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On Decay Properties of Solutions of the k-Generalized KdV Equation

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Abstract

We prove special decay properties of solutions to the initial value problem associated to the k-generalized Korteweg-de Vries equation. These are related with persistence properties of the solution flow in weighted Sobolev spaces and with sharp unique continuation properties of solutions to this equation. As an application of our method we also obtain results concerning the decay behavior of perturbations of the traveling wave solutions as well as results for solutions corresponding to special data.

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References

  1. Bergh, J., Löfström, J.: Interpolation spaces. New York and Berlin: Springer, 1970

  2. Bourgain, J.: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations I, II. Geom. Funct. Anal. 3, 107–156, 209–262 (1993)

    Google Scholar 

  3. Colliander J., Keel M., Staffilani G., Takaoka H., Tao T.: Sharp global wellposedness results for periodic and non-periodic KdV and modified KdV on \({{\mathbb{R}}}\) and \({{\mathbb{T}}}\) . J. Amer. Math. Soc. 16, 705–749 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Escauriaza L., Kenig C.E., Ponce G., Vega L.: On Uniqueness Properties of Solutions of the k-generalized KdV. J. Funct. Anal. 244(2), 504–535 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ginibre J., Tsutsumi Y.: Uniqueness for the generalized Korteweg-de Vries equations. SIAM J. Math Anal. 20, 1388–1425 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  6. Grünrock A.: A bilinear Airy type estimate with application to the 3-gkdv equation. Diff. Int. Eqs. 18, 1333–1339 (2005)

    MATH  Google Scholar 

  7. Grünrock A., Panthee M., Drumond Silva J.: A remark on global well-posedness below L 2 for the gKdV-3 equation. Diff. Int. Eqs. 20, 1229–1236 (2007)

    MATH  Google Scholar 

  8. : On the Cauchy problem for the (generalized) Korteweg-de Vries equation. Adv. in Math. Suppl. Studies, Studies in App. Math. 8, 93–128 (1983)

    Google Scholar 

  9. Kenig C.E., Ponce G., Vega L.: Well-posedness and scattering results for the generalized Korteweg-de Vries equation via contraction principle. Comm. Pure Appl. Math. 46, 527–620 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kenig C.E., Ponce G., Vega L.: A bilinear estimate with applications to the KdV equation. J. Amer. Math. Soc. 9, 573–603 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kruzhkov S.N., Faminskii A.V.: Generalized solutions of the Cauchy problem for the Korteweg-de Vries equation. Math. U.S.S.R. Sbornik 48, 93–138 (1984)

    Article  Google Scholar 

  12. Nahas J., Ponce G.: On the persistent properties of solutions to semi-linear Schrödinger equation. Comm. P.D.E. 34, 1–20 (2009)

    Article  MathSciNet  Google Scholar 

  13. Nahas, J., Ponce, G.: On the well-posedness of the modified Korteweg-de Vries equation in weighted Sobolev spaces Preprint, available at http://www.math.ucsb.edu/~ponce/jnjpz.pdf

  14. Robbiano L.: Unicité forte à l’infini pour KdV. Control Opt. and Cal. Var. 8, 933–939 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Tao T.: Scattering for the quartic generalized Korteweg-de Vries equation. J. Diff. Eqs. 232, 623–651 (2007)

    Article  ADS  MATH  Google Scholar 

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad Nacional de Colombia, A. A. 3840, Medellin, Colombia

    Pedro Isaza

  2. IMPA, Instituto Matemática Pura e Aplicada, Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, RJ, Brazil

    Felipe Linares

  3. Department of Mathematics, University of California, Santa Barbara, CA, 93106, USA

    Gustavo Ponce

Authors
  1. Pedro Isaza
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  2. Felipe Linares
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  3. Gustavo Ponce
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Correspondence to Felipe Linares.

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Communicated by W. Schlag

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Isaza, P., Linares, F. & Ponce, G. On Decay Properties of Solutions of the k-Generalized KdV Equation. Commun. Math. Phys. 324, 129–146 (2013). https://doi.org/10.1007/s00220-013-1798-7

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  • DOI: https://doi.org/10.1007/s00220-013-1798-7

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