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Segregated and Synchronized Vector Solutions for Nonlinear Schrödinger Systems

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Abstract

We consider the following nonlinear Schrödinger system in \({\mathbb{R}^3}\)

$$\left\{\begin{array}{ll}-\Delta u + P(|x|)u = \mu u^{2}u + \beta v^2u,\quad x \in \mathbb{R}^3,\\-\Delta v + Q(|x|)v = \nu v^{2}v + \beta u^2v,\quad x \in \mathbb{R}^3,\end{array}\right.$$

where P(r) and Q(r) are positive radial potentials, \({\mu > 0, \nu > 0}\) and \({\beta \in \mathbb{R}}\) is a coupling constant. This type of system arises, in particular, in models in Bose–Einstein condensates theory.

We examine the effect of nonlinear coupling on the solution structure. In the repulsive case, we construct an unbounded sequence of non-radial positive vector solutions of segregated type, and in the attractive case we construct an unbounded sequence of non-radial positive vector solutions of synchronized type. Depending upon the system being repulsive or attractive, our results exhibit distinct characteristic features of vector solutions.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Central China Normal University, 430079, Wuhan, People’s Republic of China

    Shuangjie Peng

  2. Chern Institute of Mathematics, Nankai University, 300071, Tianjin, People’s Republic of China

    Zhi-qiang Wang

  3. Department of Mathematics and Statistics, Utah State University, Logan, UT, 84322, USA

    Zhi-qiang Wang

Authors
  1. Shuangjie Peng
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  2. Zhi-qiang Wang
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Correspondence to Zhi-qiang Wang.

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Communicated by P. Rabinowitz

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Peng, S., Wang, Zq. Segregated and Synchronized Vector Solutions for Nonlinear Schrödinger Systems. Arch Rational Mech Anal 208, 305–339 (2013). https://doi.org/10.1007/s00205-012-0598-0

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  • DOI: https://doi.org/10.1007/s00205-012-0598-0

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