Analytical solutions of some partial differential equations

This thesis aimed to examine the exact solutions of the Perturbed Chen-Lee-Liu model which expresses the soliton propagation dynamics in plasma optical fiber and the Time-dependent Paraxial Equation which depicts physical phenomena such as refraction, diffraction and reflection in electromagnetic fields. In the first chapter, about nonlinear partial differential equations, soliton, solitary (singular) waves and optical fiber are given comprehensive and general information. In the second part, there are definitions necessary for our thesis. In the third chapter, the basic properties of the jacobi elliptic functions that will be mentioned in our thesis and the general lines of the jacobi elliptic function expansion method, unified auxiliariy equation method and the newly developed unified auxiliary equation method are explained. In the fourth chapter, Jacobi elliptic function expansion method is applied to the Perturbed Chen-Lee-Liu model which expresses the soliton propagation dynamics in plasma optical fiber. Also, the unified auxiliary equation method and the newly modified version of this method are applied to the Time-dependent Paraxial Equation which depicts physical phenomena such as refraction, diffraction and reflection in electromagnetic fields. In this section, the results of the newly developed unified auxiliary equation method are compared with the unified auxiliary equation method. In the next section, conclusions and recommendations are mentioned.