l.A2)Spectral Asymptotics.There exist d>0 andδ<0 such that for all i∈Z1p,Ωi(ξ)≠0,i∈Z1p,ξ∈Π, (0.0.13)Ωi(ξ)=Ω-i+Ω-i,Ω-i=▕i▕d+o(▕i▕d),Ω-i=o(▕i▕δ), (0.0.14) and the asymptotic behavior ofΩi is assumed to be as following:Ω-i-Ω-j=▕i▕d-▕j▕d+o(▕j▕-δ),▕j▕≤▕i▕. (0.0.15) Moreover,ω(ξ),Ω(ξ)are Lipschtiz continuous inξ whereA3)Regularity.The perturbation P∈fa-,p-a,p,that is,P is real analytic in the space coordinate and Lipschitz in the parameters,and for eachξ,Hamiltonian vector space field Xp= (Py,-Px,Pz-,-Pz)T defines a real analytic map XP:Pa,p→pa,p, wherea->a>0,p-≥p≥0.A4)Special form of the perturbation.The perturbation P∈A where We suppose that H=N+P satisfies Al)-A4).定理0.5 For given r,s>0,if there exits a sufficiently smallμ=μ(α,p,p-,r,s,n,τ)>0,or. equivalentlyμ*=μ*(a,p,p-,r,s,n,τ)＞0,such that▕XP▕α,p≤γμ, (0.0.16)▕Xp▕￡α-p≤Mμ(0.0.17)for all(x,y,z,z-,)∈Dα,p(r,s),ξ∈Π,then there exit a Cantor setΠγCΠ,with▕Π\Πγ▕→0, nsγ→0,and a family of C2 symplectic transformations which are Lipschitz continuous in parameterξand C2 uniformly close to the identity, such that for eachξ∈Πγ, corresponding to the unperturbed torus Tξ, the associated perturbed invariant torus can be described as where Moreover the perturbation P*=poφξis real analytic in phase variables x, Lipshchtz continuous in the parameters, and for all x€Πn,ξ∈Πγ. Namely. the unperturbed torus T=ξΠn×{0} x{0} x{0} associated to the frequencyω(ξ) persists and gives rise to an analytic, Diophantine, invariant torus of the perturbed system with the frequencyω*(ξ). Moreover, these perturbed tori form a Lipshcitz continuous family.For p=1, we consider the following Hamiltonian systemsSuppose that H= N+P satisfies A1)'-A3)', we have定理0.6 For given r, s, a> 0, if there exits a sufficiently smallμ=μ(a,p,p-,r,s,n,τ)> 0, or equivalentlyμ*=μ*(a, p, p-,r,s,n,τ)> 0, such that the perturbation P satisfies (0.0.16) and (0.0.17) for all (x,y,z,z-)∈Da,p(r,s),ξ∈Π, then there exit a Cantor setΠγCΠ, with |Π\Πγ|→0, as yγ→0, and a family of C2 symplectic transformations which are Lipschitz continuous in parameterξand C2 uniformly close to the identity, such that for eachξ∈Πγ, corresponding to the unperturbed torus Tξ. the associated perturbed invariant torus can be described as where The perturbation P*= Poφξis real analytic in phase variables x, Lipshchtz continuous in the parameters, and for all x∈Πn,ξ∈Πγ. Namely, the unperturbed torus Tξ=Πn x{0} x{0} x{0} associated to the frequencyω(ξ) and zz-space frequencyΩ(ξ) persists and gives rise to an analytic, Diophantine, invariant torus of the perturbed system with the frequencyω*(ξ) and zz-—pace frequencyΩ*(ξ). Moreover, these perturbed tori form a Lipshcitz continuous family.Apply the results above, we proof the existence of the quasi-periodic solutions for wave equations with periodic boundary condition and Schrodinger equations with special spec-trum.Consider the wave equations with periodic boundary condition where F is real analytic and ((?)F)/((?)u)=f(u2)u with f(0)=0,f'(0)=0 andξis a parameter on a closed set in Rρ. Then, we assume:Ⅰ) Then operator A=-△+Mξunder the boundary condition (0.0.20) admits the spectrum定理0.7 Under the assumptionsⅠ), for any 0<γ≤1, there exists a Cantor set∏γ(?)∏, with |∏\∏γ|= O(γ), such that for anyξ∈∏γ, wave equations (0.0.19) subjected to the boundary condition (0.0.20) admits a family of small amplitude quasi-periodic solutions u*(t,x) with respect to time t. Moreover u*(t,x)∈Wba,p for fixed t. Consider the Schrodinger equations with the linear boundary conditionwhereΩ(?)Rρis a bounded domain with smooth boundary (?)Ω. v is the normal vector to the boundary (?)Ω, F is real analytic and ((?)F)/((?)u)=f(|u|2)u with f(0)= 0, f'(0)≠0,ξis defined as before. We make the following assumption:Ⅱ) The eigenvalues of operator A=-△+V(x,ξ) under the boundary condition (0.0.22) satisfy where j=1,2,…andω= (ω1,…,ωn),Ω= (…,Ωj,…) satisfy A1)', A2)'in Theo-rem 0.6. 0.8 Under the assumptions 11), for any 0<γ<1,there exists a Cantor setΠγcΠ, with|Π\Πγ▕= O(γ), such that for anyξ∈Πγ, Schrodinger equations (0.0.21) subjected to the boundary condition (0.0.22) admits a family of small amplitude quasi-periodic solutions u*(t,x) with respect to timet. Moreover u*(t, x)∈Wa,pb for fixed t.