Computational study of thin films made from the ferroelectric materials with Paul Painlevé approach and expansion and variational methods

In this paper, the thin-film ferroelectric material equation which enables a propagation of solitary polarization in thin-film ferroelectric materials, and it also can be described using the nonlinear evolution equations. Ferroelectrics are dielectric materials explain wave propagation nonlinear behaviors. Thin films made from the ferroelectric materials are used in various modern electronics devices. The Paul-Painlevé approach is adopted for the first time to solve these nonlinear thin-film equation analytically. To investigate the characterizations of new waves, the solitary wave dynamics of the thin-film ferroelectric material equation are obtained using the standard \(\tan (\phi /2)\) -expansion technique and generalized G-expansion method. The bright and periodic solutions are obtained by semi-inverse variational principle scheme. Many alternative responses are achieved utilizing various formulaes; each of these solutions is shown by a distinct graph. The validity of such methods and solutions are demonstrated by assessing how well the relevant techniques and solutions match up. Three novel analytical and numerical techniques provide new, dependable approaches for determining and estimating responses. The effect of the free variables on the behavior of reached solutions to a few of graphs on the exact solutions is also explored depending upon the nature of nonlinearities. The simulations, which are exhibited in both two-dimensional (2D) and three-dimensional (3D), depict the behavior of a solitary solution in both the natural and digital worlds. These findings demonstrate that this strategy is the most effective way to solve nonlinear mathematical physics problems.