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Communications in Mathematical Physics

, Volume 367, Issue 2, pp 581–598 | Cite as

Breathers and the Dynamics of Solutions in KdV Type Equations

  • Claudio Muñoz
  • Gustavo PonceEmail author
Article

Abstract

In this paper our first aim is to identify a large class of non-linear functions  f(·)  for which the IVP for the generalized Korteweg–de Vries equation does not have breathers or “small” breathers solutions. Also, we prove that all uniformly in time L1H1 bounded solutions to KdV and related “small” perturbations must converge to zero, as time goes to infinity, locally in an increasing-in-time region of space of order t1/2 around any compact set in space. This set is included in the linearly dominated dispersive region xt. Moreover, we prove this result independently of the well-known supercritical character of KdV scattering. In particular, no standing breather-like nor solitary wave structures exists in this particular regime.

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Notes

Acknowledgements

We are indebted to M.A. Alejo for several interesting comments and remarks about a first version of this work.

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CNRS and Departamento de Ingeniería Matemática DIM-CMM UMI 2807-CNRSUniversidad de ChileSantiagoChile
  2. 2.Department of MathematicsUniversity of California-Santa BarbaraSanta BarbaraUSA

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