Exact Solutions of the Gardner Equation and their Applications to the Different Physical Plasmas
Traveling wave solution of the Gardner equation is studied analytically by using the two dependent (G (')/G,1/G)-expansion and (1/G ('))-expansion methods and direct integration. The exact solutions of the Gardner equations are obtained. Our analytic solutions are applied to the unmagnetized four-component and dusty plasma systems consisting of hot protons and electrons to investigate dynamical features of the solitons and shock waves produced in these systems. A wide variety of parameters of the plasma is used, and the basic features of the Gardner solitons that are beyond the existing study in literature are found. It is observed that the analytic solutions from (G (')/G,1/G)-expansion and (1/G ('))-expansion methods only produce shock waves but the solitary waves are found from the analytic solutions derived from the direct integration. It is also noted that the superhot electrons and relative mass density of the electrons significantly effect the soliton's amplitude, width, and position. We have also numerically proved that the combination of every value of nomalized density mu (1) or temperature ratio sigma (1) with the other sets of plasma parameters creates a region where the solutions have similar physical properties. The time-dependent behavior of the soliton is also studied, and a periodic motion of soliton along the phase variable eta is found during the evolution. The investigations and the limits presented in this study may be helpful for studying and understanding the nonlinear properties of the solitary and shock waves seen in various physical and astrophysical plasma systems.