Optical, soliton solutions and stability analysis of some nonlinear partial differential equations

This thesis is set as six chapters. In first chapter, the topic is introduced and technical information about its historical development are told in detail. In second chapter, some fundamental definition, theorem and mathod analysis, which will be used next chapters are given. In the third chapter, the full moving wave soliton solutions of the coupled nonlinear equations describing the pulse propagation in nonlinear optical fibers are examined with the new sub-equation method, and as a result, trigonometric and singular function solutions of bright, W-shaped bright, kink solitons are obtained and behaviors of these results is illustrated by figures. Moreover, behavior of Modulation instability gain spectra under the influence of ellipticity angle and coefficient of coupled equation is investigated. In the fourth chapter, some solutions of the periodic general Degasperis-Procesi equation are investigated. After reducing the equation to nonlinear ordinary differential equation with Lie symmetry approach, first integrals and exact solutions were found with Nucci reduction method, which is rare in the literature. On the other hand, some solutions of reduced ordinary differential equations are derived by sine expansion method for special cases. In the fifth chapter, new solutions of the periodic Hunter-Saxton equation are investigated by reducing with Lie symmetry and Nucci method. Here it is concluded that we arrive at various exact solutions for both reductions of the model presented under the two advantageous analytical solution method. In order to better understand the physical properties of the solutions obtained, some graphs are given. In the last section, the obtained results are evaluated.