Science China Mathematics

, Volume 60, Issue 4, pp 651–670 | Cite as

Multi-solitons for a generalized Davey-Stewartson system



This paper studies multi-solitons of non-integrable generalized Davey-Stewartson system in the elliptic-elliptic case. By extending the method for constructing multi-solitons of non-integrable nonlinear Schrödinger equations and systems developed by Martel et al. to the present non-integrable generalized Davey- Stewartson system and overcoming some new difficulties caused by the nonlocal operator B, we prove the existence of multi-solitons for this system. Furthermore, we also give a generalization of this result to a more general class of equations with nonlocal nonlinearities.


Davey-Stewartson system multi-solitons existence nonlocal 


35Q35 76W05 35B65 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



This work was supported by National Natural Science Foundation of China (Grant No. 11571381).


  1. 1.
    Ablowitz M, Clarkson P. Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge: Cambridge University Press, 1991CrossRefMATHGoogle Scholar
  2. 2.
    Ablowitz M, Fokas A. On the inverse scattering transform of multidimensional nonlinear equations. J Math Phys, 1984, 25: 2494–2505MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Ablowitz M, Haberman R. Nonlinear evolution equations in two and three dimensions. Phys Rev Lett, 1975, 35: 1185–1188MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ablowitz M, Segur H. On the evolution of packets of water waves. J Fluid Mech, 1979, 92: 691–715MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Anker D, Freeman N C. On the soliton solutions of the Davey-Stewartson equation for long waves. Proc R Soc Lond Ser A, 1978, 360: 529–540MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Babaoglu C, Eden A, Erbay S, et al. Global existence and nonexistence results for a generalized Davey-Stewartson system. J Phys A Math Gene Phys, 2004, 37: 11531–11546MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Boiti M, Leon J, Martina L, et al. Scattering of localized solitons in the plane. Phys Lett A, 1988, 132: 432–439MathSciNetCrossRefGoogle Scholar
  8. 8.
    Cazenave T. Semilinear Schrödinger Equations. New York: New York University, 2003CrossRefMATHGoogle Scholar
  9. 9.
    Cipolatti R. On the existence of standing waves for a Davey-Stewartson system. Comm Partial Differential Equations, 1992, 17: 967–988MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Cipolatti R. On the instability of ground states for a Davey-Stewartson system. Ann Inst H Poincaré Sec A, 1993, 58: 85–104MathSciNetMATHGoogle Scholar
  11. 11.
    Cornille H. Solutions of the generalized nonlinear Schrödinger equation in two spatial dimensions. J Math Phys, 1979, 20: 199–209MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Côte R, Le Coz S. High-speed excited multi-solitons in nonlinear Schrödinger equations. J Math Pures Appl, 2011, 96: 135–166MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Côte R, Martel Y, Merle F, et al. Construction of multi-soliton solutions for the L2-supercritical gKdV and NLS equations. Rev Mat Iberoam, 2011, 27: 273–302MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Davey A, Stewartson K. On 3-dimensional packets of surface waves. Proc R Soc Lond Ser A, 1974, 338: 101–110MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Fokas A S, Sung L Y. On the solvability of the N-wave, Davey-Stewartson and Kadomtsev-Petviashvili equations. Inverse Probl, 1992, 8: 375–419MathSciNetMATHGoogle Scholar
  16. 16.
    Gan Z, Zhang J. Sharp threshold of global existence and instability of standing wave for a Davey-Stewartson system. Comm Math Phys, 2008, 283: 93–125MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Ghidaglia J M, Saut J C. On the initial value problem for the Davey-Stewartson systems. Nonlinearity, 1990, 3: 475–506MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Guo B L, Wang B X. The Cauchy problem for Davey-Stewartson systems. Commun Pure Appl Math, 1999, 52: 1477–1490MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Ianni I, Le Coz S. Multi-Speed solitary waves solutions for nonlinear Schrödinger systems. J Lond Math Soc, 2014, 89: 623–639MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Lions P L. The concentration-compactness principle in the calculus of variations: The locally compact case 1. Ann Inst H Poincaré Anal Non Linéaire, 1984, 1: 105–145MathSciNetGoogle Scholar
  21. 21.
    Martel Y, Merle F. Multi solitary waves for nonlinear Schrödinger equations. Ann Inst H Poincaré Anal Non Linéaire, 2006, 23: 849–864MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Ohta M. Stability of standing waves for the generalized Davey-Stewartson system. J Dynam Differential Equations, 1994, 6: 325–334MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Ohta M. Instability of standing waves for the generalized Davey-Stewartson system. Ann Inst H Poincaré Anal Non Linéaire, 1995, 62: 69–80MathSciNetMATHGoogle Scholar
  24. 24.
    Ohta M. Blow-up solutions and strong instability of standing waves for the generalized Davey-Stewartson system in R2. Ann Inst H Poincaré Anal Non Linéaire, 1995, 63: 111–117MATHGoogle Scholar
  25. 25.
    Sulem C, Sulem P L. The Nonlinear Schrödinger Equation: Self-focusing and Wave Collapse. New York: Springer-Verlag, 1999MATHGoogle Scholar
  26. 26.
    Sung L Y. An inverse-scattering transform for the Davey-Stewartson II equations. Part III. J Math Anal Appl, 1994, 183: 477–494MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Wang B X, Guo B L. On the initial value problem and scattering of solutions for the generalized Davey-Stewartson systems. Sci China Ser A, 2001, 44: 994–1002MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Wang Z, Cui S B. Multi-speed solitary wave solutions for a coherently coupled nonlinear Schrödinger system. J Math Phys, 2015, 56: 021503MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.School of Mathematics and Big DataFoshan UniversityFoshanChina
  2. 2.Department of MathematicsSun Yat-Sen UniversityGuangzhouChina

Personalised recommendations