A New Fractal Modified Benjamin–Bona–Mahony Equation: Its Generalized Variational Principle And Abundant Exact Solutions

In this paper, we derive a new fractal modified Benjamin–Bona–Mahony equation (MBBME) that can model the long wave in the fractal dispersive media of the optical illusion field based on He’s fractal derivative. First, we apply the semi-inverse method (SIM) to develop its fractal generalized variational principle with the aid of the fractal two-scale transforms. The obtained fractal generalized variational principle reveals the conservation laws via the energy form in the fractal space. Second, Wang’s Bäcklund transformation-based method, which combines the Bäcklund transformation and the symbolic computation with the ansatz function schemes, is used to study the abundant exact solutions. Some new solutions in the form of the rational function-type, double-exp function-type, Sin-Cos function-type and the Sinh-Cosh function-type are successfully constructed. The impact of the fractal orders on the behaviors of the different solutions is elaborated in detail via the 3D plots, 2D contours and 2D curves, where we can find that: (1) When the fractal order 𠜀 > η, the direction of wave propagation tends to be more vertical to the x-axis, on the other hand, it tends to be more parallel to the x-axis when 𠜀 < η; (2) The fractal order cannot impact the peak amplitude of the waveform; (3) For the periodic waveform, the fractal orders can affect its period, that is, the period becomes smaller when the fractal order 𠜀,η < 1. The obtained results show that the proposed methods are effective and powerful, and can construct the abundant exact solutions, which are expected to give some new enlightenment to study the variational theory and traveling wave solutions of the fractal partial differential equations.