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

In this paper, we consider a nonlinear Schodinger equation. There are many results of the research on Schodinger equations. In particular, nonlinear Schrodinger equations of the form ivt-（-△+V（x）+m）u+（γ0+γ1γ（t））|μ|2u=0,m>0,γ0,γ1∈R,（0.8） appear in the modelling of many physical phenomena, which is an issue of direct interest to Bose-Einstein condensates in the context of the Feshbach-resonance control, and fiber-optic telecommunications as concerns periodic compensation of the nonlinearity. Presently there are many significant re-sults with respect to the above equations, see [1]-[5]. However, little is known about the dynamical behavior of equation （0.8） and the analysis of solutions, when γ（t） is a quasi-periodic function in t.In this paper, we consider the existence of quasi-periodic solutions of the following quasi-periodically forced nonlinear Schrodinger equations via KAM theory iut-uxx+mu+Φ（t）|u|2u=εg（t）（0.9） subject to periodic boundary conditions u（t,x）=u（t,x+2π）,（0.10） where ε is a small parameter, m>0, Φ>（t） and g（t） are two real analytic quasi-periodic functions in t with frequency vector ω=（ω1,ω2...,WL） and for 0<e<1,Λ>0, ω∈DΛ:={ω∈[e,2e]L:|<κ,ω>|≥Λ/|κ|L+1,0≠∈ZL}.It is easy to see that u≡0is not the solution of（0.9）for g（t）（?）0and ε≠0. We first need to study the existence quasi-periodic solutions of a complex ordinary differential equation with respect to the unknown function u（t） iu+mu+Φ（t）|u|2u=εg（t）,（0.11） then we obtain the following equation with zero equilibrium point ivt-vxx+mv+Φ（t）（2|u0|2v+u02v+2εu0|v|2+εu0v2+ε2|v|2v）=0（0.12） by letting u=uo（t）+εv（x,t）in（0.9）,here u0（t）is a nonzero quasi-periodic solution of（0.11）,and construct the invariant tori or quasi-periodic solutions of （0.9）+（0.10）by means of KAM theory.Finally,we obtain that（0.9）and（0.10） have many quasi-periodic solutions in the neighborhood of a quasi-periodic solutions to（0.11）.In the following,We introduce the organization of this paper.In Chapter1the definitions and the characters of almost integrable hamil-tonian system are included..We introduce the definition of symplectic struc-ture.We also introduce the classical KAM theory and an infinite-dimensional KAM theory.In the last we describe the research background and progress of nonlinear Schrodinger equations with quasi-periodic forcing.Chapter2is served as the proof of the main Theorem.We need to obtain the solutions of system（0.11）and（0.12）.We first obtain a solution of nonlinear ODE（0.11）with quasi-periodic forcing by Bibikov’Lemma（see[6]）,i.e., u0（t,ξ,ε）=O（ε1/4） with frequency vector ω（ξ）=（ω1,ω2,...,ωL,α（ξ））∈DΛ×Aγ,where Aγ:={α∈R:|<κ:ω>+lα|>γ（|κ|+|l|）-（L+1）,0≠（k,l）∈ZL×Z}. We Consider the equation（0.12）with a zero equilibrium point by KAM theory. We overcome some difficulties of quasi-periodically forced term via reducibility. In the end,we get the Birkhoff normal,i.e., where ξ,∈=（ξj）o≤j∈n,ξj∈[0,1]are parameters,and where o（1）�'0as ε�'0,and P=ε2G**+ε2G*+ε3K,G**=O（|ρ|2）+O（|ρ|||z||）.In order to apply the KAM theorem to our problem,we introduce a new parameter ω below.For fixed ω₌（ω1_,ω2_,...,ωL_,ωL+1_）∈（?）*and ω（ξ）=（ω1,...,ωL,α（ξ）） arbitrarily.Forω（ξ）∈=Ⅱ*={ω（ξ）∈DA×Aγ:|ωi-ωi_|≤ε,|α（ξ）-ωL+1_ξ|≤ε}, we can introduce new parameter ω=（ω1,ω2,...,ωL,ωL+1）by the following ωj=ωj+ε3εj,εj∈[0,1],j=1,...L, α（ξ）=ωL+1₊ε3ωL+1,ωL+1∈[0,1]. Hence,the Hamililtonian（0.13）becomes H=<ω（ξ）,y>+<Ω（ξ）,Z>+P （0.14） where ω（ξ）=ω（ξ）（?）ω0（?）eω with ω=α+ε3Aξ, Ω（ξ）=β+ε3Bξ, ξ=ω（?）ξ0（?）ξ, ξ=（ξ1,...,ξn）, and y=J（?）ρ0（?）ρ, α=（ω1,...,ωn）,β=（λn+1,κn+2,...）.We can verify that Hamiltonian（0.14）satisfies the conditions of KAM theorem in[7].In the end,we prove the existence of the quasi-periodic solutions of the equation（0.9）Via an infinite-dimensional KAM theory.We obtain the main result of this paper:Assume thatΦ（t）and g（t）are real analytic and quasi-periodic functions in t,with ω∈DΛ.For each index set （?）=（1,2,…,n）with n≥1,there is a small enough positive ε*such that for any0<ε<ε*,there are the subsets J （?）[1/2（m+2[Φ]）,3/2（m+2[Φ]）],（ω,α（ξ））∈（?）*（?） DA×Aγ and∑ε（?）∑DΛ×Aγ×[0,1]n+1with measJ>0and meas（∑\∑ε）≤ε,such that for any ξ∈J,（ω,α（ξ）,ξ0,ξ1,...,ξn）∈∑ε,the system（0.9）+（0.10） possess a solution of the following form with frequency vector ω=（ω,α（ξ）,（ωj）0≤j≤n）∈RL+n+2, where （1）ω is the frequency vector of Φ and g,while α,ωj are constructed in the proof,and are functions of ε and of parameters ξ,ξ=（ξ0,...,ξn）∈Rn+1. In particular, ωj=μj（ε）+ε3αj（ξ,ε）, μj=j2+m+ε1/2（cj+rj（ε））, j=0,1,...,n, where cj is a constant,|Rj（ε）|�'0as ε�'0,and|αj（ξ,ε）|≤C|ξ|；（2）u0（t）is a non-trivial solution of（0.11）depending on the parameters （ξ,ε）,it is of size O（ε1/4） of quasi-periodic with frequency（ω.α（ξ））.（3）u1（t,x）is a solution of the linear equation i（?）tu11-（?）xxu1+mu1=0, and is quasi-periodic with frequencies j2+m.In Chapter3,some measure estimates are proved. We obtain the Birkhoff normal（0.13）by some symplectic transformations.However,there exist the problems of "small divisor" in estimating symplectic transformations. Thus, in Appendix we construct all of the measure estimates.