A coupled nonlinear Schrödinger type equation and its explicit solutions

A new coupled nonlinear Schrödinger type equation is proposed and is proved to be completely integrable. Based on the resulting Lax pair, a Darboux transformation for the coupled nonlinear Schrödinger type equation is derived with the aid of a gauge transformation between spectral problems. As an application, some explicit solutions of the coupled nonlinear Schrödinger type equation are obtained, including periodic and rational solutions.