Strongly nonlinear stationary waves
In this chapter we shall start to investigate wave-plasma interaction processes where the relative density changes produced by the ponderomotive force of the high-frequency field cannot longer be considered as small. Thus, attempts to describe such processes have to start from the fully nonlinear system (3.14)–(3.16). Since there exist no general analytical methods of solving such systems of coupled partial differential equations, progress is usually made by looking for certain special solutions. One method which is often used and gives valuable insight, is that of looking for nonlinear stationary waves.
In this paper, we study a system governed by impulsive semilinear nonautonomous differential equations. We present the β –Ulam stability, β –Hyers–Ulam stability and β –Hyers–Ulam–Rassias stability for the said system on a compact interval and then extended it to an unbounded interval. We use Gronwall type inequality and evolution family as a basic tool for our results. We present an example to demonstrate the application of the main result.