Sur la dynamique explosive des solutions de l’équation de Schrödinger non linéaire

Pierre Raphaël[1]

  • [1] Université de Paris Sud et CNRS

Séminaire Équations aux dérivées partielles (2004-2005)

  • page 1-11

How to cite

top

Raphaël, Pierre. "Sur la dynamique explosive des solutions de l’équation de Schrödinger non linéaire." Séminaire Équations aux dérivées partielles (2004-2005): 1-11. <http://eudml.org/doc/11115>.

@article{Raphaël2004-2005,
affiliation = {Université de Paris Sud et CNRS},
author = {Raphaël, Pierre},
journal = {Séminaire Équations aux dérivées partielles},
language = {fre},
pages = {1-11},
publisher = {Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Sur la dynamique explosive des solutions de l’équation de Schrödinger non linéaire},
url = {http://eudml.org/doc/11115},
year = {2004-2005},
}

TY - JOUR
AU - Raphaël, Pierre
TI - Sur la dynamique explosive des solutions de l’équation de Schrödinger non linéaire
JO - Séminaire Équations aux dérivées partielles
PY - 2004-2005
PB - Centre de mathématiques Laurent Schwartz, École polytechnique
SP - 1
EP - 11
LA - fre
UR - http://eudml.org/doc/11115
ER -

References

top
  1. Berestycki, H. ; Lions, P.-L., Nonlinear scalar field equations. I. Existence of a ground state. Arch. Rational Mech. Anal. 82 (1983), no. 4, 313–345. Zbl0533.35029MR695535
  2. Bourgain, J. ; Wang, W., Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 1-2, 197–215 (1998). Zbl1043.35137MR1655515
  3. Ginibre, J. ; Velo, G., On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case. J. Funct. Anal. 32 (1979), no. 1, 1–32. Zbl0396.35028MR533218
  4. Glassey, R.T., On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys. 18, 1794-1797 (1977). Zbl0372.35009MR460850
  5. Glangetas, L. ; Merle, F., Existence of self-similar blow-up solutions for Zakharov equation in dimension two. I. Comm. Math. Phys. 160 (1994), no. 1, 173–215. Zbl0808.35137MR1262194
  6. Kwong, M. K., Uniqueness of positive solutions of Δ u - u + u p = 0 in R n . Arch. Rational Mech. Anal. 105 (1989), no. 3, 243–266. Zbl0676.35032MR969899
  7. 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 (3) 38 (1988), no. 8, 3837–3843. MR966356
  8. 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. (2) 155 (2002), no. 1, 235–280. Zbl1005.35081MR1888800
  9. Merle, F., Construction of solutions with exactly k blow up points for the Schrödinger equation with critical nonlinearity, J. Diff. Eq. 84 (1990), no. 2, 223-240. Zbl0707.35021MR1048692
  10. Merle, F., Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power. Duke Math. J. 69 (1993), no. 2, 427–454. Zbl0808.35141MR1203233
  11. Merle, F., Lower bounds for the blow up rate of solutions of the Zakharov equations in dimension two, Comm. Pure. Appl. Math. 49 (1996), n0. 8, 765-794. Zbl0856.35014MR1391755
  12. Merle, F. ; Raphaël, P., Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation, to appear in Annals of Math. Zbl1185.35263MR1968208
  13. Merle, F. ; Raphaël, P., Sharp upper bound on the blow up rate for critical nonlinear Schrödinger equation, Geom. Funct .Ana 13 (2003), 591–642. Zbl1061.35135MR1995801
  14. Merle, F. ; Raphaël, P., On Universality of Blow up Profile for L 2 critical nonlinear Schrödinger equation, Invent. Math. 156, 565-672 (2004). Zbl1067.35110MR2061329
  15. Merle, F. ; Raphaël, P., Sharp lower bound on the blow up rate for critical nonlinear Schrödinger equation, preprint. Zbl1075.35077
  16. Merle, F. ; Raphaël, P., Profiles and quantization of the blow up mass for critical non linear Schrödinger equation, to appear in Comm. Math. Phys. Zbl1062.35137MR2116733
  17. Perelman, G., On the blow up phenomenon for the critical nonlinear Schrödinger equation in 1D, Ann. Henri. Poincaré, 2 (2001), 605-673. Zbl1007.35087MR1826598
  18. Raphaël, P., Stability of the log-log bound for blow up solutions to the critical nonlinear Schrödinger equation, to appear in Math. Annalen. Zbl1082.35143
  19. Sulem, C. ; Sulem, P.L., The nonlinear Schrödinger equation. Self-focusing and wave collapse. Applied Mathematical Sciences, 139. Springer-Verlag, New York, 1999. Zbl0928.35157MR1696311
  20. Weinstein, M.I., Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1983), 567—576. Zbl0527.35023MR691044
  21. Zakharov, V.E. ; Shabat, A.B., Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in non-linear media, Sov. Phys. JETP 34 (1972), 62—69. MR406174

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.