It is known that the complex sine-Gordon (csG) and sinh-Gordon (cshG) equations belong to the first negative flow of the AKNS hierarchy. It is worthy to note that the csG/cshG equations are also called coupled dispersionless equations of focusing and defocusing type in the literature. Recently, by hodograph transformation, above equations can be transformed into a focusing and defocusing complex short pulse equation, which is viewed as an analogue of the nonlinear Schroedinger equation in the ultrashort regime. In this talk, I will firstly present the bilinear forms and Darboux transformation, which both lead to multi-soliton solutions of bright and dark type. Then, the integrable semi-discrete and fully-discrete complex sine/sinh-Gordon equation equations will be constructed, their solutions will be touched. If time permits, we will discuss several future topics such as the coupled cSP equation and the inverse scattering transform for the cSP equation. Speaker(s): Baofeng Feng (The University of Texas Rio Grande Valley)