Quasi-periodic solutions with prescribed frequency in a nonlinear Schrödinger equation
Quasi-periodic solutions with prescribed frequency in a nonlinear Schrödinger equation
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DOI：
10.1007/s11425-010-4074-8
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发表时间：
2010-09
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影响因子：
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通讯作者：
Xiufang Ren
中科院分区：
文献类型：
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作者：
Xiufang Ren
本文研究了一维非线性薛定谔方程iut −uxx+Mσu+f（|u| 2）u= 0，t，x∈ n，在周期边界条件下，非线性函数f是u = 0附近的真实的解析函数，f（0）= 0，f′（0）= 0，Floquet乘数M σ定义为M σeinx= σneinx，其中σn= σ，当n = 0，否则σn= 0。证明了对于任意给定的0 < σ < 1和任意给定的整数b> 1，上述方程存在一个Whitney光滑的小振幅拟周期解族，其频率为b维丢番图，对应的无穷维Hamilton系统的不变环面为b维不变环面.此外，这些b维丢番图频率是一个指定的丢番图向量的小膨胀。证明是基于部分Birkhoff规范形式减少和改进的KAM方法。
In this paper, one-dimensional (1D) nonlinear Schrödinger equationiut−uxx+Mσu+f(|u|2)u= 0,t,x∈ ℝ, subject to periodic boundary conditions is considered, where the nonlinearityfis a real analytic function nearu= 0 withf(0) = 0,f′(0) ≠ 0, and the Floquet multiplierMσis defined asMσeinx= σneinx, with σn= σ, whenn⩾ 0, otherwise, σn= 0. It is proved that for each given 0 < σ < 1, and each given integerb> 1, the above equation admits a Whitney smooth family of small-amplitude quasi-periodic solutions withb-dimensional Diophantine frequencies, corresponding tob-dimensional invariant tori of an associated infinite-dimensional Hamiltonian system. Moreover, theseb-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.