Global Bifurcation of Solitary Waves in a Dual Equation of the Kaup–Boussinesq System

In this paper, we study the traveling wave solutions of a dual Kaup–Boussinesq system, particularly the global bifurcation and dynamical behavior of peakon type traveling wave solutions. Based on the theory of singular traveling wave systems, this paper focuses on the exact forms of solitary wave solutions. Using bifurcation theory, we analyze the global properties of the solutions and their dynamical behavior near singular lines. It is discovered that solitary wave solutions undergo a “break” as they approach the singular line, transitioning from smooth solitons to peakons and then to cuspons. Through theoretical analysis and numerical simulations, we reveal these phenomena, providing new insights into the global behavior of water waves.