Bifurcations of traveling wave solutions for two coupled variant Boussinesq equations in shallow water waves

The bifurcations of traveling wave solutions for two coupled variant Boussinesq equations introduced as a model for water waves are studied in this paper. Transition boundaries have been presented to divide the parameter space into different regions associated with qualitatively different types of solutions. The conditions for the existence of solitary wave solutions and uncountably infinite, smooth, non-smooth and periodic wave solutions are obtained. The explicit exact traveling wave solutions are presented by using an algebraic method.