Stability and instability of solutions of nonlinear evolution equations

In the first chapter of this thesis nonlinear evolution equations, their mathematical behaviorand stability, which constitute the main subject of this thesis, are considered. In the second chapter, shallow water waves and solitary waves are investigated, and historical development of these models and related literature are given. In the third chapter, basic definitions, theorems and inequalities that will be used throughout the thesis are given. Moreover, Kato's theorem, orbital stability and wave breaking, which is a blow-up phenomena for shallow water wave models, are given. The fourth chapter is the original part of this thesis and consists of three subsections. In the first part, local well-posedness of an integrable generalized shallow water wave equation with strong dispersive term is studied. Moreover, existence and stability of solitary wave solutions of this equation is studied. In the second part, wave breaking and stability of solitary wave solutions are studied for the generalized Dullin-Gottwald-Holm equation. In the third part, local existence, existence and stability of solitary waves are studied for solutions of a generalized KdV-BBM type equation. In the fifth chapter, the obtained results are summarized and suggestions are presented.