数学物理学报(英文版) ›› 2015, Vol. 35 ›› Issue (1): 45-70.doi: 10.1016/S0252-9602(14)60138-7

• 论文 • 上一篇    下一篇

EXISTENCE AND STABILITY OF STANDING WAVES FOR A COUPLED NONLINEAR SCHR¨|ODINGER SYSTEM

曾小雨|张贻民|周焕松   

  1. Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China
  • 收稿日期:2014-01-09 修回日期:2014-08-25 出版日期:2015-01-20 发布日期:2015-01-20
  • 基金资助:

    This work was supported by NSFC (11471331, 11101418 and 11271360).

EXISTENCE AND STABILITY OF STANDING WAVES FOR A COUPLED NONLINEAR SCHR¨|ODINGER SYSTEM

ZENG XiaoYu,ZHANG Yi Ming, ZHOU Huang Song   

  1. Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China
  • Received:2014-01-09 Revised:2014-08-25 Online:2015-01-20 Published:2015-01-20
  • Supported by:

    This work was supported by NSFC (11471331, 11101418 and 11271360).

摘要:

We study the existence and stability of the standing waves of two coupled Schr¨odinger equations with potentials |x|bi (bi 2 R, i = 1, 2). Under suitable conditions on the growth of the nonlinear terms, we first establish the existence of standing waves of the  Schr¨odinger system by solving a L2-normalized minimization problem, then prove that the set of all minimizers of this minimization problem is stable. Finally, we obtain the least energy solutions by the Nehari method and prove that the orbit sets of these least energy solutions are unstable, which generalizes the results of [11] where b1 = b2 = 2.

关键词: nonlinear Schr¨odinger system, constrained variational problem, standing waves, orbital stubility

Abstract:

We study the existence and stability of the standing waves of two coupled Schr¨odinger equations with potentials |x|bi (bi 2 R, i = 1, 2). Under suitable conditions on the growth of the nonlinear terms, we first establish the existence of standing waves of the Schr¨odinger system by solving a L2-normalized minimization problem, then prove that the set of all minimizers of this minimization problem is stable. Finally, we obtain the least energy solutions by the Nehari method and prove that the orbit sets of these least energy solutions are unstable, which generalizes the results of [11] where b1 = b2 = 2.

Key words: nonlinear Schr¨odinger system, constrained variational problem, standing waves, orbital stubility

中图分类号: 

  • 32J20