This thesis is dedicated to the analysis of certain nonlinear partial differential equations of different orders. We begin with the method of characteristics. We show how the method is used for solving quasilinear partial differential equations. We also state the existence theorem which tells us under which conditions there exists a unique local solution. Using the example of Euler equation, the problem of non-uniqueness of the solution, constructed by means of the characteristics, is presented. The equation is written in the form of a conservation law and the solution is constructed by introducing the shock waves. We continue by focusing on the equations with dispersion and we study its impact on the shape of shock waves. We present the solution of Burgers equation using the Hopf-Cole transformation. The thesis is concluded by the presentation of a simple singlesoliton solution of the Korteweg-deVries equation.