Exact explicit solitary wave and periodic wave solutions and their dynamical behaviors for the Schamel-Korteweg-de Vries equation

The Schamel-Korteweg-de Vries equation is investigated by the approach of dynamics. The existences of solitary wave including ω-shape solitary wave and periodic wave are proved via investigating the dynamical behaviors with phase space analyses. The sufficient conditions to guarantee the existences of the above solutions in different regions of the parametric space are given. All possible exact explicit parametric representations of the waves are also presented. Along with the details of the analyses, the analytical results are numerically simulated lastly.