The Solitary Solution Of A Type Of Nonlinear Evolution Equations

In this paper, we study exact solutions of a type of nonlinear evolution equations, for examples compacton solution, peakon solution, kink solution, solitary periodic solution and so on. First, using the mathematic software, we apply the Adomian decomposition method to a type of full nonlinear sine-gordon equation, and obtain compacton solution, kink solution, Mult-compacton solution, compacton-kink solution. In addition, some new types of solutions are also generated by combining different kinds of solutions. Next, using the bifurcation theory of dynamical systems, we consider the full nonlinear (n+1)-dimensional double sine-gordon equation and its approximate equation, and investigate the bifurcation and phase portraits in parameter space and the existence of the bounded traveling wave solutions, then give the dynamical behavior and some exact solutions. In the next section, we discuss the D-P equation and obtain some approximate periodic solutions by the homotopy analysis method, and show that they are consistent with the known exact solutions. At last, we research the discrete KdV equation and derive some discrete solutions of hyperbolic type.