On the wave solutions of Shrödinger type equations

Nonlinear partial differential equations (PDEs) have been the focus of many studies due to their frequent occurrence in various applications such as physics, biology, engineering, signal processing, control theory, finance, and dynamics. Recently, many methods for constructing solutions of nonlinear partial differential equations of physical interest are available in the literature. Among these, the most widely used method in recent times is the method of auxiliary equations. The nonlinear Schrödinger equation (NLSE), which plays an important role in quantum mechanics, is used to model physical phenomena. The Rogue Wave (RW) solutions of the Schrödinger equation represent a finite fraction of an infinite period soliton, which implies that the soliton and RW solutions can transform into each other under certain conditions. Solitons are waves with certain mathematical properties that can carry energy intact. RW, on the other hand, refers to giant waves that usually occur in large and unexpected ways in the oceans. These waves can often be much larger, more sudden, and more dangerous than other waves. The formation of RWs is often due to the interaction of complex water motions, wind and other factors. In the first part of this paper, we present the methods we use to solve the equations. In the second part, RW and soliton solutions of the third-order nonlinear Schrödinger equation are obtained. In the third part, different reduced forms of the fifth order nonlinear Schrödinger equation are considered and RW and soliton solutions are obtained for these equations. In the last part, solutions for systems of nonlinear Schrödinger equations are obtained.