Application of the ITEM for the system of equations for the ion sound and Langmuir waves

Abstract

In this paper, by introducing new approach, the improved \(\tan (\phi (\xi )/2)\)-expansion method is further extended into the system of equations for the ion sound and Langmuir waves in plasma. As a result, the hyperbolic function solution, the trigonometric function solution, the exponential solution and the rational solution with free parameters are obtained. When the parameters are taken as special values the solitary wave solutions and the periodic wave solutions are also derived from the traveling wave solutions. Moreover, it is observed that the suggested technique is compatible of such problems. We obtained the further solutions comparing with other methods. The exact traveling wave solutions including solitons, kink, periodic and rational solutions have been found. The results of applying this procedure (Figs. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13) to the studied cases show the high efficiency of the new technique. It is shown that this method is a powerful mathematical tool for solving problems in the electron gas of the plasma.