Bifurcation analysis, chaotic phenomena, variational principle, Hamiltonian, solitary and periodic wave solutions of the time-fractional Benjamin Ono equation

The major goal of this work is to seek the exact wave solutions, and give the bifurcation and chaotic analysis of the time-fractional Benjamin Ono equation in the conformable sense for shallow-water waves. First, the traveling wave transformation is used for the considered model and the variational principle (VP) is extracted by the semi-inverse method (SIM). Based on the VP, the Hamiltonian is derived. Applying Galilean transformation, the corresponding planar dynamical system is obtained, and the bifurcation and chaotic analysis are presented in detail. In the end, the variational method, which is based on the VP and Ritz method, and the Hamiltonian-based method are employed to develop the abundant wave solutions, including the bright solitary, dark solitary and periodic wave solutions. The graphic depictions of the obtained diverse wave solutions are presented with the aid of Maple. In the meantime, the impact of the fractional order on the structure of the extracted waveforms are elaborated. As far as we all know, the findings of this research have not been reported and can enable us to gain a deeper understanding of the nonlinear dynamics of the considered equation.