Resumen de Expressions and Evolution of Traveling wave Solutions in a Generalized Two-Component Rotation b-Family System
Feiting Fan, Xingwu Chen
In this paper we investigate traveling waves for a generalized two-component rotation b-family (R-b-family) system with b > 1. Based on qualitative theory and bifurcation method of dynamical systems, the traveling wave problem is converted into the dynamical analysis of the corresponding traveling wave system with 5 parameters and 2 distinct singular lines. We systematically analyze this traveling wave system with the help of the three-step method to obtain 6 bifurcation curves of its phase portraits, which enables us to draw all the phase portraits. Combining these phase portraits, we get that the R-b-family system has exactly 6 kinds of bounded traveling wave solutions and give all the explicit conditions for their existence as well as their expressions and coexistence. Finally, discussing dynamical behavior of these traveling waves, we not only provide the bifurcation wave velocities for solitary wave solutions of peak and valley type, for solitary cusp wave solutions of peak and valley type and for periodic cusp wave solutions of peak and valley type, respectively, but also find 3 other types of traveling wave evolution, namely kink wave bifurcation, solitary cusp bifurcation and periodic wave bifurcation.
