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Nonexistence of small, odd breathers for a class of nonlinear wave equations

Abstract

In this note, we show that for a large class of nonlinear wave equations with odd nonlinearities, any globally defined odd solution which is small in the energy space decays to 0 in the local energy norm. In particular, this result shows nonexistence of small, odd breathers for some classical nonlinear Klein Gordon equations, such as the sine-Gordon equation and \(\phi ^4\) and \(\phi ^6\) models. It also partially answers a question of Soffer and Weinstein (Invent Math 136(1): 9–74, p 19 1999) about nonexistence of breathers for the cubic NLKG in dimension one.

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Acknowledgements

M. Kowalczyk was partially supported by Chilean research Grants FONDECYT 1130126, Fondo Basal CMM-Chile. C. Muñoz was partly funded by Chilean research Grants FONDECYT 1150202, Fondo Basal CMM-Chile, and Millennium Nucleus Center for Analysis of PDE NC130017 Part of this work was done, while the authors were hosted at the IHES during the program Nonlinear Waves 2016. Their stay was partly funded by the Grant ERC 291214 BLOWDISOL.

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Correspondence to Yvan Martel.

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Kowalczyk, M., Martel, Y. & Muñoz, C. Nonexistence of small, odd breathers for a class of nonlinear wave equations. Lett Math Phys 107, 921–931 (2017). https://doi.org/10.1007/s11005-016-0930-y

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Keywords

  • Nonlinear wave equation
  • Sine-Gordon
  • Klein-Gordon
  • Breathers
  • Asymptotic behaviour

Mathematics Subject Classification

  • 35J61