Skip to main content
Log in

Asymptotic stability of solitons of the gKdV equations with general nonlinearity

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We consider the generalized Korteweg-de Vries equation (gKdV)

$$\partial_t u + \partial_x (\partial_x^2 u + f(u)) = 0, \quad (t, x) \in [0, T) \times {\mathbb{R}},$$

with general C 3 nonlinearity f. Under an explicit condition on f and c > 0, there exists a solution in the energy space H 1 of the type u(t, x) = Q c (xx 0ct), called soliton. In this paper, under general assumptions on f and Q c , we prove that the family of solitons around Q c is asymptotically stable in some local sense in H 1, i.e. if u(t) is close to Q c (for all t ≥  0), then u(t) locally converges in the energy space to some Q c+ as t → +∞. Note in particular that we do not assume the stability of Q c . This result is based on a rigidity property of the gKdV equation around Q c in the energy space whose proof relies on the introduction of a dual problem. These results extend the main results in Martel (SIAM J. Math. Anal. 38:759–781, 2006); Martel and Merle (J. Math. Pures Appl. 79:339–425, 2000), (Arch. Ration. Mech. Anal. 157:219–254, 2001), (Nonlinearity 1:55–80), devoted to the pure power case.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Berestycki H. and Lions P.-L. (1983). Nonlinear scalar field equations. I. Existence of a ground state. Arch. Ration. Mech. Anal. 82: 313–345

    MATH  MathSciNet  Google Scholar 

  2. Bona J.L., Souganidis P.E. and Strauss W.A. (1987). Stability and instability of solitary waves of Korteweg-de Vries type. Proc. R. Soc. Lond. 411: 395–412

    Article  MATH  MathSciNet  Google Scholar 

  3. Buslaev, V.S., Perelman, G.S.: On the stability of solitary waves for nonlinear Schrödinger equations. Nonlinear evolution equations, pp. 75–98, Am. Math. Soc. Transl. Ser. 2, 164. Am. Math. Soc., Providence, RI (1995)

  4. Eckhaus W. and Schuur P. (1983). The emergence of solutions of the Korteweg-de Vries equation from arbitrary initial conditions. Math. Meth. Appl. Sci. 5: 97–116

    Article  MATH  MathSciNet  Google Scholar 

  5. Grillakis M. (1990). Existence of nodal solutions of semilinear equations in \({\mathbb{R}}\). J. Diff. Eq. 85: 367–400

    Article  MATH  MathSciNet  Google Scholar 

  6. Kato T. (1983). On the Cauchy problem for the (generalized) Korteweg-de Vries equation. Studies in applied mathematics, pp. 93–128, Adv. Math. Suppl. Stud., 8. Academic Press, New York

    Google Scholar 

  7. Kenig C.E., Ponce G. and Vega L. (1989). On the (generalized) Korteweg-de Vries equation. Duke Math. J. 59: 585–610

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  9. Krieger J. and Schlag W. (2006). Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension. J. Am. Math. Soc. 19: 815–920

    Article  MATH  MathSciNet  Google Scholar 

  10. Martel Y. (2005). Asymptotic N-soliton like solutions of the subcritical and critical generalized Korteweg- de Vries equations. Am. J. Math. 127: 1103–1140

    MATH  MathSciNet  Google Scholar 

  11. Martel Y. (2006). Linear problems related to asymptotic stability of solitons of the generalized KdV equations. SIAM J. Math. Anal. 38: 759–781

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  13. Martel Y. and Merle F. (2001). Asymptotic stability of solitons for subcritical generalized KdV equations. Arch. Ration. Mech. Anal. 157: 219–254

    Article  MATH  MathSciNet  Google Scholar 

  14. Martel Y. and Merle F. (2001). Instability of solitons for the critical generalized Korteweg-de Vries equation. Geom. Funct. Anal. 11: 74–123

    Article  MATH  MathSciNet  Google Scholar 

  15. Martel Y. and Merle F. (2005). Asymptotic stability of solitons of the subcritical gKdV equations revisited. Nonlinearity 18(1): 55–80

    Article  MATH  MathSciNet  Google Scholar 

  16. Martel, Y., Merle, F.: Description of two soliton collision for the quartic KdV equation, arXiv:0709. 2672v1

  17. Martel, Y., Merle, F.: Stability of two soliton collision for the gKdV equations, arXiv:0709.2677v1

  18. Martel, Y., Merle, F.: Refined asymptotics around solitons for the gKdV equations. Discret. Contin. Dyn. Syst. Ser. A (to appear)

  19. Martel Y. and Merle F. (2002). Tai-Peng Tsai, stability and asymptotic stability in the energy space of the sum of N solitons for subcritical gKdV equations. Commun. Math. Phys. 231: 347–373

    Article  MATH  MathSciNet  Google Scholar 

  20. Martel Y. and Merle F. (2006). Tai-Peng Tsai, Stability in H 1 of the sum of K solitary waves for some nonlinear Schrödinger equations. Duke Math. J. 133: 405–466

    Article  MATH  MathSciNet  Google Scholar 

  21. Mizumachi T. (2001). Large time asymptotics of solutions around solitary waves to the generalized Korteweg-de Vries equations. SIAM J. Math. Anal. 32: 1050–1080

    Article  MATH  MathSciNet  Google Scholar 

  22. Pego R.L. and Weinstein M.I. (1994). Asymptotic stability of solitary waves. Commun. Math. Phys. 164: 305–349

    Article  MATH  MathSciNet  Google Scholar 

  23. Perelman G.S. (2004). Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations. Comm. Partial Diff. Equ. 29: 1051–1095

    Article  MATH  MathSciNet  Google Scholar 

  24. Rodnianski, I., Schlag, W., Soffer, A.D.: Asymptotic stability of N-soliton states of NLS. Comm. Pure. Appl. Math. (To appear)

  25. Sulem C. and Sulem P.-L. (1999). The Nonlinear Schrödinger Equation. Self-Focusing and Wave Collapse. Applied Mathematical Sciences, 139. Springer, New York

    Google Scholar 

  26. Weinstein M.I. (1985). Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16: 472–491

    Article  MATH  MathSciNet  Google Scholar 

  27. Weinstein M.I. (1986). Lyapunov stability of ground states of nonlinear dispersive evolution equations. Comm. Pure Appl. Math. 39: 51–68

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yvan Martel.

Additional information

This research was supported in part by the Agence Nationale de la Recherche (ANR ONDENONLIN).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Martel, Y., Merle, F. Asymptotic stability of solitons of the gKdV equations with general nonlinearity. Math. Ann. 341, 391–427 (2008). https://doi.org/10.1007/s00208-007-0194-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-007-0194-z

Mathematics Subject Classification (2000)

Navigation