Bäcklund transformations and Painlevé analysis: Exact soliton solutions for the unstable nonlinear Schrödinger equation modeling electron beam plasma

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Abstract
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.