Advertisement

Annales Henri Poincaré

, Volume 16, Issue 2, pp 535–567 | Cite as

Null Structure in a System of Quadratic Derivative Nonlinear Schrödinger Equations

  • Masahiro Ikeda
  • Soichiro Katayama
  • Hideaki SunagawaEmail author
Article

Abstract

We consider the initial value problem for a three-component system of quadratic derivative nonlinear Schrödinger equations in two space dimensions with the masses satisfying the resonance relation. We present a structural condition on the nonlinearity under which small data global existence holds. It is also shown that the solution is asymptotically free. Our proof is based on the commuting vector field method combined with smoothing effects.

Keywords

Nonlinear Term Space Dimension Global Existence Null Condition Unique Global Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Agemi R.: Global existence of nonlinear elastic waves. Invent. Math. 142(2), 225–250 (2000)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bernal-Vílchis F., Hayashi N., Naumkin P.I.: Quadratic derivative nonlinear Schrödinger equations in two space dimensions. NoDEA Nonlinear Differ. Equ. Appl. 18(3), 329–355 (2011)CrossRefzbMATHGoogle Scholar
  3. 3.
    Chihara H.: Local existence for the semilinear Schrödinger equations in one space dimension. J. Math. Kyoto Univ. 34(2), 353–367 (1994)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Chihara H.: Local existence for semilinear Schrödinger equations. Math. Jpn. 42(1), 35–51 (1995)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Chihara H.: Gain of regularity for semilinear Schrödinger equations. Math. Ann. 315(4), 529–567 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Christodoulou D.: Global solutions of nonlinear hyperbolic equations for small initial data. Commun. Pure Appl. Math. 39(2), 267–282 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Colin M., Colin T.: On a quasilinear Zakharov system describing laser–plasma interactions. Differ. Integral Equ. 17(3-4), 297–330 (2004)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Colin M., Colin T., Ohta M.: Stability of solitary waves for a system of nonlinear Schrödinger equations with three wave interaction. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(6), 2211–2226 (2009)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Delort, J.-M.: Global solutions for small nonlinear long range perturbations of two dimensional Schrödinger equations. Mém. Soc. Math. Fr., no.91 (2002)Google Scholar
  10. 10.
    Delort J.-M., Fang D., Xue R.: Global existence of small solutions for quadratic quasilinear Klein–Gordon systems in two space dimensions. J. Funct. Anal. 211(2), 288–323 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Doi S.: On the Cauchy problem for Schrödinger type equations and the regularity of solutions. J. Math. Kyoto Univ. 34(2), 319–328 (1994)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Friedman A.: Partial Differential Equations. Holt, Rinehart and Winston, New York (1969)zbMATHGoogle Scholar
  13. 13.
    Georgiev V.: Global solution of the system of wave and Klein–Gordon equations. Math. Z. 203(4), 683–698 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Germain P., Masmoudi N., Shatah J.: Global solutions for 2D quadratic Schrödinger equations. J. Math. Pures Appl. 97(5), 505–543 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Hayashi N., Kato K.: Global existence of small analytic solutions to Schrödinger equations with quadratic nonlinearity. Commun. Partial Differ. Equ. 22(5–6), 773–798 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Hayashi N., Li C., Naumkin P.I.: On a system of nonlinear Schrödinger equations in 2d. Differ. Integral Equ. 24(5–6), 417–434 (2011)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Hayashi N., Li C., Naumkin P.I.: Modified wave operator for a system of nonlinear Schrödinger equations in 2d. Commun. Partial Differ. Equ. 37(6), 947–968 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Hayashi N., Li C., Ozawa T.: Small data scattering for a system of nonlinear Schrödinger equations. Differ. Equ. Appl. 3(3), 415–426 (2011)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Hayashi N., Naumkin P.I.: Asymptotics of small solutions to