Skip to main content
SpringerLink
Log in

High-speed excited multi-solitons in competitive power nonlinear Schrödinger equations

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

This paper deals with the competitive power nonlinear Schrödinger equation, which originates from the cubic-quintic model in physics. The equation admits infinitely many excited solitons, and the Cauchy problem is globally well-posed in the energy space. In terms of Côte and Le Coz’s argument, high-speed excited multi-solitons of the equation are constructed, which extend Côte and Le Coz’s results from the focusing nonlinear cases to the competitive nonlinear cases combining the focusing nonlinearities and defocusing nonlinearities.

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. Berestycki, H., Lions, P.L.: Nonlinear scalar field equations. Arch. Ration. Math. Anal. 82, 313–375 (1983)

    Article  Google Scholar 

  2. Cazenave, T.: Semilinear Schrödinger Equations. Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI (2003)

  3. Côte, R., Le Coz, S.: High-speed excited multi-solitons in nonlinear Schrödinger equations. J. Math. Pures Appl. 96, 135–166 (2011)

    Article  MathSciNet  Google Scholar 

  4. Côte, R., Martel, Y., Merle, F.: Construction of multi-soliton solutions for the \(L^{2}\)-supercritical gKdv and NLS equations. Rev. Mat. Iberoam. 27(1), 273–302 (2011)

    Article  MathSciNet  Google Scholar 

  5. Desyatnikov, A., Maimistov, A., Malomed, B.: Three-dimensional spinning solitons in dispersive media with the cubic-quintic nonlinearity. Phys. Rev. E. 61(3), 3107–3113 (2000)

    Article  Google Scholar 

  6. Fibich, G.: The Nonlinear Schrödinger Equation: Singular Solutions and Optical Collapse. Applied Mathematical Sciences, vol. 192. Springer, Cham (2015)

    Book  Google Scholar 

  7. Fukaya, N., Hayashi, M.: Instability of algebraic standing waves for nonlinear Schrödinger equations with double power nonlinearities. Trans. Am. Math. Soc. 374, 1421–1447 (2021)

    Article  Google Scholar 

  8. Jones, C., Küpper, T.: On the infinitely many solutions of a semilinear elliptic equation. SIAM J. Math. Anal. 17(4), 803–835 (1986)

    Article  MathSciNet  Google Scholar 

  9. Killip, R., Oh, T., Pocovnicu, O., Visan, M.: Solitons and scattering for the cubic-quintic nonlinear Schrödinger equation on \(\mathbb{R}^{3}\). Arch. Ration. Mech. Anal. 225(1), 469–548 (2017)

    Article  MathSciNet  Google Scholar 

  10. Le Coz, S.: Standing waves in nonlinear Schrödinger equations. In: Emmrich, E., Wittbold, P. (eds.) Analytical and Numerical Aspects of Partial Differential Equations, pp. 151–192. Walter de Gruyter, Berlin (2009)

    MATH  Google Scholar 

  11. Martel, Y., Merle, F.: Multi solitary waves for nonlinear Schrödinger equations. Ann. I. H. Poincaré-AN 23, 849–864 (2006)

    Article  Google Scholar 

  12. Martel, Y., Merle, F., Tsai, T.P.: Stability in \(H^1\) of the sum of \(K\) solitary waves for some nonlinear Schrödinger equations. Duke Math. J. 133(3), 405–466 (2006)

    Article  MathSciNet  Google Scholar 

  13. Martel, Y., Raphaël, P.: Strongly interacting blow up bubbles for the mass critical NLS. Ann. Sci. Éc. Norm. Supér 51, 701–737 (2018)

    Article  MathSciNet  Google Scholar 

  14. Merle, F.: Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity. Commun. Math. Phys. 129(2), 223–240 (1990)

    Article  MathSciNet  Google Scholar 

  15. Mihalache, D., Mazilu, D., Crasovan, L.C., Towers, I., Malomed, B.A., Buryak, A.V., Torner, L., Lederer, F.: Stable three-dimensional spinning optical solitons supported by competing quadratic and cubic nonlinearities. Phys. Rev. E. 66, 016613–7 (2002)

    Article  MathSciNet  Google Scholar 

  16. Mihalache, D., Mazilu, D., Crasovan, L.C., Malomed, B.A., Lederer, F.: Three-dimensional spinning solitons in the cubic-quintic nonlinear medium. Phys. Rev. E. 61(6), 7142–7145 (2000)

    Article  Google Scholar 

  17. Sulem, C., Sulem, P.-L.: The Nonlinear Schrödinger Equation: Self-focusing and Wave Collapse. Applied Mathematical Sciences, vol. 139. Springer, New York (1999)

    MATH  Google Scholar 

  18. Strauss, W.A.: Existence of solitary waves in higher dimensions. Commun. Math. Phys. 55(2), 149–162 (1977)

    Article  MathSciNet  Google Scholar 

  19. Tao, T.: Why are solitons stable? Bull. Am. Math. Soc. 46(1), 1–33 (2009)

    Article  MathSciNet  Google Scholar 

  20. Nguy\(\tilde{\hat{e}}\)n, T.V.: Existence of multi-solitary waves with logarithmic relative distances for the NLS equation. C. R. Acad. Sci. Paris. Ser. I 357, 13–58 (2019)

  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. JETP 34, 62–69 (1972)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

This research is supported by the National Natural Science Foundation of China 11871138.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China

    Mengxue Bai & Jian Zhang

Authors
  1. Mengxue Bai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jian Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jian Zhang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bai, M., Zhang, J. High-speed excited multi-solitons in competitive power nonlinear Schrödinger equations. Z. Angew. Math. Phys. 73, 141 (2022). https://doi.org/10.1007/s00033-022-01774-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-022-01774-0

Keywords

Mathematics Subject Classification

Search

Navigation