New Exact Solutions of the System of Equations for the Ion Sound and Langmuir Waves by ETEM

Abstract

This manuscript focuses attention on new exact solutions of the system of equations for the ion sound wave under the action of the ponderomotive force due to high-frequency field and for the Langmuir wave. The extended trial equation method (ETEM), which is one of the analytical methods, has been handled for finding exact solutions of the system of equations for the ion sound wave and the Langmuir wave. By using this method, exact solutions including the rational function solution, traveling wave solution, soliton solution, Jacobi elliptic function solution, hyperbolic function solution and periodic wave solution of this system of equations have been obtained. In addition, by using Mathematica Release 9, some graphical simulations were done to see the behavior of these solutions.Keywords: the system of equations for the ion sound wave and the Langmuir wave; extended trial equation method; exact solutions; Mathematica Release 9

New structures for closed-form wave solutions for the dynamical equations model related to the ion sound and Langmuir waves

“…Numerous new approaches have been exploited to study this kind of solution for the NPDEs. Some crucial processes include the extended direct algebraic method [11], the sech-tanh technique [12], the sine-Gordon expansion scheme [13,14], the finite series Jacobi elliptic cosine function ansatz [15], the extended tanh expansion method [16], the generalized exponential rational function method [17,18], the improved tan (f(ξ)/2)-expansion scheme [19,20], the novel ¢ G G ( )-expansion approach [21,22], the extended trial equation method (ETEM) [23], the improved tan (f(ξ)/2)-expansion method (ITEM) [24], optimal homotopy and the differential transform method [25], the homogeneous balance method [26], the modified Kudryashov method [27], Riccati-Bernoulli sub-ODE method [28], the exp (− f(ξ))-expansion method [29,30], and the unified method and its generalized form [31][32][33][34][35][36][37].…”