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On the theory of Weak Turbulence for the Nonlinear Schrödinger Equation


About this Title

M. Escobedo and J. J. L. Velázquez

Publication: Memoirs of the American Mathematical Society
Publication Year: 2015; Volume 238, Number 1124
ISBNs: 978-1-4704-1434-4 (print); 978-1-4704-2611-8 (online)
DOI: http://dx.doi.org/10.1090/memo/1124
Published electronically: April 15, 2015
Keywords:amsbook, Weak turbulence, finite time blow up, condensation, pulsating solution

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. Well-Posedness Results
  • Chapter 3. Qualitative behaviors of the solutions
  • Chapter 4. Solutions without condensation: Pulsating behavior
  • Chapter 5. Heuristic arguments and open problems
  • Chapter 6. Auxiliary results

Abstract


We study the Cauchy problem for a kinetic equation arising in the weak turbulence theory for the cubic nonlinear Schrödinger equation. We define suitable concepts of weak and mild solutions and prove local and global well posedness results. Several qualitative properties of the solutions, including long time asymptotics, blow up results and condensation in finite time are obtained. We also prove the existence of a family of solutions that exhibit pulsating behavior.

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