Soliton solution and conservation laws of the Zakharov equation in plasmas with power law nonlinearity

There are several nonlinear evolution equations (NLEEs) that appears in various areas of applied mathematics and theoretical physics [1–13]. These NLEEs are a key to the understanding of various physical phenomena that governs the world today. Some of these commonly studied NLEEs are the nonlinear Schrödinger’s equation (NLSE), Korteweg– de Vries (KdV) equation, sine–Gordon equation (SGE), just to name a few. Some of these equations appear in the real domain while others appear in the complex domain. The NLSE appears in nonlinear optics, while KdV equation is studied in fluid dynamics and the SGE is seen in theoretical physics. There are various vector valued coupled equations that describe many physical phenomena. One such equation is the Zakharov equation (ZE) that is studied in the context of plasma physics [1]. NLEEs are studied by several authors and there are several interesting aspects and issues that have been addressed in the past. A systematical discussion on the secant type function was conducted earlier [6]. Additionally, the traveling wave solutions to nonlinear evolution equations by the transformed rational function method was displayed in 2009 [8]. Moreover, a hierarchy of conservation can be easily generated from conserved densities of Hamiltonian structures behind Lax pairs. This aspect has been addressed on