The bifurcations of solitary waves and kink waves for variant Boussinesq equations are studied by using the bifurcation theory of planar dynamical systems. The bifurcation sets and the numbers of solitary waves and kink waves for the variant Boussinesq equations are presented. Several types explicit formulas of solitary waves solutions and kink waves solutions are obtained. In the end, several formulas of periodic wave solutions are presented.
By using the theory of planar dynamical systems to the ion acoustic plasma equations, we obtain the existence of the solutions of the smooth and non-smooth solitary waves and the uncountably infinite smooth and non-smooth periodic waves. Under the given parametric conditions, we present the sufficient conditions to guarantee the existence of the above solutions.
The travelling wave solutions (TWS) in a class of P.D.E. is studied. The travelling wave equation of this P. D.E. is a planar cubic polynomial system in three-parameter space. The study for TWS became the topological classifications of bifurcations of phase portraits defined by the planar system. By using the theory of planar dynamical systems to do qualitative analysis, all topological classifications of the cubic polynomial system can be obtained. Returning the results of the phase plane analysis to TWS, u(~) , and considering discontinuity of the right side of the equation of TWS when ~ = x -ct is varied along a phase orbit and passing through a singular curve, all conditions of existence of smooth and nonsmooth travelling waves are given.
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