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Pitchfork Bifurcation at Line Solitons for Nonlinear Schrödinger Equations on the Product Space \({\mathbb {R}}\times {\mathbb {T}}\)

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Abstract

In this paper, we study the bifurcation problem from a line soliton for a stationary nonlinear Schrödinger equation on the product space \({\mathbb {R}}\times {\mathbb {T}}\). We extend earlier results to a larger class of the nonlinearity in the equation. The salient point of our analysis relies on a lower bound of solution to the “auxiliary equation” and then on the application of the Crandall–Rabinowitz argument.

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References

  1. Berestycki, H., Nirenberg, L.: Some qualitative properties of solutions of semilinear elliptic equations in cylindrical domains. In: Analysis, et Cetera, pp. 115–164. Academic Press, Cambridge (1990)

    Chapter  Google Scholar 

  2. Chang, S.-M., Gustafson, S., Nakanishi, K., Tsai, T.-P.: Spectra of linearized operators for NLS solitary waves. SIAM J. Math. Anal. 39, 1070–1111 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Crandall, M.G., Rabinowitz, P.H.: Bifurcation from simple eigenvalues. J. Funct. Anal. 8, 321–340 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  4. Kirr, E., Kevrekidis, P.G., Pelinovsky, D.E.: Symmetry-breaking bifurcation in the nonlinear Schrödinger equation with symmetric potentials. Commun. Math. Phys. 308, 795–844 (2011)

    Article  ADS  MATH  Google Scholar 

  5. Lieb, E.H., Loss, M.: Analysis, 2nd edn. American Mathematical Society, Providence (2001)

    MATH  Google Scholar 

  6. Yamazaki, Y.: Stability for line standing waves near the bifurcation point for nonlinear Schrödinger equations. Kodai Math. J. 38, 65–96 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

TA was supported by JSPS KAKENHI Grant Number (20K03697). YB was supported by PIMS grant and NSERC grant (371637-2019). SI was supported by NSERC grant (371637-2019). HK was supported by JSPS KAKENHI Grant Number JP20K03706.

Funding

Canadian Network for Research and Innovation in Machining Technology, Natural Sciences and Engineering Research Council of Canada (371637-2019).

Author information

Authors and Affiliations

  1. Department of Engineering, Shizuoka University, Jyohoku 3-5-1, Hamamatsu-Shi, Shizuoka, 432-8561, Japan

    Takafumi Akahori

  2. Department of Mathematics and Statistics, University of Victoria, 3800 Finnerty Road, Victoria, BC, V8P 5C2, Canada

    Yakine Bahri & Slim Ibrahim

  3. Department of Mathematics, Tsuda University, 2-1-1 Tsuda-machi, Kodaira-shi, Tokyo, 187-8577, Japan

    Hiroaki Kikuchi

Authors
  1. Takafumi Akahori
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  2. Yakine Bahri
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  3. Slim Ibrahim
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  4. Hiroaki Kikuchi
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Corresponding author

Correspondence to Slim Ibrahim.

Additional information

Communicated by Claude-Alain Pillet.

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Akahori, T., Bahri, Y., Ibrahim, S. et al. Pitchfork Bifurcation at Line Solitons for Nonlinear Schrödinger Equations on the Product Space \({\mathbb {R}}\times {\mathbb {T}}\). Ann. Henri Poincaré (2023). https://doi.org/10.1007/s00023-023-01370-6

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  • DOI: https://doi.org/10.1007/s00023-023-01370-6

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