On Darboux-B\"acklund Transformations for the Q-Deformed Korteweg-de   Vries Hierarchy
  We study Darboux-B\"acklund transformations (DBTs) for the $q$-deformed Korteweg-de Vries hierarchy by using the $q$-deformed pseudodifferential operators. We identify the elementary DBTs which are triggered by the gauge operators constructed from the (adjoint) wave functions of the hierarchy. Iterating these elementary DBTs we obtain not only $q$-deformed Wronskian-type but also binary-type representations of the tau-function to the hierarchy. 
Ming-Hsien Tu, Jiin-Chang Shaw and Chin-Rong Lee