Periodic wave solutions and solitary wave solutions of generalized modified Boussinesq equation and evolution relationship between both solutions

In this paper, we focus on studying the exact solitary wave solutions and periodic wave solutions of the generalized modified Boussinesq equation utt−δuttxx−(a1u+a2u2+a3u3)xx=0, as well as the evolution relationship between these solutions. Detailed qualitative analysis is conducted on traveling wave solutions of this equation, and global phase portraits in various parameter conditions are proposed. Various significant results about the existence of both solutions, including three forms of solitary wave solutions and four exact bounded periodic wave solutions in different conditions are obtained. Then, we further discuss the relationship between energy of Hamiltonian system corresponding to this equation and the periodic wave solutions and solitary wave solutions. It is concluded that the essential reason of periodic wave solutions and solitary wave solutions is the different values for the energy of Hamiltonian system corresponding to this equation. In addition, the limited relations of periodic wave solutions and solitary wave solutions with the energy of Hamiltonian system are proposed, and the schematic diagram of evolution from periodic wave solutions to solitary wave solutions is drawn.