On existence of dark solitons in cubic-quintic nonlinear Schr\"{o}dinger   equation with a periodic potential
  A proof of existence of stationary dark soliton solutions of the cubic-quintic nonlinear Schr\"{o}dinger equation with a periodic potential is given. It is based on the interpretation of the dark soliton as a heteroclinic on the Poincar\'e map. 
Pedro J. Torres, Vladimir V. Konotop