Thirty traveling wave solutions to the systems of ion sound and Langmuir waves

Abstract

Nine kinds of general solutions of certain popular first-order ordinary differential equation are obtained by direct calculations. Compared with complete discrimination system for polynomials, our classifications of solutions straightly depend on the coefficients of the ordinary differential equation and hence are easier to be employed. According to the nine kinds of solutions, we establish thirty traveling wave solutions to the coupled systems of ion sound and Langmuir waves, including rational function solution, exponential function solution, trigonometric function solutions, hyperbolic function solutions and Jacobi elliptic function solutions. To the best of our knowledge, many of them are new. Solutions discussed in two recent articles are showed to be equivalent to some special cases of our thirty traveling wave solutions. Our solutions include the kink and anti-kink soliton solutions, dark and bright soliton solutions as well as periodic soliton solutions and solitary wave solutions. We believe that these solutions will be of great use to researchers concerning with nonlinear physical phenomena. In addition, the famous (1+1)-dimensional dispersive long wave equations is taken to illustrate the applicability of these thirty solutions.