This paper addresses controlled g-frames which are extensions of g-frames and controlled frames. Firstly, we discuss the characterization of controlled g-frames, obtain some equivalent conditions of them. Then, we introduce the concept of controlled dual g-frames and controlled dual g-frames operator, get some properties of them. Finally, we characterize all controlled dual g-frames of a controlled generalized frame.2010 MS Classification: 42C15, 42A38

The theory of regular wavelet/Gabor frames has been extensively investigated, including the sufficient and necessary conditions for regular wavelet/Gabor systems to be frames and the perturbation theorem. This paper addresses the irregular wavelet/Gabor systems in Sobolev space Hs(R). The sufficient and necessary conditions for irregular wavelet/Gabor systems to be frames are presented. As corollaries, under certain restrictions on the support, we also give the characterization of irregular wavelet/Gabor systems to be frames. Finally, we discuss the perturbation theorem of irregular wavelet/Gabor frames.

In this paper, we investigate the coupled modified nonlinear Schödinger equation. Through the traditional Darboux transformation, we construct the first-order breather solution which can exhibit Akhmediev breather and general breather. To obtain the higher-order localized wave solution, N-fold generalized Darboux transformation is given. Under the condition that the characteristic equation admits a double-root, we present the expression of first-order interactional solution. Then we graphically analyze the dynamics of breather and rogue wave. Due to the simultaneous existence of nonlinear and derivative terms in the equation, there presents different profile in two components for the breathers.