Quasi-periodic solutions with prescribed frequency in a nonlinear Schrdinger equation  

作　　者：REN Xiu-Fang Department of Mathematics, Nanjing University, Nanjing 210093, China 

出　　处：《Science China Mathematics》2010年第12期3067-3084,共18页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation (Grant Nos.10531050, 10771098);National Basic Research Program of China (973 Projects) (Grant No. 2007CB814800)

摘　　要：In this paper, one-dimensional (1D) nonlinear Schrdinger equation iut-uxx + Mσ u + f ( | u | 2 )u = 0, t, x ∈ R , subject to periodic boundary conditions is considered, where the nonlinearity f is a real analytic function near u = 0 with f (0) = 0, f (0) = 0, and the Floquet multiplier Mσ is defined as Mσe inx = σne inx , with σn = σ, when n 0, otherwise, σn = 0. It is proved that for each given 0 【 σ 【 1, and each given integer b 】 1, the above equation admits a Whitney smooth family of small-amplitude quasi-periodic solutions with b-dimensional Diophantine frequencies, corresponding to b-dimensional invariant tori of an associated infinite-dimensional Hamiltonian system. Moreover, these b-dimensional Diophantine frequencies are the small dilation of a prescribed Diophantine vector. The proof is based on a partial Birkhoff normal form reduction and an improved KAM method.In this paper, one-dimensional (1D) nonlinear Schrdinger equation iut-uxx + Mσ u + f ( | u | 2 )u = 0, t, x ∈ R , subject to periodic boundary conditions is considered, where the nonlinearity f is a real analytic function near u = 0 with f (0) = 0, f (0) = 0, and the Floquet multiplier Mσ is defined as Mσe inx = σne inx , with σn = σ, when n 0, otherwise, σn = 0. It is proved that for each given 0 < σ < 1, and each given integer b > 1, the above equation admits a Whitney smooth family of small-amplitude quasi-periodic solutions with b-dimensional Diophantine frequencies, corresponding to b-dimensional invariant tori of an associated infinite-dimensional Hamiltonian system. Moreover, these b-dimensional Diophantine frequencies are the small dilation of a prescribed Diophantine vector. The proof is based on a partial Birkhoff normal form reduction and an improved KAM method.

关 键 词：Schrdinger equation HAMILTONIAN system BIRKHOFF NORMAL FORM QUASI-PERIODIC solution 

分 类 号：O175.29[理学—数学]