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A Simple Proof of the Stability of Solitary Waves in the Fermi-Pasta-Ulam Model Near the KdV Limit

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Infinite Dimensional Dynamical Systems

Part of the book series: Fields Institute Communications ((FIC,volume 64))

Abstract

By combining results of Mizumachi and Pego on the stability of solutions for the Toda lattice with a simple rescaling and a careful control of the KdV limit we give a simple proof that small amplitude, long-wavelength solitary waves in the Fermi-Pasta-Ulam (FPU) model are linearly stable and hence by the results of Friesecke and Pego that they are also nonlinearly, asymptotically stable.

Mathematics Subject Classification (2010): Primary 37K60; Secondary 37K10, 37K90

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References

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Acknowledgements

A. Hoffman was supported in part by the NSF grant DMS-0603589. C. Eugene Wayne was supported in part by the NSF grant DMS-0405724. Any findings, conclusions, opinions, or recommendations are those of the authors, and do not necessarily reflect the views of the NSF.The work reported here was completed while A. Hoffman was a member of the Department of Mathematics and Statistics, Boston University. The authors also wish to thank T. Mizumachi and R. Pego for very helpful discussions.

Received 9/12/2009; Accepted 8/23/2010

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Authors and Affiliations

  1. Franklin W. Olin College of Engineering, Olin Way, Needham, MA, 02492, USA

    A. Hoffman

  2. Department of Mathematics and Statistics and Center for BioDynamics, Boston University, 111 Cummington Street, Boston, MA, 02215, USA

    C. Eugene Wayne

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  1. A. Hoffman
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  2. C. Eugene Wayne
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Correspondence to A. Hoffman .

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Editors and Affiliations

  1. Division of Applied Mathematics, Brown University, Providence, RI, USA

    John Mallet-Paret

  2. Department of Mathematics and Statistics, York University, Toronto, ON, Canada

    Jianhong Wu  & Huaiping Zhu  & 

  3. Georgia Institute of Technology, School of Mathematics, Atlanta, GA, USA

    Yingfei Yi

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Hoffman, A., Wayne, C.E. (2013). A Simple Proof of the Stability of Solitary Waves in the Fermi-Pasta-Ulam Model Near the KdV Limit. In: Mallet-Paret, J., Wu, J., Yi, Y., Zhu, H. (eds) Infinite Dimensional Dynamical Systems. Fields Institute Communications, vol 64. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4523-4_7

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