Singular wave solutions associated with a singular parabola

In this paper, the bifurcations and exact singular wave solutions of an integrable model equation have been systematically investigated using the dynamical systems approach. By analyzing the phase portraits of the associated traveling wave system, it is revealed that the presence of a singular parabola in the phase plane plays a crucial role in generating various types of singular waves. Specifically, when the phase orbits are tangent to or intersect the singular parabola, a variety of non-smooth traveling wave solutions emerge, including singular periodic wave solutions, two-peaked solitary wave solutions, and multi-peaked periodic wave solutions. Exact parametric representations of these solutions have been derived, and numerical simulations are provided to illustrate their dynamical behaviors. Some of these solutions are new, while others extend previous results in the literature.