In this paper the Bäcklund transformations technique and Painlevé analysis are used to generate classes of exact soliton solutions for some nonlinear evolution equations. For the (1+1)-dimensional problem, the unstable system of plasma equations where an electron beam is injected under a high-frequency electric field is reduced to the unstable nonlinear Schrödinger (UNLS) equation. Using the Darboux–Bargmann technique, we obtain the Bäcklund transformations for UNLS equation solvable by the inverse scattering method of Zakharov–Shabat/Ablowitz–Kaup–Newell–Segur (ZS/AKNS) and the ZS/AKNS wave functions corresponding to the soliton solutions of this equation.
A small-amplitude slow ion acoustic monotonic double layer in an unmagnetized plasma consisting of relativistic drifting cold electrons and nonrelativistic drifting thermal ions is investigated. By using the reductive perturbation method, Schamel–Korteweg–de Vries (SKdV) and Schamel equations are derived. We used the linearization transformation to obtained the solutions of the SKdV and Schamel equations. The method is based upon a linearization principle that can be applied on nonlinearities which have a polynomial form. We illustrate the potential of the method by finding solutions of the SKdV and Schamel equations. Furthermore, we show that the monotonic double-layer solution is a nonlinear extension of the slow ion acoustic solitary hole having a negative trapping parameter in a semi relativistic plasma.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.