Interaction between soliton and periodic solutions and the stability analysis to the Gilson–Pickering equation by bilinear method and \(\exp (-\theta (\alpha ))\)-function approach arising plasma physics

The Gilson–Pickering equation, which describes wave propagation in plasma physics and the structure of crystal lattice theory that is most frequently used. The discussed model is converted into a bilinear form utilizing the Hirota bilinear technique. Sets of case study are kink wave solutions; breather solutions; collision between soliton and periodic waves; soliton and periodic waves. The \(\exp (-\theta (\alpha ))\) $exp(-\theta (\alpha ))$ -function approach is employed to discover travelling wave solutions including five classes of solutions. In addition, it has been confirmed that the established findings are stable, and it has been helpful to validate the computations. The effect of the free variables on the behavior of obtained solutions to some plotted graphs for the exact cases is also explored depending upon the nature of nonlinearities. The exact solutions are utilized to demonstrate the physical natures of 3D, density, and 2D graphs.