Optical and other soliton solutions, Lie point symmetries, conservation laws and modulation instability analysis of some nonlinear partial differential equations

This thesis deals with the investigation of Lie point symmetries, optical and other solitons, conservation laws (Cls) and modulation instability analysis (MI) of some of nonlinear Schrodinger equations (NLSEs). We present a detailed analysis for the existence of dark, bright, dark-bright or combined and singular soliton solutions of some NLSEs. Furthermore, the soliton solutions and Cls of some nonlinear partial differential equations (NLPDEs) are investigated. For some of the NLPDEs, we present a detailed analysis for the existence of topological, non-topological, hyperbolic, function, singular and periodic soliton solutions. Seven different integration schemes are used to study the nonlinear models, namely; the complex envelope ansatz, sine-Gordon expansion method (SGEM), the Riccati Bernoulli sub-ODE (RBSO), modified F-expansion, the generalized tanh, generalized projective Riccati equation, Jaccobi elliptic function ansatz and the undetermined coefficient methods. The problem on nonlinear self-adjointness of several nonlinear models has not been studied in previous time. In this thesis, we solve this problem for several nonlinear models and find an explicit form of the differential substitution satisfying the nonlinear self-adjoint condition. Then we use this fact to construct a set of conserved vectors using the classical symmetries admitted by the models and by invoking the general Cls theorem due to Ibragimov. Moreover, we investigate MI of some NLSEs by using the concept of linear stability analysis. Some figures are plotted to show the physical interpretations of the obtained results.