New traveling wave solutions for Gilson–Pickering equation in plasma via bifurcation analysis and direct method

Abstract

This work aims to study the wave propagation for the Gilson–Pickering equation appearing in plasma. By using an appropriate wave transformation, this equation is converted into a singular dynamical system that can then be converted into a regular dynamical system through a suitable point transformation for the independent variable. We demonstrate that both systems have the same first integral. In light of the topological equivalence between the phase orbits of both systems, we give a brief description of each system's phase plane. On the basis of the bifurcation analysis, we introduce two theorems summarizing the conditions on the parameters giving rise to periodic and solitary solutions, besides the conditions of unbounded wave solutions. Consequently, we construct some parametric wave solutions which are periodic, solitary, and unbounded wave solutions. We perform a numerical study to clarify these solutions graphically and to confirm the efficacy of the analytical study. We also analyze the influence of the parameters on some of the obtained solutions numerically.