Abstract
In optical solitons, the propagation is usually governed by the nonlinear Schrödinger equations. In nonlinear Schrödinger equations (NLSEs), the unstable NLSE is a universal equation of the class of nonlinear integrable systems, which describes the disturbances in time evolution of marginally stable or unstable media. The aim of this article is to study the unstable NLSE analytically. Several explicit new exact solutions such as soliton, solitary wave, elliptic function and periodic solutions of unstable NLSE are constructed by using proposed modified extended mapping method, which have important applications in applied science and engineering. The movement of obtained soliton solutions are presented graphically, which helps to understand the physical phenomenas of the unstable equation. The stability analysis of the constructed solutions and the movement role of waves are examined by employing modulation instability analysis. All solutions are stable and analytical.