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A remark on normal forms and the “upside-down” I-method for periodic NLS: Growth of higher Sobolev norms

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Abstract

We study growth of higher Sobolev norms of solutions of the onedimensional periodic nonlinear Schrödinger equation (NLS). By a combination of the normal form reduction and the upside-down I-method, we establish

$${\left\| {u(t)} \right\|_{{H^s}}} \le {(1 + \left| t \right|)^{a(s - 1) + }}$$

with α = 1 for a general power nonlinearity. In the quintic case, we obtain the above estimate with α = 1/2 via the space-time estimate due to Bourgain [4, 5]. In the cubic case, we compute concretely the terms arising in the first few steps of the normal form reduction and prove the above estimate with α = 4/9. These results improve the previously known results (except for the quintic case). In the 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.

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4, Canada

    James Colliander

  2. Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, 335 Gwahangno (373-1 Guseong-Dong), Yuseong-gu, Daejeon, 305-701, Republic of Korea

    Soonsik Kwon

  3. Department of Mathematics, Princeton University, Princeton, NJ, 08544-1000, USA

    Tadahiro Oh

Authors
  1. James Colliander
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  2. Soonsik Kwon
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  3. Tadahiro Oh
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Corresponding author

Correspondence to James Colliander.

Additional information

J. C. is supported in part by NSERC grant RGP250233-07.

S. K. is supported in part by NRF 2010-0024017.

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Colliander, J., Kwon, S. & Oh, T. A remark on normal forms and the “upside-down” I-method for periodic NLS: Growth of higher Sobolev norms. JAMA 118, 55–82 (2012). https://doi.org/10.1007/s11854-012-0029-z

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  • DOI: https://doi.org/10.1007/s11854-012-0029-z

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