On the integrability of some nonlinear partial differential equations with Painleve analysis and soliton solutions

This thesis, consisting of six chapters, has originality in the 3rd, 4th and 5th chapters, and the following issues have been investigated in each part. In the first chapter, fundamental knowledge about the integrability and solitary waves have been given. In the second chapter, for integrability by the means of Painleve, the ARS algorithm for differential equations and WTC algorithm for partial differential equations have been investigated in details. In the third chapter, Painleve analysis has been applied to Ostrovsky equation and shown that this equation is integrable. Besides, some soliton solutions of Ostrovsky equation have been obtained by using auxiliary equation method. In the fourth chapter, Painleve analysis has been applied to a nonlinear evolution equation with respect to the situation of m and p. Finding the general formulations of first term behavior and resonance values for different values of p and m, it has been discussed that the mentioned equation whether possesses Painleve property or conditional Painleve property. In the fifth chapter, Differential transform method has been applied to KdV, mKdV equations, Hirota-Satsuma Coupled KdV and Coupled mKdV equation systems and the approximate solutions have been compared with the analytical solutions available in the literature. Hence, the accuracy of the approximate solutions has been investigated. In the last section, outcomes and some suggestions have been ranked.