Lower dimensional invariant tori with prescribed frequency for nonlinear wave equation |
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Authors: | Jiansheng Geng Xiufang Ren |
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Affiliation: | Department of Mathematics and Institute of Mathematical Science, Nanjing University, Nanjing 210093, PR China |
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Abstract: | In this paper, one-dimensional (1D) nonlinear wave equation utt−uxx+mu+u3=0, subject to Dirichlet boundary conditions is considered. We show that for each given m>0, and each prescribed integer b>1, the above equation admits a Whitney smooth family of small-amplitude quasi-periodic solutions with b-dimensional Diophantine frequencies, which correspond to b-dimensional invariant tori of an associated infinite-dimensional dynamical system. In particular, these 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. |
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Keywords: | Wave equation Hamiltonian system Birkhoff normal form KAM theory Invariant tori |
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