Yan, ZY (reprint author), Chinese Acad Sci, Inst Syst Sci, AMSS, Key Lab Math Mech, Beijing 100190, Peoples R China.
We study the existence of dark solitons of the defocusing cubic nonlinear Schrodinger (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 Poincare 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.
National Natural Science Foundation of China 
; National Key Basic Research Program of China [2011CB302400]
; State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, China [Y4KF211CJ1]
Yan, ZY,Yan, FC. Dark Solitons for the Defocusing Cubic Nonlinear Schrodinger Equation with the Spatially Periodic Potential and Non linearity[J]. COMMUNICATIONS IN THEORETICAL PHYSICS,2015,64(3):309-319.