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Breathers and the Dynamics of Solutions in KdV Type Equations

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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 L1∩ H1 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 x≪ t. 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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References

  1. Ablowitz, M.J., Segur, H.: Solitons and the Inverse Scattering Transform. SIAM Studies in Applied Mathematics, vol. 4, x+425 pp. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1981). ISBN 0-89871-174-6

  2. Alejo M.A.: Nonlinear stability of Gardner breathers. J. Differ. Eqs. 264, 1192–1230 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Alejo M.A., Muñoz C.: Nonlinear stability of mKdV breathers. Commun. Math. Phys. 324(1), 233–262 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Alejo M.A., Muñoz C.: Dynamics of complex-valued modified KdV solitons with applications to the stability of breathers. Anal. PDE. 8(3), 629–674 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Alejo M.A., Muñoz C: Almost sharp nonlinear scattering in one-dimensional Born–Infeld equations arising in nonlinear electrodynamics. Proc. AMS 146(5), 2225–2237 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  6. Alejo M.A., Muñoz C., Palacios J.M.: On the variational structure of breather solutions II. Periodic mKdV equation. EJDE 2017(56), 1–26 (2017)

    MATH  Google Scholar 

  7. Alejo M.A., Muñoz C., Vega L.: The Gardner equation and the L 2-stability of the N-soliton solution of the Korteweg–de Vries equation. Trans. AMS 365(1), 195–212 (2013)

    Article  MATH  Google Scholar 

  8. Christ F.M., Weinstein M.I.: Dispersion of small amplitude solutions of the generalized Korteweg–de Vries equation. J. Funct. Anal. 100(1), 87–109 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  9. Côte R.: Construction of solutions to the subcritical gKdV equations with a given asymptotical behavior. J. Funct. Anal. 241(1), 143–211 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Deift P., Venakides S., Zhou X.: The collisionless shock region for the long-time behavior of solutions of the KdV equation. Commun. Pure Appl. Math. 47, 199–206 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Eckhaus W., Schuur P.: The emergence of solitons of the Korteweg–de Vries equation from arbitrary initial conditions. Math. Methods Appl. Sci. 5, 97–116 (1983)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. 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 

  13. Germain P., Pusateri F., Rousset F.: Asymptotic stability of solitons for mKdV. Adv. Math. 299, 272–330 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hayashi N., Naumkin P.I.: Large time asymptotics of solutions to the generalized Korteweg–de Vries equation. J. Funct. Anal. 159(1), 110–136 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hayashi N., Naumkin P.I.: Large time behavior of solutions for the modified Korteweg–de Vries equation. Int. Math. Res. Not. 8, 395–418 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. Harrop-Griffiths B.: Long time behavior of solutions to the mKdV. Commun. Partial Differ. Equ. 41(2), 282–317 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Isaza P., Linares F., Ponce G.: On decay properties of solutions of the k-generalized KdV equation. Commun. Math. Phys. 324(1), 129–146 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Kato T.: On the Cauchy problem for the (generalized) Korteweg–de Vries equation. Adv. Math. Suppl. Stud. Stud. Appl. Math. 8, 93–128 (1983)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Kenig C.E., Ponce G., Vega L.: Lower bounds for non-trivial traveling wave solutions of equations of KdV type. Nonlinearity 25(5), 1235–1245 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Koch H., Marzuola J.: Small data scattering and soliton stability in H −1/6 for the quartic KdV equation. Anal. PDE 5(1), 145–198 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kowalczyk M., Martel Y., Muñoz C.: Kink dynamics in the ϕ4 model: asymptotic stability for odd perturbations in the energy space. J. Am. Math. Soc. 30, 769–798 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kowalczyk M., Martel Y., Muñoz C.: Nonexistence of small, odd breathers for a class of nonlinear wave equations. Lett. Math. Phys. 107(5), 921–931 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Kowalczyk, M., Martel, Y., Muñoz, C.: On asymptotic stability of nonlinear waves, Seminaire Laurent Schwartz—EDP et applications, vol (2016–2017), Exp. No. 18. https://doi.org/10.5802/SLSEdp.111 (2017)

  25. Kwak, C., Muñoz, C.: Extended decay properties for generalized BBM equations. arXiv:1802.01925 (2018)

  26. Kwak, C., Muñoz, C., Poblete, F., Pozo, J. C.: The scattering problem for the abcd Boussinesq system in the energy space. arXiv:1712.09256 (2017)

  27. Martel Y., Merle F.: A Liouville theorem for the critical generalized Korteweg-de Vries equation. J. Math. Pures Appl. (9) 79(4), 339–425 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  28. Martel Y., Merle F.: Blow up in finite time and dynamics of blow up solutions for the critical generalized KdV equation. J. Am. Math. Soc. 15, 617–664 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  29. Martel Y., Merle F.: Asymptotic stability of solitons for subcritical gKdV equations revisited. Nonlinearity 18(1), 55–80 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Martel Y., Merle F.: Description of two soliton collision for the quartic gKdV equation. Ann. Math. (2) 174(2), 757–857 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Merle F., Raphaël P.: The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation. Ann. Math. (2) 161(1), 157–222 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  32. Muñoz, C.: On the inelastic 2-soliton collision for generalized KdV equations. IMRN 9, 1624–1719 (2010). arXiv:0903.1240

  33. Muñoz, C., Poblete, F., Pozo, J.C.: Scattering in the energy space for Boussinesq equations. arXiv:1707.02616. To appear in Commun. Math. Phys.

  34. Pelinovsky D., Grimshaw R.: Structural transformation of eigenvalues for a perturbed algebraic soliton potential. Phys. Lett. A 229(3), 165–172 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. Ponce G., Vega L.: Nonlinear small data scattering for the generalized Korteweg–de Vries equation. J. Funct. Anal. 90(2), 445–457 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  36. Tao T.: Scattering for the quartic generalised Korteweg–de Vries equation. J. Differ. Eqs. 232(2), 623–651 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. Wadati M.: The modified Korteweg–de Vries equation. J. Phys. Soc. Jpn. 34(5), 1289–1296 (1973)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. Weinstein M.I.: Nonlinear Schrödinger equations and sharp interpolation estimartes. Commun. Math. Phys. 87, 567–576 (1983)

    Article  ADS  MATH  Google Scholar 

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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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Authors and Affiliations

  1. CNRS and Departamento de Ingeniería Matemática DIM-CMM UMI 2807-CNRS, Universidad de Chile, Santiago, Chile

    Claudio Muñoz

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

    Gustavo Ponce

Authors
  1. Claudio Muñoz
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  2. Gustavo Ponce
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Correspondence to Gustavo Ponce.

Additional information

Communicated by W. Schlag

CM was partially supported by Fondecyt No. 1150202, Millennium Nucleus Center for Analysis of PDE NC130017, Fondo Basal CMM, and MathAmSud EEQUADD collaboration Math16-01. Part of this work was done while the first author was visiting Fields Institute (Toronto, Canada), as part of the “Focus Program on Nonlinear Dispersive Partial Differential Equations and Inverse Scattering”.

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Muñoz, C., Ponce, G. Breathers and the Dynamics of Solutions in KdV Type Equations. Commun. Math. Phys. 367, 581–598 (2019). https://doi.org/10.1007/s00220-018-3206-9

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