Skip to main content
SpringerLink
Log in

Blow up for the critical generalized Korteweg–de Vries equation. I: Dynamics near the soliton

  • Published:
Acta Mathematica

Abstract

We consider the quintic generalized Korteweg–de Vries equation (gKdV)

$$u_t + (u_{xx} + u^5)_x =0,$$

which is a canonical mass critical problem, for initial data in H 1 close to the soliton. In earlier works on this problem, finite- or infinite-time blow up was proved for non-positive energy solutions, and the solitary wave was shown to be the universal blow-up profile, see [16], [26] and [20]. For well-localized initial data, finite-time blow up with an upper bound on blow-up rate was obtained in [18].

In this paper, we fully revisit the analysis close to the soliton for gKdV in light of the recent progress on the study of critical dispersive blow-up problems (see [31], [39], [32] and [33], for example). For a class of initial data close to the soliton, we prove that three scenarios only can occur: (i) the solution leaves any small neighborhood of the modulated family of solitons in the scale invariant L 2 norm; (ii) the solution is global and converges to a soliton as t → ∞; (iii) the solution blows up in finite time T with speed

$$\|u_x(t)\|_{L^2} \sim \frac{C(u_0)}{T-t} \quad {\rm as}\, t\to T.$$

Moreover, the regimes (i) and (iii) are stable. We also show that non-positive energy yields blow up in finite time, and obtain the characterization of the solitary wave at the zero-energy level as was done for the mass critical non-linear Schrödinger equation in [31].

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bourgain, J. & Wang, W., Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity. Ann. Sc. Norm. Super. Pisa Cl. Sci., 25 (1997), 197–215 (1998).

    Google Scholar 

  2. Côte R.: Construction of solutions to the L 2-critical KdV equation with a given asymptotic behaviour. Duke Math. J., 138, 487–531 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  3. Côte, R., Martel, Y. & Merle, F., Construction of multi-soliton solutions for the L 2-supercritical gKdV and NLS equations. Rev. Mat. Iberoam., 27 (2011), 273–302.

    Google Scholar 

  4. Duyckaerts, T. & Merle, F., Dynamics of threshold solutions for energy-critical wave equation. Int. Math. Res. Pap. IMRP, 2008 (2008), Art ID rpn002, 67 pp.

  5. Duyckaerts T., Merle F.: Dynamic of threshold solutions for energy-critical NLS. Geom. Funct. Anal., 18, 1787–1840 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. Duyckaerts T., Roudenko S.: Threshold solutions for the focusing 3D cubic Schrödinger equation. Rev. Mat. Iberoam., 26, 1–56 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  7. Fibich, G., Merle, F. & Raphaël, P., Proof of a spectral property related to the singularity formation for the L 2 critical nonlinear Schrödinger equation. Phys. D, 220 (2006), 1–13.

    Google Scholar 

  8. Hillairet, M. & Raphaël, P., Smooth type II blow-up solutions to the four-dimensional energy-critical wave equation. Anal. PDE, 5 (2012), 777–829.

    Google Scholar 

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

  10. Kenig, C. E. & Merle, F., Global well-posedness, scattering and blow-up for the energycritical, focusing, non-linear Schrödinger equation in the radial case. Invent. Math., 166 (2006), 645–675.

    Google Scholar 

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

    Google Scholar 

  12. Krieger, J., Nakanishi, K. & Schlag, W., Global dynamics away from the ground state for the energy-critical nonlinear wave equation. Amer. J. Math., 135 (2013), 935–965.

    Google Scholar 

  13. Krieger J., Schlag W.: Non-generic blow-up solutions for the critical focusing NLS in 1-D. J. Eur. Math. Soc. (JEMS), 11, 1–125 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  14. Krieger, J., Schlag, W. & Tataru, D., Renormalization and blow up for charge one equivariant critical wave maps. Invent. Math., 171 (2008), 543–615.

    Google Scholar 

  15. Landman, M. J., Papanicolaou, G. C., Sulem, C. & Sulem, P.-L., Rate of blowup for solutions of the nonlinear Schrödinger equation at critical dimension. Phys. Rev. A, 38:8 (1988), 3837–3843.

    Google Scholar 

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

    Google Scholar 

  17. Martel, Y. & Merle, F., Instability of solitons for the critical generalized Korteweg–de Vries equation. Geom. Funct. Anal., 11 (2001), 74–123.

    Google Scholar 

  18. Martel, Y. & Merle, F., Blow up in finite time and dynamics of blow up solutions for the L 2-critical generalized KdV equation. J. Amer. Math. Soc., 15 (2002), 617–664.

    Google Scholar 

  19. Martel, Y. & Merle, F., Nonexistence of blow-up solution with minimal L 2-mass for the critical gKdV equation. Duke Math. J., 115 (2002), 385–408.

    Google Scholar 

  20. Martel, Y. & Merle, F., Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation. Ann. of Math., 155 (2002), 235–280.

