Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Well-posedness for the Schrödinger-Korteweg-de Vries system


Authors: A. J. Corcho and F. Linares
Journal: Trans. Amer. Math. Soc. 359 (2007), 4089-4106
MSC (2000): Primary 35Q55, 35Q60, 35B65
DOI: https://doi.org/10.1090/S0002-9947-07-04239-0
Published electronically: April 11, 2007
MathSciNet review: 2309177
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study well-posedness of the Cauchy problem associated to the Schrödinger-Korteweg-de Vries system. We obtain local well-posedness for weak initial data, where the best result obtained is for data in the Sobolev space $ L^2({\mathbb{R}})\times H^{-\tfrac{3}{4}+} $. This result implies in particular the global well-posedness in the energy space $ H^1({\mathbb{R}})\times H^1({\mathbb{R}})$. Both results considerably improve the previous ones by Bekiranov, Ogawa and Ponce (1997), Guo and Miao (1999), and Tsutsumi (1993).


References [Enhancements On Off] (What's this?)

  • 1. D. Bekiranov, T. Ogawa and G. Ponce, Weak solvability and well-posedness of a coupled Schrödinger-Korteweg de Vries equation for capillary-gravity wave interactions, Proceedings of the AMS., 125 (10), (1997), 2907-2919.MR 1403113 (97m:35238)
  • 2. D. Bekiranov, T. Ogawa and G. Ponce, Interaction equations for short and long dispersive waves, J. Funct. Anal., 158, (1998), 357-388.MR 1648479 (99i:35143)
  • 3. E. S. Benilov and S. P. Burtsev, To the integrability of the equations describing the Langmuir-wave-ion-acoustic-wave interaction, Phys. Let., 98A (1983), 256-258.MR 0720816 (85f:76120)
  • 4. J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Geometric and Functional Anal., 3 (1993), 107-156, 209-262. MR 1209299 (95d:35160a); MR 1215780 (95d:35160b)
  • 5. M. Christ, J. Colliander, and T. Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math., 125 (2003), no. 6, 1235-1293. MR 2018661 (2005d:35223)
  • 6. M. Christ, J. Colliander, and T. Tao, Ill-posedness for nonlinear Schrödinger and Wave equations, preprint.
  • 7. A. J. Corcho, On Some Nonlinear Dispersive Systems, Ph.D. Thesis. Informes de Matemática. IMPA, Rio de Janeiro, 18 (2003).
  • 8. M. Funakoshi and M. Oikawa, The resonant interaction between a long internal gravity wave and a surface gravity wave packet, J. Phys. Soc. Japan, 52 (1983), 1982-1995.MR 0710730 (84k:76030)
  • 9. J. Ginibre, Y.Tsutsumi and G. Velo, On the Cauchy Problem for the Zakharov system, J. Funct. Anal., 151 (1997), 384-436.MR 1491547 (2000c:35220)
  • 10. B. Guo and Ch. Miao, Well-posedness of the Cauchy problem for the coupled system of the Schrödinger-KdV equations, Acta Math. Sinica, Engl. Series, 15 (1999), 215-224. MR 1714079 (2000e:35207)
  • 11. H. Hojo, H. Ikezi, K. Mima and K. Nishikawa, Coupled nonlinear electron-plasma and ion-acoustic waves, Phys. Rev. Lett., 33 (1974), 148-151.
  • 12. T. Kakutani, T. Kawahara and N. Sugimoto, Nonlinear interaction between short and long capillary-gravity waves, J. Phys. Soc. Japan, 39 (1975), 1379-1386.
  • 13. C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21.MR 1230283 (94g:35196)
  • 14. C. E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603.MR 1329387 (96k:35159)
  • 15. C. E. Kenig, G. Ponce and L. Vega, Quadratic Forms for the 1-D semilinear Schrödinger equation, Trans. Amer. Math. Soc., 348 (1996), no. 8, 3323-3353. MR 1357398 (96j:35233)
  • 16. C. E. Kenig, G. Ponce and L. Vega, On ill-posedness of some canonical dispersive equations, Duke Math. J., 106 (2001), 617-633. MR 1813239 (2002c:35265)
  • 17. J. Satsuma and N. Yajima, Soliton solutions in a diatomic lattice system, Progr. Theor. Phys., 62 (1979), 370-378.
  • 18. Y. Tsutsumi, $ L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcialaj Ekvacioj, 30 (1987), 115-125.MR 0915266 (89c:35143)
  • 19. M. Tsutsumi, Well-posedness of the Cauchy problem for a coupled Schrödinger-KdV equation, Math. Sciences Appl., 2 (1993), 513-528.MR 1370488 (96k:35163)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35Q55, 35Q60, 35B65

Retrieve articles in all journals with MSC (2000): 35Q55, 35Q60, 35B65


Additional Information

A. J. Corcho
Affiliation: Departamento de Matemática, Universidade Federal de Alagoas, Campus A. C. Simões, Tabuleiro dos Martins, Maceió-AL, 57072-970, Brazil
Email: adan@mat.ufal.br

F. Linares
Affiliation: IMPA, Estrada Dona Castorina 110, Rio de Janeiro, 22460–320, Brazil
Email: linares@impa.br

DOI: https://doi.org/10.1090/S0002-9947-07-04239-0
Keywords: Well-posedness, Schr\"odinger-Korteweg-de Vries system
Received by editor(s): February 4, 2005
Published electronically: April 11, 2007
Additional Notes: The first author was supported by CNPq and FAPEAL, Brazil
The second author was partially supported by CNPq, Brazil
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society