Random modulation of solitons for the stochastic Korteweg–de Vries equation

A. de Bouard; A. Debussche

Annales de l'I.H.P. Analyse non linéaire (2007)

  • Volume: 24, Issue: 2, page 251-278
  • ISSN: 0294-1449

How to cite

top

de Bouard, A., and Debussche, A.. "Random modulation of solitons for the stochastic Korteweg–de Vries equation." Annales de l'I.H.P. Analyse non linéaire 24.2 (2007): 251-278. <http://eudml.org/doc/78734>.

@article{deBouard2007,
author = {de Bouard, A., Debussche, A.},
journal = {Annales de l'I.H.P. Analyse non linéaire},
keywords = {stochastic partial differential equations; white noise; central limit theorem; solitary waves},
language = {eng},
number = {2},
pages = {251-278},
publisher = {Elsevier},
title = {Random modulation of solitons for the stochastic Korteweg–de Vries equation},
url = {http://eudml.org/doc/78734},
volume = {24},
year = {2007},
}

TY - JOUR
AU - de Bouard, A.
AU - Debussche, A.
TI - Random modulation of solitons for the stochastic Korteweg–de Vries equation
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 2007
PB - Elsevier
VL - 24
IS - 2
SP - 251
EP - 278
LA - eng
KW - stochastic partial differential equations; white noise; central limit theorem; solitary waves
UR - http://eudml.org/doc/78734
ER -

References

top
  1. [1] Abdullaev F., Darmanyan S., Khabibullaev P., Optical Solitons, Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1993. 
  2. [2] Ablowitz M.J., Segur H., Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981. Zbl0472.35002MR642018
  3. [3] Bass G.G., Kivshar Y.S., Konotop V.V., Sinitsyn Y.A., Dynamics of solitons under random perturbations, Phys. Rep.157 (1988) 63-181. MR920335
  4. [4] Benjamin T.B., The stability of solitary waves, Proc. Roy. Soc. London Ser. A328 (1972) 153-183. MR338584
  5. [5] de Bouard A., Debussche A., On the stochastic Korteweg–de Vries equation, J. Funct. Anal.154 (1998) 215-251. Zbl0912.60074
  6. [6] de Bouard A., Debussche A., The stochastic nonlinear Schrödinger equation in H 1 , Stochastic Anal. Appl.21 (2003) 97-126. Zbl1027.60065MR1954077
  7. [7] de Bouard A., Debussche A., Tsutsumi Y., White noise driven Korteweg–de Vries equation, J. Funct. Anal.169 (1999) 532-558. Zbl0952.60062
  8. [8] A. de Bouard, A. Debussche, Y. Tsutsumi, Periodic solutions of the Korteweg–de Vries equation driven by white noise, SIAM J. Math. Anal., in press. Zbl1075.35121
  9. [9] Da Prato G., Zabczyk J., Stochastic equations in infinite dimensions, in: Encyclopedia of Mathematics and its Application, Cambridge University Press, Cambridge, 1992. Zbl0761.60052MR1207136
  10. [10] Debussche A., Printems J., Numerical simulation of the stochastic Korteweg–de Vries equation, Phys. D134 (1999) 200-226. Zbl0948.76038
  11. [11] Falkovich G.E., Kolokolov I., Lebedev V., Turitsyn S.K., Statistics of soliton-bearing systems with additive noise, Phys. Rev. E63 (2001) 025601(R). 
  12. [12] Freidlin M.I., Wentzell A.D., Random Perturbations of Dynamical Systems, Comprehensive Studies in Mathematics, Springer-Verlag, Berlin, 1984. Zbl0922.60006MR722136
  13. [13] Garnier J., Long-time dynamics of Korteweg–de Vries solitons driven by random perturbations, J. Statist. Phys.105 (2001) 789-833. Zbl0999.35086
  14. [14] Grillakis M., Shatah J., Strauss W., Stability theory of solitary waves in the presence of symmetry I, J. Funct. Anal.74 (1987) 160-197. Zbl0656.35122MR901236
  15. [15] Gyöngy I., Krylov N.V., Existence of strong solutions for Itô's stochastic equations via approximations, Probab. Theory Related Fields105 (1996) 143-158. Zbl0847.60038MR1392450
  16. [16] Herman R., The stochastic, damped Korteweg–de Vries equation, J. Phys. A.23 (1990) 1063-1084. Zbl0706.60068
  17. [17] Konotop V.V., Vasquez L., Nonlinear Random Waves, World Scientific, Singapore, 1994. Zbl1058.76500MR1425880
  18. [18] Martel Y., Merle F., Asymptotic stability of solitons for subcritical generalized KdV equations, Arch. Ration. Mech. Anal.157 (2001) 219-254. Zbl0981.35073MR1826966
  19. [19] Martel Y., Merle F., Asymptotic stability of solitons of the subcritical gKdV equations revisited, Nonlinearity18 (2005) 55-80. Zbl1064.35171MR2109467
  20. [20] Martel Y., Merle F., Tsai T.P., Stability and asymptotic stability in the energy space of the sum of N solitons for subcritical gKdV equations, Comm. Math. Phys.231 (2002) 347-373. Zbl1017.35098MR1946336
  21. [21] Miura R.M., The Korteweg–de Vries equation: a survey of results, SIAM Rev.18 (1976) 412-459. Zbl0333.35021
  22. [22] Pego R.L., Weinstein M.I., Eigenvalues, and instabilities of solitary waves, Philos. Trans. Roy. Soc. London Ser. A340 (1992) 47-94. Zbl0776.35065MR1177566
  23. [23] Pego R.L., Weinstein M.I., Asymptotic stability of solitary waves, Comm. Math. Phys.164 (1994) 305-349. Zbl0805.35117MR1289328
  24. [24] J. Printems, Aspects Théoriques et numériques de l'équation de Korteweg–de Vries stochastique, thèse, Université Paris-Sud, Orsay, France, 1998. 
  25. [25] Scalerandi M., Romano A., Condat C.A., Korteweg–de Vries solitons under additive stochastic perturbations, Phys. Rev. E58 (1998) 4166-4173. 
  26. [26] Temam R., Sur un problème non linéaire, J. Math. Pures Appl.48 (1969) 159-172. Zbl0187.03902MR261183
  27. [27] Wadati M., Stochastic Korteweg–de Vries equation, J. Phys. Soc. Japan52 (1983) 2642-2648. 
  28. [28] Wadati M., Akutsu Y., Stochastic Korteweg–de Vries equation with and without damping, J. Phys. Soc. Japan53 (1984) 3342-3350. 

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.