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  22. Martel, Y., Merle, F. & Raphaël, P., Blow up for the critical gKdV equation II: minimal mass dynamics. Preprint, 2012. arXiv:1204.4624 [math.AP].

  23. Martel, Y., Merle, F. & Raphaël, P., Blow up for the critical gKdV equation III: exotic regimes. To appear in Ann. Sc. Norm. Super. Pisa Cl. Sci.. arXiv:1209.2510 [math.AP].

  24. Martel, Y., Merle, F. & Tsai, T.-P., Stability in H 1 of the sum of K solitary waves for some nonlinear Schrödinger equations. Duke Math. J., 133 (2006), 405–466.

    Google Scholar 

  25. Merle F.: Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power. Duke Math. J., 69, 427–454 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  26. Merle F.: Existence of blow-up solutions in the energy space for the critical generalized KdV equation. J. Amer. Math. Soc., 14, 555–578 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  27. Merle, F. & Raphaël, P., Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation. Geom. Funct. Anal., 13 (2003), 591–642.

    Google Scholar 

  28. Merle, F. & Raphaël, P., On universality of blow-up profile for L 2 critical nonlinear Schrödinger equation. Invent. Math., 156 (2004), 565–672.

    Google Scholar 

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

    Google Scholar 

  30. Merle, F. & Raphaël, P., Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation. Comm. Math. Phys., 253 (2005), 675–704.

    Google Scholar 

  31. Merle, F. & Raphaël, P., On a sharp lower bound on the blow-up rate for the L 2 critical nonlinear Schrödinger equation. J. Amer. Math. Soc., 19 (2006), 37–90.

    Google Scholar 

  32. Merle, F., Raphaël, P. & Rodnianski, I., Blowup dynamics for smooth data equivariant solutions to the critical Schrödinger map problem. Invent. Math., 193 (2013), 249–365.

    Google Scholar 

  33. Merle, F., Raphaël, P. & Szeftel, J., The instability of Bourgain–Wang solutions for the L 2. critical NLS. Amer. J. Math., 135 (2013), 967–1017.

    Google Scholar 

  34. Nakanishi, & K., Schlag, W., Global dynamics above the ground state energy for the focusing nonlinear Klein–Gordon equation. J. Differential Equations, 250:5 (2011), 2299–2333.

    Google Scholar 

  35. Nakanishi, K. & Schlag, W., Global dynamics above the ground state for the nonlinear Klein–Gordon equation without a radial assumption. Arch. Ration. Mech. Anal., 203 (2012), 809–851.

    Google Scholar 

  36. Nakanishi, K. & Schlag, W., Global dynamics above the ground state energy for the cubic NLS equation in 3D. Calc. Var. Partial Differential Equations, 44 (2012), 1–45.

  37. Perelman G.: On the formation of singularities in solutions of the critical nonlinear Schrödinger equation. Ann. Henri Poincaré, 2, 605–673 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  38. Raphaël, P., Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation. Math. Ann., 331 (2005), 577–609.

    Google Scholar 

  39. Raphaël, P. & Rodnianski, I., Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang–Mills problems. Publ. Math. Inst. Hautes Études Sci., 115 (2012), 1–122.

    Google Scholar 

  40. Raphaël, P. & Schweyer, R., Stable blowup dynamics for the 1-corotational energy critical harmonic heat flow. Comm. Pure Appl. Math., 66 (2013), 414–480.

    Google Scholar 

  41. Raphaël, P. & Szeftel, J., Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS. J. Amer. Math. Soc., 24 (2011), 471–546.

    Google Scholar 

  42. Rodnianski, I. & Sterbenz, J., On the formation of singularities in the critical O(3) σ.-model. Ann. of Math., 172 (2010), 187–242.

    Google Scholar 

  43. Weinstein M. I.: Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys., 87, 567–576 (1982/83)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

Download references

Author information

Author notes

  1. Yvan Martel

    Present address: École Polytechnique, CMLS, CNRS UMR7640, FR-91128, Palaiseau Cedex, France

  2. Pierre Raphaël

    Present address: Université Nice Sophia Antipolis, CNRS UMR 7351, FR-06108, Nice Cedex 02, France

Authors and Affiliations

  1. Université de Versailles St-Quentin and Institut Universitaire de France, LMV CNRS UMR8100, FR-78035, Versailles Cedex, France

    Yvan Martel

  2. Université de Cergy Pontoise and Institut des Hautes Études Scientifiques, AGM CNRS UMR8088, FR-75002, Paris, France

    Frank Merle

  3. Université Paul Sabatier and Institut Universitaire de France, IMT CNRS UMR 5219, FR-31062, Toulouse, France

    Pierre Raphaël

Authors
  1. Yvan Martel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Frank Merle
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pierre Raphaël
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Frank Merle.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Martel, Y., Merle, F. & Raphaël, P. Blow up for the critical generalized Korteweg–de Vries equation. I: Dynamics near the soliton. Acta Math 212, 59–140 (2014). https://doi.org/10.1007/s11511-014-0109-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11511-014-0109-2

Search

Navigation