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- W2951465684 abstract "We study growth of higher Sobolev norms of solutions to the one-dimensional periodic nonlinear Schrodinger equation (NLS). By a combination of the normal form reduction and the upside-down I-method, we establish |u(t)|_{H^s} lesssim (1+|t|)^{alpha (s-1)+} with alpha = 1 for a general power nonlinearity. In the quintic case, we obtain the above estimate with alpha = 1/2 via the space-time estimate due to Bourgain [4], [5]. In the cubic case, we concretely compute the terms arising in the first few steps of the normal form reduction and prove the above estimate with alpha = 4/9. These results improve the previously known results (except for the quintic case.) In Appendix, we also show how Bourgain's idea in [4] on the normal form reduction for the quintic nonlinearity can be applied to other powers." @default.
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- W2951465684 date "2010-10-12" @default.
- W2951465684 modified "2023-09-27" @default.
- W2951465684 title "A remark on normal forms and the upside-down I-method for periodic NLS: growth of higher Sobolev norms" @default.
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