Acta Applicandae Mathematicae

, Volume 113, Issue 1, pp 75–100 | Cite as

Analyticity and Smoothing Effect for the Coupled System of Equations of Korteweg-de Vries Type with a Single Point Singularity

  • Margareth S. Alves
  • Bianca M. R. Calsavara
  • Jaime E. Muñoz Rivera
  • Mauricio SepúlvedaEmail author
  • Octavio Vera Villagrán


Using Bourgain spaces and the generator of dilation P=3t t +x x , which almost commutes with the linear Korteweg-de Vries operator, we show that a solution of the initial value problem associated for the coupled system of equations of Korteweg-de Vries type which appears as a model to describe the strong interaction of weakly nonlinear long waves, has an analyticity in time and a smoothing effect up to real analyticity if the initial data only have a single point singularity at x=0.


Evolution equations Bourgain space Smoothing effect 

Mathematics Subject Classification (2000)



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alves, M., Villagrán, O. Vera: Smoothing properties for a coupled system of nonlinear evolution dispersive equations. Indag. Math. 20(2), 285–327 (2009) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Ash, J.M., Cohen, J., Wang, G.: On strongly interacting internal solitary waves. J. Fourier Anal. Appl. 2(5), 507–517 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bisognin, E., Bisognin, V., Perla Menzala, G.: Asymptotic behaviour in time of the solutions of a coupled system of KdV equations. Funkc. Ekvacioj 40, 353–370 (1997) zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bisognin, E., Bisognin, V., Sepúlveda, M., Vera, O.: Coupled system of Korteweg-de Vries equations type in domains with moving boundaries. J. Comput. Appl. Math. 220, 290–321 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bona, J., Ponce, G., Saut, J.C., Tom, M.M.: A model system for strong interaction between internal solitary waves. Commun. Math. Phys. Appl. Math. 143, 287–313 (1992) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bourgain, J.: Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, I Schrödinger equation. Geom. Funct. Anal. 3, 107–156 (1993) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Craig, W., Kappeler, T., Strauss, W.: Gain of regularity for equations of Korteweg-de Vries type. Ann. Inst. Henri Poincaré 2, 147–186 (1992) MathSciNetGoogle Scholar
  8. 8.
    Dávila, M.: Continuação única para um Sistema Acoplado de Equações do tipo Korteweg-de Vries e para as Equações de Benjamin-Bona-Mahony e de Boussinesq. Tese de Doutorado, IM-UFRJ, Brazil (1995) Google Scholar
  9. 9.
    Gear, J.A., Grimshaw, R.: Weak and strong interactions between internal solitary waves. Commun. Math. 70, 235–258 (1984) zbMATHMathSciNetGoogle Scholar
  10. 10.
    Kato, K., Ogawa, T.: Analyticity and smoothing effect for the Korteweg-de Vries equation with a single point singularity. Math. Ann. 316(3), 577–608 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kato, T.: On the Cauchy problem for the (generalized) Korteweg-de Vries equations. Adv. Math. Suppl. Stud., Stud. Appl. Math. 8, 93–128 (1983) Google Scholar
  12. 12.
    Kato, T., Masuda, K.: Nonlinear evolution equations and analyticity. I. Ann. Inst. Henri Poincaré Anal. Non-linéaire 3, 455–467 (1986) zbMATHMathSciNetGoogle Scholar
  13. 13.
    Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier-Stokes equations. Commun. Pure Appl. Math. 41, 891–907 (1988) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kenig, C., Ponce, G., Vega, L.: A bilinear estimate with applications to the KdV equation. J. Am. Math. Soc. 9, 573–603 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kenig, C., Ponce, G., Vega, L.: Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction mapping principle. Commun. Pure Appl. Math. 46, 527–620 (1993) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Kenig, C., Ponce, G., Vega, L.: On the (generalized) Korteweg-de Vries equation. Duke Math. J. 59(3), 585–610 (1989) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Kenig, C., Ponce, G., Vega, L.: Oscillatory integrals and regularity equations. Indiana Univ. Math. J. 40, 33–69 (1991) zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Linares, F., Panthee, M.: On the Cauchy problem for a coupled system of KdV equations. Commun. Pure Appl. Ann. 3(3), 417–431 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Vera, O.: Gain of regularity for a generalized coupled system of nonlinear evolution dispersive equations type. Ph.D. Thesis, UFRJ, Rio de Janeiro, Brazil (2001) Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • Margareth S. Alves
    • 1
  • Bianca M. R. Calsavara
    • 2
  • Jaime E. Muñoz Rivera
    • 3
  • Mauricio Sepúlveda
    • 4
    Email author
  • Octavio Vera Villagrán
    • 5
  1. 1.Departamento de MatemáticaUniversidade Federal de Viçosa-UFVViçosaBrazil
  2. 2.Universidade Estadual de Campinas-UnicampLimeiraBrazil
  3. 3.National Laboratory for Scientific ComputationRio de JaneiroBrazil
  4. 4.CI²MA and Departamento de Ingeniería MatemáticaUniversidad de ConcepciónConcepciónChile
  5. 5.Departamento de MatemáticaUniversidad del Bío-BíoConcepciónChile

Personalised recommendations