Singular and non-singular complexiton solutions of partial differential equations

In recent years, various methods have been developed to obtain wave solutions of nonlinear partial differential equations and examine structures of obtained solutions. One of the newest members of solution types derived by these methods is complexiton wave. Complexiton wave can easily be distinguished among other waves by their characteristic shapes and structures. In this thesis, firstly, some fundamental concepts which are necessary to understand simplified Hirota method, are given. These fundamental concepts are exact solutions, types of wave solutions, Hirota bilinear derivatives, Hirota bilinear forms, generalized bilinear derivatives and Bell polynomials. In following section, by simplified Hirota method, singular and non-singular complexiton solutions, soliton solutions and interaction solutions of (1+1)-dimensional sixth-order Ramani equation, (2+1)-dimensional fourth-order extended Calogero-Bogoyavlenskii-Schiff equation and (2+1)-dimensional fourth-order Hirota-Satsuma-Ito equation are derived. During the application process of simplified Hirota method, the case N=3 has also been studied which is and additional case compared with existing studies in literature. The more dimension is, the more number of types of derived solutions are. Derived solutions are illustrated by giving some specific values to parameters. These presented illustrations of solutions help us in understanding propagations and characteristic structures of derived wave solutions.