Skip to main content
Log in

On solutions of the initial value problem for the nonlinear Schrödinger equations in one space dimension

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  1. Baillon, J.B., Cazenave, T., Figueira, M.: Équation de Schrödinger nonlinéaire. C.R. Acad. Sci. Paris284, 869–872 (1977)

    Google Scholar 

  2. Barab, J.E.: Nonexistence of asymptotic free solutions for a nonlinear Schrödinger equation. J. Math. Phys.25, 3270–3273 (1984)

    Google Scholar 

  3. Friedman, A.: Partial Differential Equations. New York: Holt-Rinehart and Winston 1969

    Google Scholar 

  4. Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura, R.M.: Korteweg-de Vries equation and generalization, VI. Methods for exact solution. Commun. Pure Appl. Math.27, 97–133 (1974)

    Google Scholar 

  5. Ginibre, J., Velo, G.: On a class of nonlinear Schrödinger equations I, II. J. Funct. Anal.32, 1–32, 33–71 (1979); III. Ann. Inst. Henri Poincare, Sect. A28, 287–316 (1978)

    Google Scholar 

  6. Ginibre, J., Velo, G.: The global Cauchy problem for the nonlinear Schrödinger equation revisited. Ann. Inst. Henri Poincare Analyse non linéaire2, 309–327 (1985)

    Google Scholar 

  7. Ginibre, J., Velo, G.: Scattering theory in the energy space for a class of nonlinear Schrödinger equations. Preprint (1984)

  8. Glassey, R.T.: On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations. J. Math. Phys.18, 1794–1797 (1977)

    Google Scholar 

  9. Hayashi, N.: Classical solutions of nonlinear Schrödinger equation. Manuscripta Math.55, 171–190 (1986)

    Google Scholar 

  10. Hayashi, N., Tsutsumi, M.:L -decay of classical solutions for nonlinear Schrödinger equations. Submitted (1985)

  11. Hayashi, N., Nakamitsu, K., Tsutsumi, M.: On solutions of the initial value problem for the nonlinear Schrödinger equations. To appear in J. Funct. Anal.

  12. Kadekawa, S.: Ph.D. Thesis, Indiana University (1980)

  13. Kametaka, Y.: On some seri-linear Schrödinger equations. Sûrikaisekikenkyusho Kôkyûroku106, 145–152 (1971) (in Japanese)

    Google Scholar 

  14. Kato, T.: The Cauchy problem for the Korteweg-de Vries equations, in “Nonlinear partial differential equations and their applications”. College de France Seminair, Vol. I, 293–306. Boston: Pitman 1979

    Google Scholar 

  15. Strauss, W.A.: Decay and asymptotic for □u=F(u), J. Funct. Anal.2, 409–457 (1968)

    Google Scholar 

  16. Strichartz, R.S.: Restriction of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J.44, 705–714 (1977)

    Google Scholar 

  17. Tsutsumi, M.: Weighted Sobolev spaces and rapidly decreasing solutions of some nonlinear dispersive wave equations. J. Differ. Equations42, 260–281 (1981)

    Google Scholar 

  18. Tsutsumi, M.: Nonexistence of global solutions to the Cauchy problem for damped nonlinear Schrödinger equations. SIAM J. Math. Anal.15, 357–366 (1984)

    Google Scholar 

  19. Tsutsumi, Y.: Global existence and asymptotic behavior of solutions for nonlinear Schrödinger equations, D. Thesis, University of Tokyo (1985)

  20. Tsutsumi, Y., Yajima, K.: The asymptotic behavior of nonlinear Schrödinger equations. Bull. Amer. Math. Soc., New Ser.11, 186–188 (1984)

    Google Scholar 

  21. Zakharov, V.E., Shabat, A.B.: Exact theory of two dimensional self-focusing and one dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP34, 62–69 (1972)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

This work was supported in part by Grant-in-Aid for Scientific Research (No. 60540124), Ministry of Education and by W.U. Grant for Special Research Projects.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hayashi, N., Nakamitsu, K. & Tsutsumi, M. On solutions of the initial value problem for the nonlinear Schrödinger equations in one space dimension. Math Z 192, 637–650 (1986). https://doi.org/10.1007/BF01162710

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01162710

Keywords

Navigation