Abundant dynamical behaviors of bounded traveling wave solutions to generalized $\theta$-equation

We study existence and dynamics of bounded traveling wave solutions to generalized $\theta$-equation from the perspective of dynamical systems. We obtain bifurcation of traveling wave solutions for the equation, prove the existence of several types of bounded traveling wave solutions, including solitary wave solutions, periodic wave solutions, peakons, periodic cusp waves, compactons and kink-like (antikink-like) waves, and derive some of their exact expressions. Most importantly, we confirm abundant dynamical behaviors of the traveling wave solutions to the equation, which are summarized as follows: (1) We confirm that three types of orbits give rise to solitary wave solutions, that is, the homoclinic orbit passing the singular point, the composed homoclinic orbit which is comprised of three heteroclinic orbits of the associated system, and the composed homoclinic orbit which is comprised of two heteroclinic orbits and tangent to the singular line at the singular point of the associated system. (2) We confirm that four types of orbits correspond to periodic wave solutions, that is, the periodic orbit surrounding a center, the periodic orbit surrounding two connected homoclinic orbits, the composed periodic orbit which is comprised of two heteroclinic orbits of the associated system, and the homoclinic orbit of the associated system which is tangent to the singular line at the singular point of the associated system. (3) We confirm that two types of orbits correspond to periodic cusp waves, that is, the semiellipse orbit surrounding a center, and the semiellipse-like orbit surrounding two connected homoclinic orbits. (4) We confirm that two families of periodic orbits, which surround two connected homoclinic orbits and are comprised of two heteroclinic orbits of associated system, respectively, and the composed homoclinic orbit, which is comprised of two heteroclinic orbits and tangent to the singular line at the singular point of associated system, have envelope.