The Cauchy problem for the Gross–Pitaevskii equation

P. Gérard

Annales de l'I.H.P. Analyse non linéaire (2006)

  • Volume: 23, Issue: 5, page 765-779
  • ISSN: 0294-1449

How to cite

top

Gérard, P.. "The Cauchy problem for the Gross–Pitaevskii equation." Annales de l'I.H.P. Analyse non linéaire 23.5 (2006): 765-779. <http://eudml.org/doc/78711>.

@article{Gérard2006,
author = {Gérard, P.},
journal = {Annales de l'I.H.P. Analyse non linéaire},
keywords = {nonlinear Schrödinger equation; Gross-Pitaevskii equation; global well posedness; Strichartz estimates},
language = {eng},
number = {5},
pages = {765-779},
publisher = {Elsevier},
title = {The Cauchy problem for the Gross–Pitaevskii equation},
url = {http://eudml.org/doc/78711},
volume = {23},
year = {2006},
}

TY - JOUR
AU - Gérard, P.
TI - The Cauchy problem for the Gross–Pitaevskii equation
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 2006
PB - Elsevier
VL - 23
IS - 5
SP - 765
EP - 779
LA - eng
KW - nonlinear Schrödinger equation; Gross-Pitaevskii equation; global well posedness; Strichartz estimates
UR - http://eudml.org/doc/78711
ER -

References

top
  1. [1] Béthuel F., Orlandi G., Smets D., Vortex rings for the Gross–Pitaevskii equation, J. Eur. Math. Soc.6 (2004) 17-94. Zbl1091.35085MR2041006
  2. [2] Béthuel F., Saut J.C., Travelling waves for the Gross–Pitaevskii equation I, Ann. Inst. H. Poincaré Phys. Théor.70 (1999) 147-238. Zbl0933.35177MR1669387
  3. [3] Brezis H., Gallouët T., Nonlinear Schrödinger evolution equations, Nonlinear Anal.4 (1980) 677-681. Zbl0451.35023MR582536
  4. [4] Cazenave T., Semilinear Schrödinger Equations, Courant Lecture Notes in Math., vol. 10, New York University, American Mathematical Society, Providence, RI, 2003. Zbl1055.35003MR2002047
  5. [5] Frisch T., Pomeau Y., Rica S., Transition to dissipation in a model of superflow, Phys. Rev. Lett.69 (1992) 1644-1647. 
  6. [6] Gallo C., Schrödinger group on Zhidkov spaces, Adv. Differential Equations9 (2004) 509-538. Zbl1103.35093MR2099970
  7. [7] Ginibre J., Velo G., On a class of nonlinear Schrödinger equations, J. Funct. Anal.32 (1979) 1-71. Zbl0396.35029MR533219
  8. [8] Ginibre J., Velo G., The global Cauchy problem for the nonlinear Schrödinger equation, Ann. Inst. H. Poincaré Anal. Non Linéaire2 (1985) 309-327. Zbl0586.35042MR801582
  9. [9] O. Goubet, Two remarks on solutions of Gross–Pitaevskii equations on Zhidkov spaces, Preprint, 2005. Zbl1128.35096MR2317389
  10. [10] Gravejat P., A non-existence result for supersonic travelling waves in the Gross–Pitaevskii equation, Comm. Math. Phys.243 (2003) 93-103. Zbl1044.35087MR2020221
  11. [11] Gravejat P., Decay of travelling waves in the Gross–Pitaevskii equation, Ann. Inst. H. Poincaré Anal. Non Linéaire21 (2004) 591-637. Zbl1057.35060MR2086751
  12. [12] Gravejat P., Limit at infinity and non-existence result for sonic travelling waves in the Gross–Pitaevskii equation, Differential Integral Equations17 (2004) 1213-1232. Zbl1150.35301MR2100023
  13. [13] Gross E.P., J. Math. Phys.4 (1963) 195. 
  14. [14] Hörmander L., The Analysis of Linear Partial Differential Operators, vol. 1, Springer-Verlag, 1983. Zbl0521.35002
  15. [15] Kato T., On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor.46 (1987) 113-129. Zbl0632.35038MR877998
  16. [16] Keel M., Tao T., Endpoint Strichartz estimates, Amer. J. Math.120 (1998) 955-980. Zbl0922.35028MR1646048
  17. [17] Nore C., Abid M., Brachet M., Decaying Kolmogorov turbulence in a model of superflow, Phys. Fluids9 (1997) 2644-2669. Zbl1185.76669MR1467686
  18. [18] Pitaevskii L.P., Sov. Phys. JETP13 (1961) 451. 
  19. [19] Sulem C., Sulem P.L., The Nonlinear Schrödinger Equation. Self-Focusing and Wave Collapse, Appl. Math. Sci., vol. 139, Springer-Verlag, 1999. Zbl0928.35157MR1696311
  20. [20] P.E. Zhidkov, The Cauchy problem for a nonlinear Schrödinger equation, Dubna, 1987. 
  21. [21] Zhidkov P.E., Korteweg–de Vries and Nonlinear Schrödinger Equations: Qualitative Theory, Lecture Notes in Math., vol. 1756, Springer-Verlag, 2001. Zbl0987.35001MR1831831

Citations in EuDML Documents

top
  1. Fabrice Béthuel, Philippe Gravejat, Didier Smets, Stability in the energy space for chains of solitons of the one-dimensional Gross-Pitaevskii equation
  2. V. Banica, L. Vega, On the Dirac delta as initial condition for nonlinear Schrödinger equations
  3. Thomas Alazard, Rémi Carles, WKB analysis for the Gross-Pitaevskii equation with non-trivial boundary conditions at infinity
  4. Evelyne Miot, Dynamique des points vortex dans une équation de Ginzburg-Landau complexe
  5. Mihai Mariş, Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity: some results and open problems
  6. Fabrice Béthuel, Philippe Gravejat, Jean-Claude Saut, Ondes progressives pour l’équation de Gross-Pitaevskii

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.