Construction of modulated amplitude waves via averaging in collisionally   inhomogeneous Bose-Einstein condensates
  We apply the averaging method to analyze spatio-temportal structures in nonlinear Schr\"odinger equations and thereby study the dynamics of quasi-one-dimensional collisionally inhomogeneous Bose-Einstein condensates with the scattering length varying periodically in spatial and crossing zero. Infinitely many (positive measure set) modulated amplitude waves (periodic and quasi-periodic), which are instable, can be proved to exist by adjusting the intergration constant c on some open interval. Finally, some numerical simulations support our results. 
Qihuai Liu and Dingbian Qian
nlin.PS