The exponential decay of solutions and traveling wave solutions for a modified Camassa–Holm equation with cubic nonlinearity

The present paper is devoted to the study of persistence properties, infinite propagation, and the traveling wave solutions for a modified Camassa–Holm equation with cubic nonlinearity. We first show that persistence properties of the solution to the equation provided the initial datum is exponential decay and the initial potential satisfies a certain sign condition. Next, we get the infinite propagation if the initial datum satisfies certain compact conditions, while the solution to Eq. (1.1) instantly loses compactly supported, the solution has exponential decay as |x| goes to infinity. Finally, we prove Eq. (1.1) has a family traveling wave solutions.