The analysis of bifurcation solutions of the Camassa–Holm equation by angular singularities

This paper studies bifurcation solutions of the Camassa–Holm equation by using the local Lyapunov–Schmidt method. The Camassa–Holm equation is studied by reduction to an ODE. We find the key function that corresponds to the functional related to this equation and defined on a new domain. The bifurcation analysis of the key function is investigated by the angular singularities. We find the parametric equation of the bifurcation set (caustic) with its geometric description. Also, the bifurcation spreading of the critical points is found.