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On the rigidity of solitary waves for the focusing mass-critical NLS in dimensions d⩾2

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Abstract

For the focusing mass-critical NLS \(iu_t + \Delta u = - \left| u \right|^{\tfrac{4} {d}} u\), it is conjectured that the only global non-scattering solution with ground state mass must be a solitary wave up to symmetries of the equation. In this paper, we settle the conjecture for H 1 x initial data in dimensions d = 2, 3 with spherical symmetry and d ⩾ 4 with certain splitting-spherically symmetric initial data.

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Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z3, Canada

    Dong Li

  2. Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Beijing, 100190, China

    XiaoYi Zhang

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  1. Dong Li
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  2. XiaoYi Zhang
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Correspondence to XiaoYi Zhang.

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Dedicated to the NSFC-CNRS Chinese-French summer institute on fluid mechanics in 2010

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Li, D., Zhang, X. On the rigidity of solitary waves for the focusing mass-critical NLS in dimensions d⩾2 . Sci. China Math. 55, 385–434 (2012). https://doi.org/10.1007/s11425-012-4359-1

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