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Blowing Up Solutions to the Zakharov System for Langmuir Waves

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Laser Filamentation

Part of the book series: CRM Series in Mathematical Physics ((CRM))

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Abstract

Langmuir waves take place in a quasi-neutral plasma and are modeled by the Zakharov system. The phenomenon of collapse, described by blowing up solutions, plays a central role in their dynamics. We present in this article a review of the main mathematical properties of blowing up solutions. They include conditions for blowup in finite or infinite time, description of self-similar singular solutions and lower bounds for the rate of blowup of certain norms associated with the solutions.

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Acknowledgements

MC is partially supported by grant #246255 from the Simons Foundation. CS is partially supported by NSERC through grant number 46179–13 and Simons Foundation Fellowship #265059.

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Authors and Affiliations

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

    Yuri Cher & Catherine Sulem

  2. Department of Mathematical Sciences, Binghamton University, Binghamton, NY, 13902-6000, USA

    Magdalena Czubak

Authors
  1. Yuri Cher
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  2. Magdalena Czubak
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  3. Catherine Sulem
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Corresponding author

Correspondence to Catherine Sulem .

Editor information

Editors and Affiliations

  1. Faculté des Sciences, Université de Sherbrooke, Sherbrooke, Québec, Canada

    Andre D. Bandrauk

  2. School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada

    Emmanuel Lorin

  3. Arizona Center for Mathematical Sciences, University of Arizona, Tucson, Arizona, USA

    Jerome V. Moloney

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Cher, Y., Czubak, M., Sulem, C. (2016). Blowing Up Solutions to the Zakharov System for Langmuir Waves. In: Bandrauk, A., Lorin, E., Moloney, J. (eds) Laser Filamentation. CRM Series in Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-23084-9_3

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