摘要
We study the existence of dark solitons of the defocusing cubic nonlinear Schr¨odinger(NLS) eqaution with the spatially-periodic potential and nonlinearity. Firstly, we propose six families of upper and lower solutions of the dynamical systems arising from the stationary defocusing NLS equation. Secondly, by regarding a dark soliton as a heteroclinic orbit of the Poincar′e map, we present some constraint conditions for the periodic potential and nonlinearity to show the existence of stationary dark solitons of the defocusing NLS equation for six different cases in terms of the theory of strict lower and upper solutions and the dynamics of planar homeomorphisms. Finally, we give the explicit dark solitons of the defocusing NLS equation with the chosen periodic potential and nonlinearity.
We study the existence of dark solitons of the defocusing cubic nonlinear Schr¨odinger(NLS) eqaution with the spatially-periodic potential and nonlinearity. Firstly, we propose six families of upper and lower solutions of the dynamical systems arising from the stationary defocusing NLS equation. Secondly, by regarding a dark soliton as a heteroclinic orbit of the Poincar′e map, we present some constraint conditions for the periodic potential and nonlinearity to show the existence of stationary dark solitons of the defocusing NLS equation for six different cases in terms of the theory of strict lower and upper solutions and the dynamics of planar homeomorphisms. Finally, we give the explicit dark solitons of the defocusing NLS equation with the chosen periodic potential and nonlinearity.
基金
Supported by the National Natural Science Foundation of China under Grant No.61178091
the National Key Basic Research Program of China under Grant No.2011CB302400
the Open Project Program of State Key Laboratory of Theoretical Physics,Institute of Theoretical Physics,Chinese Academy of Sciences,China under Grant No.Y4KF211CJ1
