Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity

Jean Bourgain; W. Wang

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (1997)

  • Volume: 25, Issue: 1-2, page 197-215
  • ISSN: 0391-173X

How to cite

top

Bourgain, Jean, and Wang, W.. "Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 25.1-2 (1997): 197-215. <http://eudml.org/doc/84284>.

@article{Bourgain1997,
author = {Bourgain, Jean, Wang, W.},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {behavior of the blowup solution; nonlinear Schrödinger equation},
language = {eng},
number = {1-2},
pages = {197-215},
publisher = {Scuola normale superiore},
title = {Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity},
url = {http://eudml.org/doc/84284},
volume = {25},
year = {1997},
}

TY - JOUR
AU - Bourgain, Jean
AU - Wang, W.
TI - Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 1997
PB - Scuola normale superiore
VL - 25
IS - 1-2
SP - 197
EP - 215
LA - eng
KW - behavior of the blowup solution; nonlinear Schrödinger equation
UR - http://eudml.org/doc/84284
ER -

References

top
  1. [Bo] J. Bourgain, On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE, International Math. Res. Notices6 (1996), 277-304. Zbl0934.35166MR1386079
  2. [G] R. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger operators, J. Math. Phys.8 (1977), 1794-1797. Zbl0372.35009MR460850
  3. [GV] J. Ginibre - G. Velo, On a class of nonlinear Schrödinger equations I. The Cauchy problem, general case; II. Scattering theory, general case, J. Func. Anal.32 (1979), 1-71. Zbl0396.35028MR533218
  4. [Ka] T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Physique Theorique46 (1987), 113-129. Zbl0632.35038MR877998
  5. [Kw] M. Kwong, Uniqueness of positive solutions of Δu - u + up = 0 in RN, Arch. Rat. Mech. Anal.105 (1989), 243-266. Zbl0676.35032
  6. [M1] F. Merle, Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equation with critical power, Duke Math. J.69 (1993), 427-453. Zbl0808.35141MR1203233
  7. [M2] F. Merle, Construction of solutions with exact k blow-up points for the Schrödinger equation with critical power nonlinearity, Comm. Math. Phys.149 (1992), 205-214. MR1048692
  8. [M3] F. Merle, Limit of the solution of a nonlinear Schrödinger equation at blow-up time, J. Functional Anal.84 (1989), 201-214. Zbl0681.35078MR999497
  9. [MT] F. Merle - Y. Tsutsumi, L2 concentration of blow-up solutions for the nonlinear Schrödingerequation with critical power nonlinearity, J. Diff. Eq.84 (1990), 205-214. Zbl0722.35047MR1047566
  10. [N] H. Nawa, Asymptotic profiles of blow-up solutions of the nonlinear Schrödinger equation with critical power nonlinearity, J. Math. Soc. Japan46 (1994), 557-586. Zbl0829.35121MR1291107
  11. [SSP] P.L. Sulem - C. Sulem - A. Patera, Numerical simulations ofsingular solutions to the two-dimensional cubic Schrödinger equation, J. Comp. Phus.37 (1984), 755-778. Zbl0543.65081MR762872
  12. [Wein] M. Weinstein, On the structure and formation of singularities in solutions to the nonlinear dispersive evolution equation, Comm. PDE11 (1986), 545-565. Zbl0596.35022MR829596
  13. [Wein1] M. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal.16 (1985), 472-491. Zbl0583.35028MR783974
  14. [Wein2] M. Weinstein, The nonlinear Schrödinger equation: singularity formation stability and dispersion, in "AMS-SIAM conference on the connection between infinite dimensional and finite dimensional dynamical systems" (1987), 25-40. Zbl0703.35159
  15. [Z] V. Zakharov - V. Synakh, The nature of self-focusing singularity, Zh. Eksp. Teir. Fiz.68 (1975), 940-947; Sov. Phys. JETP41 (1975), 465-468. 

Citations in EuDML Documents

top
  1. Frank Merle, Pierre Raphael, Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation
  2. Pierre Raphaël, Sur la dynamique explosive des solutions de l’équation de Schrödinger non linéaire
  3. Yvan Martel, Frank Merle, Pierre Raphaël, Blow up and near soliton dynamics for the L 2 critical gKdV equation
  4. Galina Perelman, On the blow up phenomenon for the critical nonlinear Schrödinger equation in 1D
  5. Serge Alinhac, Solutions explosives exceptionnelles
  6. Frank Merle, Pierre Raphaël, Jérémie Szeftel, Two blow-up regimes for L 2 supercritical nonlinear Schrödinger equations
  7. Nicolas Burq, Explosion pour l’équation de Schrödinger au régime du “log log”
  8. Rémi Carles, Changing blow-up time in nonlinear Schrödinger equations

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.