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Self-similar Blow-Up Profiles for Slightly Supercritical Nonlinear Schrödinger Equations

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Abstract

We construct radially symmetric self-similar blow-up profiles for the mass supercritical nonlinear Schrödinger equation \(\text {i}\partial _t u + \Delta u + |u|^{p-1}u=0\) on \(\mathbb {R}^d\), close to the mass critical case and for any space dimension \(d\ge 1\). These profiles bifurcate from the ground-state solitary wave. The argument relies on the classical matched asymptotics method suggested in Sulem and Sulem (The nonlinear Schrödinger equation. Self-focusing and wave collapse. Applied mathematical sciences, 139, Springer, New York, 1999) which needs to be applied in a degenerate case due to the presence of exponentially small terms in the bifurcation equation related to the log–log blow-up law observed in the mass critical case.

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Acknowledgements

Y.B. is partially supported by the ERC-2014-CoG 646650 SingWave. P.R. is supported by the ERC-2014-CoG 646650 SingWave. Y.M. would like to thank the DPMMS, University of Cambridge for its hospitality. P.R. would like to thank the Université de la Côte d’Azur where part of this work was done for its kind hospitality. The authors thank E. Lombardi (Toulouse) and T. Cazenave (Paris 6) for enlightening discussions. The authors are also grateful to S. Aryan (École polytechnique) for his careful reading of the manuscript.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Victoria, Victoria, Canada

    Yakine Bahri

  2. CMLS, Ecole polytechnique, CNRS, Institut Polytechnique de Paris, 91128, Palaiseau Cedex, France

    Yvan Martel

  3. DPMMS, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK

    Pierre Raphaël

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  1. Yakine Bahri
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Correspondence to Yvan Martel.

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Communicated by Nader Masmoudi.

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Bahri, Y., Martel, Y. & Raphaël, P. Self-similar Blow-Up Profiles for Slightly Supercritical Nonlinear Schrödinger Equations. Ann. Henri Poincaré 22, 1701–1749 (2021). https://doi.org/10.1007/s00023-020-01006-z

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