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Partial symmetry of normalized solutions for a doubly coupled Schrödinger system

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Abstract

We consider the normalized solutions of a Schrödinger system which arises naturally from nonlinear optics, the Hartree–Fock theory for Bose–Einstein condensates. And we investigate the partial symmetry of normalized solutions to the system and their symmetry-breaking phenomena. More precisely, when the underlying domain is bounded and radially symmetric, we develop a kind of polarization inequality with weight to show that the first two components of the normalized solutions are foliated Schwarz symmetric with respect to the same point, while the latter two components are foliated Schwarz symmetric with respect to the antipodal point. Furthermore, by analyzing the singularly perturbed limit profiles of these normalized solutions, we prove that they are not radially symmetric at least for large nonlinear coupling constant \(\beta \), which seems a new method to prove the symmetry-breaking phenomenons of normalized solutions.

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Funding

H. Luo is supported by National Natural Science Foundation of China, 11901182, and by the Fundamental Research Funds of the Central Universities, 531118010205. Z. Zhang is supported by National Natural Science Foundation of China, 11771428, 11926335.

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Authors and Affiliations

  1. School of Mathematics, Hunan University, Changsha, 410082, Hunan, People’s Republic of China

    Haijun Luo

  2. Academy of Mathematics and Systems Science, The Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Zhitao Zhang

  3. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, People’s Republic of China

    Zhitao Zhang

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  1. Haijun Luo
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  2. Zhitao Zhang
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Correspondence to Haijun Luo.

Additional information

This article is part of the topical collection dedicated to Prof. Dajun Guo for his 85th birthday, edited by Yihong Du, Zhaoli Liu, Xingbin Pan, and Zhitao Zhang.

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Luo, H., Zhang, Z. Partial symmetry of normalized solutions for a doubly coupled Schrödinger system. SN Partial Differ. Equ. Appl. 1, 24 (2020). https://doi.org/10.1007/s42985-020-00016-0

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