In this paper, the generalized Zakharov equations, which describe interactions between high- and low-frequency waves in plasma physics are studied from the perspective of Lie symmetry analysis and conservation laws. Based on some subalgebras of symmetries, several reductions and numerous new exact solutions are obtained. All of these solutions represent modified traveling waves. The obtained solutions include expressions involving Airy functions, Bessel functions, Whittaker functions, and generalized hypergeometric functions. Previously unknown conservation laws are constructed for the generalized Zakharov equations using the direct method. Profiles are presented for some of these new solutions.

The pull-in instability is the inherent property of a micro-electromechanical system (MEMS) when the voltage is larger than its threshold value. Recently, a fractal MEMS system was proposed to overcome the pull-in instability with great success, and it has opened a total new path for the so-called pull-in stability. This paper suggests a pull-in plateau, a novel concept for qualifying the pull-in stability. The plateau’s basic properties are elucidated, and the effect of the fractal dimensions on the plateau width is elucidated, and the paper concludes that there exists a critical condition for an ever pull-in stability when both the acceleration and the speed of the system equal zero.

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