Stability in the energy space for chains of solitons of the one-dimensional Gross-Pitaevskii equation

Fabrice Béthuel[1]; Philippe Gravejat[2]; Didier Smets[1]

  • [1] Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Boîte Courrier 187, 75252 Paris Cedex 05, France.
  • [2] Centre de Mathématiques Laurent Schwartz, École Polytechnique, 91128 Palaiseau Cedex, France.

Annales de l’institut Fourier (2014)

  • Volume: 64, Issue: 1, page 19-70
  • ISSN: 0373-0956

Abstract

top
We establish the stability in the energy space for sums of solitons of the one-dimensional Gross-Pitaevskii equation when their speeds are mutually distinct and distinct from zero, and when the solitons are initially well-separated and spatially ordered according to their speeds.

How to cite

top

Béthuel, Fabrice, Gravejat, Philippe, and Smets, Didier. "Stability in the energy space for chains of solitons of the one-dimensional Gross-Pitaevskii equation." Annales de l’institut Fourier 64.1 (2014): 19-70. <http://eudml.org/doc/275571>.

@article{Béthuel2014,
abstract = {We establish the stability in the energy space for sums of solitons of the one-dimensional Gross-Pitaevskii equation when their speeds are mutually distinct and distinct from zero, and when the solitons are initially well-separated and spatially ordered according to their speeds.},
affiliation = {Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Boîte Courrier 187, 75252 Paris Cedex 05, France.; Centre de Mathématiques Laurent Schwartz, École Polytechnique, 91128 Palaiseau Cedex, France.; Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Boîte Courrier 187, 75252 Paris Cedex 05, France.},
author = {Béthuel, Fabrice, Gravejat, Philippe, Smets, Didier},
journal = {Annales de l’institut Fourier},
keywords = {Gross-Pitaevskii equation; sums of solitons; stability},
language = {eng},
number = {1},
pages = {19-70},
publisher = {Association des Annales de l’institut Fourier},
title = {Stability in the energy space for chains of solitons of the one-dimensional Gross-Pitaevskii equation},
url = {http://eudml.org/doc/275571},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Béthuel, Fabrice
AU - Gravejat, Philippe
AU - Smets, Didier
TI - Stability in the energy space for chains of solitons of the one-dimensional Gross-Pitaevskii equation
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 1
SP - 19
EP - 70
AB - We establish the stability in the energy space for sums of solitons of the one-dimensional Gross-Pitaevskii equation when their speeds are mutually distinct and distinct from zero, and when the solitons are initially well-separated and spatially ordered according to their speeds.
LA - eng
KW - Gross-Pitaevskii equation; sums of solitons; stability
UR - http://eudml.org/doc/275571
ER -

References

top
  1. F. Béthuel, P. Gravejat, J.-C. Saut, Existence and properties of travelling waves for the Gross-Pitaevskii equation, Stationary and time dependent Gross-Pitaevskii equations 473 (2008), 55-104, FarinaA.A. Zbl1216.35132MR2522014
  2. F. Béthuel, P. Gravejat, J.-C. Saut, D. Smets, Orbital stability of the black soliton for the Gross-Pitaevskii equation, Indiana Univ. Math. J 57 (2008), 2611-2642 Zbl1171.35012MR2482993
  3. F. Béthuel, P. Gravejat, J.-C. Saut, D. Smets, On the Korteweg-de Vries long-wave approximation of the Gross-Pitaevskii equation I, Int. Math. Res. Not. 2009 (2009), 2700-2748 Zbl1183.35240MR2520771
  4. F. Béthuel, P. Gravejat, J.-C. Saut, D. Smets, On the Korteweg-de Vries long-wave approximation of the Gross-Pitaevskii equation II, Comm. Partial Differential Equations 35 (2010), 113-164 Zbl1213.35367MR2748620
  5. D. Chiron, Travelling waves for the nonlinear Schrödinger equation with general nonlinearity in dimension one, Nonlinearity 25 (2012), 813-850 Zbl1278.35226MR2887994
  6. D. Chiron, F. Rousset, The KdV/KP-I limit of the nonlinear Schrödinger equation, SIAM J. Math. Anal. 42 (2010), 64-96 Zbl1210.35229MR2596546
  7. N. Dunford, J.T. Schwartz, Linear operators. Part II. Spectral theory. Self-adjoint operators in Hilbert space, 7 (1963), Interscience Publishers, John Wiley and Sons, New York-London-Sydney Zbl0635.47002MR188745
  8. L.D. Faddeev, L.A. Takhtajan, Hamiltonian methods in the theory of solitons, (2007), Springer-Verlag, Berlin-Heidelberg-New York Zbl1111.37001MR2348643
  9. C. Gallo, Schrödinger group on Zhidkov spaces, Adv. Differential Equations 9 (2004), 509-538 Zbl1103.35093MR2099970
  10. P. Gérard, The Cauchy problem for the Gross-Pitaevskii equation, Ann. Inst. Henri Poincaré, Analyse Non Linéaire 23 (2006), 765-779 Zbl1122.35133
  11. P. Gérard, Z. Zhang, Orbital stability of traveling waves for the one-dimensional Gross-Pitaevskii equation, J. Math. Pures Appl. 91 (2009), 178-210 Zbl1232.35152MR2498754
  12. M. Grillakis, J. Shatah, W.A. Strauss, Stability theory of solitary waves in the presence of symmetry I, J. Funct. Anal. 74 (1987), 160-197 Zbl0656.35122MR901236
  13. Z. Lin, Stability and instability of traveling solitonic bubbles, Adv. Differential Equations 7 (2002), 897-918 Zbl1033.35117MR1895111
  14. Y. Martel, F. Merle, Stability of two soliton collision for nonintegrable gKdV equations, Commun. Math. Phys. 286 (2009), 39-79 Zbl1179.35291MR2470923
  15. Y. Martel, F. Merle, Inelastic interaction of nearly equal solitons for the quartic gKdV equation, Invent. Math. 183 (2011), 563-648 Zbl1230.35121MR2772088
  16. Y. Martel, F. Merle, T.-P. Tsai, Stability and asymptotic stability in the energy space of the sum of N solitons for subcritical gKdV equations, Commun. Math. Phys. 231 (2002), 347-373 Zbl1017.35098MR1946336
  17. Y. Martel, F. Merle, T.-P. Tsai, Stability in H 1 of the sum of K solitary waves for some nonlinear Schrödinger equations, Duke Math. J. 133 (2006), 405-466 Zbl1099.35134MR2228459
  18. R.M. Miura, The Korteweg- de Vries equation: a survey of results, SIAM Rev. 18 (1976), 412-459 Zbl0333.35021MR404890
  19. H.M. Tartousi 
  20. A.H. Vartanian, Long-time asymptotics of solutions to the Cauchy problem for the defocusing nonlinear Schrödinger equation with finite-density initial data. II. Dark solitons on continua, Math. Phys. Anal. Geom. 5 (2002), 319-413 Zbl1080.35060MR1942685
  21. V.E. Zakharov, A.B. Shabat, Interaction between solitons in a stable medium, Sov. Phys. JETP 37 (1973), 823-828 
  22. P.E. Zhidkov, Korteweg-De Vries and nonlinear Schrödinger equations : qualitative theory, 1756 (2001), Springer-Verlag, Berlin Zbl0987.35001MR1831831

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.