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Dynamics of some coupled nonlinear Schrödinger systems in R 3
Author(s) -
Xu Guixiang
Publication year - 2013
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.3015
Subject(s) - mathematics , symmetrization , nonlinear system , rigidity (electromagnetism) , compact space , schrödinger's cat , ground state , scattering , breather , mathematical analysis , mathematical physics , physics , quantum mechanics
In this paper, we show the scattering and blow up result of the solution for some coupled nonlinear Schrödinger system iE ⃗t + Δ E ⃗ + | E ⃗| 2E ⃗ = 0 with static energy less than that of the ground state in H : = H 1R 3× H 1R 3, whereE ⃗ : R × R 3 → C 2 . We first use the Nehari manifold approach and the Schwarz symmetrization technique to construct the ground state and obtain the threshold energy of scattering solution, then use Payne–Sattinger's potential well argument and Kenig–Merle's compactness‐rigidity argument to show the aforementioned dichotomy result. As we know, it is the first attempt to show the scattering result for the coupled nonlinear Schrödinger system. Copyright © 2013 John Wiley & Sons, Ltd.