When the integral constant is zero,the existence of smooth solitary wave solutions,uncountably infinite,many smooth periodic wave solutions,and kink and anti-kink wave solutions are proved.

Under the condition g≠0,where g is the integral constant,the existence of smooth solitary wave solutions and uncountable infinite smooth periodic wave solutions are proved.

The existence of smooth solitary wave solutions and periodic wave solutions are proved.

The existence of smooth solitary wave solutions,kink and anti-kink wave solutions and periodic wave solutions has then proved.

Solitary wave solutions and periodic wave solutions for Zhiber-Shabat equation;

The periodic wave solutions of the integrable Davey-Stewartson equations

Then,the bifurcation phase portraits of the traveling wave system are drawn,and the special orbits corresponding to the explicit periodic wave solutions are detected by numerical simulation.

In particular,Kink Compacton(solutions,) solitary wave solution,periodic wave solution,solitary pattern solution and Compacton solutions with one and two peaks are developed.

One type of five-order fully nonlinear dispersive equations such as u~(m-1)u_t±a(u~n)_x+(b(u~k)_(xxx)+)c(u~q)_(xxxxx)=0(nkq≠0) are studied and compacton solutions, periodic wave solutions and solitary solutions are obtained by using ansatzs method.

Several exact analytical solutions are obtained for the combined KdV mKdV equation u t+2αuu x+3βu 2u x+γu xxx =0 by using a new function transformation, which contain bell solitary wave solution, kink solitary wave solution, new combining bell and kink solitary wave solution and periodic wave solutions.

By using the theory of bifurcations of planar dynamical systems to the coupled Jaulent-Miodek equations,the existence of smooth solitary travelling wave solutions and uncountably infinite many smooth periodic travelling wave solutions is studied and the bifurcation parametric sets are shown.