Skip to main content
SpringerLink
Log in

On solutions of nonlinear Schrödinger equations with Cantor-type spectrum

  • Published:
Journal d’Analyse Mathematique Aims and scope

Abstract

The solvability of the Cauchy problem for the Nonlinear Nonfocusing Schrödinger equation (NNSE) with almost periodic initial data satisfying certain conditions is studied. It is shown that solutions are uniform almost periodic functions with respect to each variable. An example of initial data with Cantor-type spectrum is given. The Cauchy problem for NNSE is solved in the class of limit periodic functions which are well approximated by periodic ones.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Ablowitz and Y. Ma,The periodic cubic Schrödinger equation, Stud. Appl. Math.65 (1981), 113–158.

    MATH  MathSciNet  Google Scholar 

  2. C. Concini and R. Johnson,The algebraic-geometric AKNS potentials, Ergodic Theory Dynam. Systems7 (1987), 1–24.

    MATH  MathSciNet  Google Scholar 

  3. W. Craig,The trace formula for Schrödinger operators on the line, Comm. Math. Phys.126 (1989), 379–407.

    Article  MATH  MathSciNet  Google Scholar 

  4. A. Dubrovin, V. Matveev and S. Novikov,Nonlinear equations ofKorteweg de Vries type, finite zone linear operators and Abelian varieties, Russian Math. Surveys31 (1976), 59–146.

    Article  MATH  MathSciNet  Google Scholar 

  5. I. Egorova,A complete collection of spectral parameters for the Dirac operator with limit periodic coefficients, Teor. Funktsii Funktsional. Anal, i Prilozhen.47 (1987), 636–644 (Russian) (J.Soviet Math.48(1990), 636–644).

    Google Scholar 

  6. I. Egorova,The Cauchy problem for the KdV equation with almost periodic initial data whose spectrum is nowhere dense, Advances in Soviet Mathematics19 (1994), 181–208.

    MathSciNet  Google Scholar 

  7. I. Egorova and A. Surkova,On the almost periodicity of “reflectionless” Dirac operator with the Cantor type spectrum, Ukrainian Math. Reports12 (1995), 13–15 (Russian).

    MathSciNet  Google Scholar 

  8. A. Its,The canonical system with finite band spectrum and the periodic solutions of nonlinear Schrödinger equation, Vestnik Leningrad Univ.7 (1976), 39–46 (Russian).

    MathSciNet  Google Scholar 

  9. B. Levitan,Inverse Sturm-Liouville Problems, Science Press, The Netherlands, 1987.

    MATH  Google Scholar 

  10. B. Levitan,On closure of the set of finite gap potential, Mat. Sb.123 (1984), 69–91 (Russian).

    MathSciNet  Google Scholar 

  11. B. Levitan and A. Mamatov,The estimate of the Cauchy matrix-function for the Dirac system in the case of finite band potentials, Mat. Zametki53 (1993), 62–76 (Russian).

    MATH  MathSciNet  Google Scholar 

  12. V. Marchenko,Sturm-Liouville Operators and.Applications, BirkhÄuser, Basel, 1986.

    MATH  Google Scholar 

  13. V. Marchenko,The periodic Korteweg-de Vries problem. Mat. Sb.95 (1974), 331–356 (Russian) (Math. USSR Sb.24 (1974), 319–344).

    MathSciNet  Google Scholar 

  14. H. McKean and E. Trubowitz,Hill’s operator and hyperelliptic function theory in the presence of infinitely many branch points, Comm. Pure Appl. Math.29 (1976), 143–226.

    Article  MATH  MathSciNet  Google Scholar 

  15. T. Misyura,The characterization of spectra of periodic and nonperiodic boundary problems generated by Dirac operation (II), Teor. Funktsii Funktsional. Anal, i Prilozhen.31 (1979), 102–109 (Russian).

    MATH  MathSciNet  Google Scholar 

  16. L. Pastur and A. Figotin,Spectra of Random and Almost Periodic Operators, Springer-Verlag, Berlin, 1992.

    MATH  Google Scholar 

  17. L. Pastur and V. Tkachenko,Spectral theory of a class of one-dimensional Schrödinger operators with limit periodic potential, Trudy Mosk. Mat. Obsh. (1984) (Russian) (Trans. Moscow Math. Soc.51 (1988), 114–168).

    Google Scholar 

  18. A. Savin,Some questions of spectral theory of Schrödinger and Dirac operators with almost periodic potentials, PhD Thesis, Moscow, 1984.

  19. M. Sodin and P. Yuditskii,Almost periodic Sturm Liouville operators with Cantor homogeneous spectrum. Comment. Math. Helv.70 (1995), 639–658.

    Article  MATH  MathSciNet  Google Scholar 

  20. M. Sodin and P. Yuditskii,Infinite dimensional Jacobi inversion problem, almost-periodic Jacobi matrices with homogeneous spectrum, and Hardy classes of character-automorphic functions, Geom. Funct. Anal., to appear.

  21. M. Zamonov,Infinite band potentials of one dimensional Dirac operator. Reports of AN Tadzh. SSR28 (1984), 17–21.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut de Mathématiques de Jussieu, CNRS UMR 9994 Laboratoire de Physique Mathématique et Géométrie, Case 7012, Université Paris 7 Denis Diderot, 2 Place Jussieu, F-75251, Paris Cedex 05, France

    Anne Boutet de Monvel & Ira Egorova

  2. Mathematical Division, Institute for Low Temperature Physics, 47 Lenin Ave, 310164, Kharkiv, Ukraine

    Anne Boutet de Monvel & Ira Egorova

Authors
  1. Anne Boutet de Monvel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ira Egorova
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

de Monvel, A.B., Egorova, I. On solutions of nonlinear Schrödinger equations with Cantor-type spectrum. J. Anal. Math. 72, 1–20 (1997). https://doi.org/10.1007/BF02843151

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02843151

Keywords

Search

Navigation