Variational principle, Hamiltonian, bifurcation analysis, chaotic behaviors and the diverse solitary wave solutions of the simplified modified Camassa–Holm equation

The aim of this study is to give a deep investigation into the dynamics of the simplified modified Camassa–Holm equation (CHe) for shallow water waves. Taking advantage of the semi-inverse method, we develop the variational principle, based on which the Hamiltonian of the system is extracted. By means of the Galilean transformation, the governing equation is transformed into a planar dynamical system. Then, the bifurcation analysis is presented via employing the theory of the planar dynamical system. Correspondingly, the quasi-periodic and chaotic behaviors of the system are also discussed by introducing two different kinds of perturbed terms. Finally, the variational method is based on the variational principle and Ritz method, and the Kudryashov method is used to construct the diverse solitary wave solutions, which include the bright solitary, dark solitary, kink solitary and the bright–dark solitary wave solutions. The graphic depictions of the obtained diverse solitary wave solutions are presented to elucidate the physical properties. The findings of this research enable us to gain a deeper understanding of the nonlinear dynamic characteristics of the considered equation.