nonlinear Schrödinger equations with cubic nonlinearities. Int. J. Pure Appl. Math. 3(3), 255–273 (2002)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Hayashi N., Naumkin P.I.: Global existence for two dimensional quadratic derivative nonlinear Schrödinger equations. Commun. Partial Differ. Equ. 37(4), 732–752 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Hayashi N., Naumkin P.I., Sunagawa H.: On the Schrödinger equation with dissipative nonlinearities of derivative type. SIAM J. Math. Anal. 40(1), 278–291 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Hörmander L.: Lectures on Nonlinear Hyperbolic Differential Equations. Springer, Berlin (1997)zbMATHGoogle Scholar
  23. 23.
    John F.: Nonlinear Wave Equations, Formation of Singularities University Lecture Series 2. American Mathematical Society, Providence (1990)Google Scholar
  24. 24.
    Katayama, S., Li, C., Sunagawa, H.: A remark on decay rates of solutions for a system of quadratic nonlinear Schrödinger equations in 2D. Differ. Integral Equ. 27(3–4), 301–312 (2014)Google Scholar
  25. 25.
    Katayama S., Ozawa T., Sunagawa H.: A note on the null condition for quadratic nonlinear Klein–Gordon systems in two space dimensions. Commun. Pure Appl. Math. 65(9), 1285–1302 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Katayama S., Tsutsumi Y.: Global existence of solutions for nonlinear Schrödinger equations in one space dimension. Commun. Partial Differ. Equ. 19(11–12), 1971–1997 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Kawahara Y., Sunagawa H.: Remarks on global behavior of solutions to nonlinear Schrödinger equations. Proc. Jpn. Acad. Ser. A 82(8), 117–122 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Kawahara Y., Sunagawa H.: Global small amplitude solutions for two-dimensional nonlinear Klein–Gordon systems in the presence of mass resonance. J. Differ. Equ. 251(9), 2549–2567 (2011)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Kenig C.E., Ponce G., Vega L.: Small solutions to nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 10(3), 255–288 (1993)zbMATHMathSciNetGoogle Scholar
  30. 30.
    Klainerman, S.: The null condition and global existence to nonlinear wave equations. In: Nonlinear Systems of Partial Differential Equations in Applied Mathematics, Part 1, Lectures in Applied Mathematics, vol. 23, pp. 293–326. AMS, Providence (1986)Google Scholar
  31. 31.
    Mizohata, S.: The Theory of Partial Differential Equations (K. Miyahara, transl.). Cambridge University Press, New York (1973)Google Scholar
  32. 32.
    Ozawa T., Sunagawa H.: Small data blow-up for a system of nonlinear Schrödinger equations. J. Math. Anal. Appl. 399(1), 147–155 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Racke R.: Lectures on Nonlinear Evolution Equations, Initial Value Problems, Aspects of Math. E19. Friedr. Vieweg & Sohn, Braunschweig (1992)CrossRefGoogle Scholar
  34. 34.
    Sideris T.: The null condition and global existence of nonlinear elastic waves. Invent. Math. 123(2), 323–342 (1996)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Sunagawa H.: On global small amplitude solutions to systems of cubic nonlinear Klein–Gordon equations with different mass terms in one space dimension. J. Differ. Equ. 192(2), 308–325 (2003)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Sunagawa H.: Lower bounds of the lifespan of small data solutions to the nonlinear Schrödinger equations. Osaka J. Math. 43(4), 771–789 (2006)zbMATHMathSciNetGoogle Scholar
  37. 37.
    Tsutsumi Y.: The null gauge condition and the one dimensional nonlinear Schrödinger equation with cubic nonlinearity. Indiana Univ. Math. J. 43(1), 241–254 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Tsutsumi Y.: Stability of constant equilibrium for the Maxwell–Higgs equations. Funkcial. Ekvac. 46(1), 41–62 (2003)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Masahiro Ikeda
    • 1
  • Soichiro Katayama
    • 2
  • Hideaki Sunagawa
    • 3
    Email author
  1. 1.Mathematical InstituteTohoku UniversityAoba-kuJapan
  2. 2.Department of MathematicsWakayama UniversityWakayamaJapan
  3. 3.Department of Mathematics, Graduate School of ScienceOsaka UniversityToyonaka, OsakaJapan

Personalised recommendations