Diverse wave structures to the modified Benjamin–Bona–Mahony equation in the optical illusions field

In this work, the modified Benjamin–Bona–Mahony equation (MBBME), which has an important role in the optical illusions field, is investigated by three analytical methods, namely, the Bernoulli sub-equation function method, variational method and the Hamiltonian method. Abundant wave solutions expressed in the form of the exponential function, hyperbolic function and trigonometric function are obtained. Finally, the dynamic behaviors of the solutions are illustrated via the 3D plots, 2D contours and 2D curves with the help of Mathematica software, and their corresponding physical interpretations are also given. The ideas in this work are expected to provide some new enlightenment for the study of the exact solutions of the PDEs in optical physics.