A trilinear estimate with application to the perturbed nonlinear Schrödinger equations with the Kerr law nonlinearity

Journal of Evolution Equations (2020)Cite this article

Abstract

In this paper, we investigate the initial value problem (IVP henceforth) associated with the perturbed nonlinear Schrödinger equations with the Kerr law nonlinearity. First, by using Fourier restriction norm method and Tao’s [k, Z]-multiplier method, we establish a trilinear estimate on the Bourgain space \(X_{s,b}.\) Then, combining the trilinear estimate with the contraction mapping principle, we prove that IVP is locally well-posed for the initial data \((u_0(x),v_0(x))\in H^s({\mathbb {R}})\times H^s({\mathbb {R}})\) with \(s\ge \frac{1}{4}\).

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Affiliations

  1. School of Mathematics, Hunan Institute of Science and Technology, Yueyang, 414006, Hunan, China

    Zaiyun Zhang

  2. School of Mathematics and Statistics, Yulin Normal University, Yulin, 537000, Guangxi, China

    Zhenhai Liu

  3. College of Sciences, Guangxi University for Nationalities, Nanning, 530006, Guangxi Province, China

    Zhenhai Liu

  4. School of Mathematics and Statistics, Central South University, Changsha, 410083, Hunan, China

    Youjun Deng

  5. School of Mathematics, Hunan Institute of Science and Technology, Yueyang, 414006, Hunan, People’s Republic of China

    Limei Li & Fan He

  6. Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, 410014, Hunan, People’s Republic of China

    Chuangxia Huang

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  1. Zaiyun Zhang
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Correspondence to Zaiyun Zhang.

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This work was supported by Hunan Provincial Natural Science Foundation of China Nos. 2016JJ2061, 2020JJ2038, Scientific Research Fund of Hunan Provincial Education Department Nos. 18A325, 17A087, 17C0711, NNSF of China Grant Nos. 11671101, 11971487, NSF of Guangxi (2018GXNSFDA138002), the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska–Curie (823731CONMECH) and Special Funds of Guangxi Distinguished Experts Construction Engineering. Also, this work was partially supported by Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering of Changsha University of Science and Technology Grant No. 2018MMAEZD05, Open project of Hainan Key Laboratory of Computing Science and Application No. JSKX201905 and NNSF of China Grant Nos. 71471020, 51839002.

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Zhang, Z., Liu, Z., Deng, Y. et al. A trilinear estimate with application to the perturbed nonlinear Schrödinger equations with the Kerr law nonlinearity. J. Evol. Equ. (2020). https://doi.org/10.1007/s00028-020-00631-9

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Keywords

  • Perturbed nonlinear Schrödinger equations with Kerr law nonlinearity
  • Fourier restriction norm method
  • Low regularity
  • Local well-posedness (LWP)

Mathematics Subject Classification

  • 35A07
  • 35Q53