Skip to main content
SpringerLink
Log in

Nonlinear Schrödinger Equation with Singularities

  • Conference paper
  • First Online:
Qualitative Properties of Dispersive PDEs (INdAM 2021)

Part of the book series: Springer INdAM Series ((SINDAMS,volume 52))

Included in the following conference series:

  • 198 Accesses

Abstract

We describe results on certain types of nonlinear Schrödinger equations, mainly the cubic equation with or without potential. We are interested in singular initial conditions and equations with a delta potential in three dimensions. The existence and uniqueness of solutions are proved in the Colombeau algebra setting and the notion of compatibility of solutions is explored.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Adami, R., Boni, F., Carlone, R., Tentarelli, L.: Ground states for the planar NLSE with a point defect as minimizers of the constrained energy (2021). arXiv:2109.09482. https://link.springer.com/article/10.1007/s00526-022-02310-8

  2. Albeverio, S., Gesztesy, F., Høegh-Krohn, R., Holden, H.: Solvable Models in Quantum Mechanics. Texts and Monographs in Physics. Springer, New York (1988)

    Google Scholar 

  3. Biagioni, H.A., Oberguggenberger, M.: Generalized solutions to the Korteweg–de Vries and the regularized long-wave equations. SIAM J. Math. Anal. 23, 923–940 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  4. Biagioni, H.A., Oberguggenberger, M.: Generalized solutions to Burgers’ equation. J. Differ. Equ. 97, 263–287 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bourgain, J.: Scattering in the energy space and below for 3D NLS. J. Anal. Math. 75, 267–297 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bourgain, J.: Global solutions of nonlinear Schrödinger equations. American Mathematical Soc. (1999)

    Google Scholar 

  7. Cacciapuoti, C., Finco, D., Noja, D.: Well posedness of the nonlinear Schröodinger equation with isolated singularities. J. Differ. Equ. 305, 288–318 (2021)

    Article  MATH  Google Scholar 

  8. Cazenave, T.: Semilinear Schrödinger Equations. American Mathematical Soc. (2003)

    Google Scholar 

  9. Christ, M., Colliander, J., Tao, T.: Instability of the periodic nonlinear Schrödinger equation (2003). arXiv preprint math/0311227

    Google Scholar 

  10. Colliander, J., Delort, J.M., Kenig, C., Staffilani, G.: Bilinear estimates and applications to 2D NLS. Trans. Am. Math. Soc. 353, 3307—3325 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Colliander, J., Keel, M., Staffilani, G., Takaoka, H., Tao, T.: Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on \(\mathbb {R}^3\). Commun. Pure Appl. Math: A Journal Issued by the Courant Institute of Mathematical Sciences 57, 987—1014 (2004)

    Google Scholar 

  12. Colombeau J.F.: New Generalized Functions and Multiplication of Distributions. North Holland, Amsterdam (1984)

    MATH  Google Scholar 

  13. Colombeau J.F.: Elementary Introduction to New Generalized Functions. North Holland, Amsterdam (1985)

    MATH  Google Scholar 

  14. Colombeau J.F.: Multiplication of Distributions, A Tool in Mathematics, Numerical Engineering and Theoretical Physics. Springer, Berlin-Heidelberg (1992)

    Google Scholar 

  15. Dugandžija, N., Nedeljkov, M.: Generalized solution to multidimensional cubic Schrödinger equation with delta potential. Monatshefte für Mathematik 190, 481–499 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dugandžija, N., Vojnović I.: Singular solution of the Hartree equation with a delta potential. Manuscript submitted for publication

    Google Scholar 

  17. Ferreira, L.C., Pava, J.A.: On the Schrödinger equation with singular potentials. Differ. Integr. Equ. 27, 767–800 (2014)

    MATH  Google Scholar 

  18. Georgiev, V., Michelangeli, A., Scandone, R.: Standing waves and global well–posedness for the 2d Hartree equation with a point interaction. arXiv:2204.05053

    Google Scholar 

  19. Grosser, M., Kunzinger, M., Oberguggenberger, M., Steinbauer, R: Geometric Theory of Generalized Functions with Applications to General Relativity. Springer Science, Berlin (2013)

    MATH  Google Scholar 

  20. Haraux, A., Weissler, F.B.: Non-uniqueness for a semilinear initial value problem. Indiana Univ. Math. J. 31, 167–189 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hörmann, G.: The Cauchy problem for Schrödinger-type partial differential operators with generalized functions in the principal part and as data. Monatshefte für Mathematik 163, 445–460 (2011). https://doi.org/10.1007/s00605-010-0232-x

    Article  MathSciNet  MATH  Google Scholar 

  22. Michelangeli, A., Olgiati, A., Scandone, R.: Singular Hartree equation in fractional perturbed Sobolev spaces. J. Nonlinear Math. Phys. 25, 558–588 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  23. Nedeljkov, M., Oberguggenberger, M., Pilipović, S.: Generalized solutions to a semilinear wave equation. Nonlinear Anal. Theory Methods Appl. 61(3), 461–475 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Nedeljkov, M., Pilipović, S., Rajter, D: Semigroups in Generalized Function Algebras. Heat Equation with Singular Potential and Singular Data. Proc. of Edinburg Royal Soc (2003)

    Google Scholar 

  25. Oberguggenberger, M. : Multiplication of Distributions and Applications to Partial Differential Equations. Pitman Research Notes Math., vol. 259. Longman, Harlow (1992)

    Google Scholar 

  26. Oberguggenberger, M., Wang, Y.G.: Generalized solutions to conservation laws. Zeitschrift für Analysis und ihre Anwendungen 13, 7–18 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  27. Scandone, R., Luperi Baglini L., Simonov, K.: A characterization of singular Schröodinger operators on the half–line. Can. Math. Bull. 64, 923–941 (2021)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

The authors acknowledge the financial support of the Ministry of Education, Science and Technological Development of the Republic of Serbia (Grant No. 451-03-68/2022-14/200125). The second author acknowledges the financial support of the Croatian Science Foundation under project 2449 MiTPDE.

Author information

Authors and Affiliations

  1. Faculty of Sciences, Novi Sad, Serbia

    Nevena Dugandžija & Ivana Vojnović

Authors
  1. Nevena Dugandžija
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ivana Vojnović
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nevena Dugandžija .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Pisa, Pisa, Italy

    Vladimir Georgiev

  2. Institute for Applied Mathematics and Hausdorff Centre for Mathematics, University of Bonn, Bonn, Germany

    Alessandro Michelangeli

  3. Gran Sasso Science Institute, L’Aquila, Italy

    Raffaele Scandone

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Dugandžija, N., Vojnović, I. (2022). Nonlinear Schrödinger Equation with Singularities. In: Georgiev, V., Michelangeli, A., Scandone, R. (eds) Qualitative Properties of Dispersive PDEs. INdAM 2021. Springer INdAM Series, vol 52. Springer, Singapore. https://doi.org/10.1007/978-981-19-6434-3_4

Download citation

Publish with us

Policies and ethics

Search

Navigation