This paper considers deriving new exact solutions of a nonlinear complex generalized Zakharov dynamical system for two different definitions of derivative operators called conformable and $M-$ truncated. The system models the spread of the Langmuir waves in ionized plasma. The extended rational $sine-cosine$ and $sinh-cosh$ methods are used to solve the considered system. The paper also includes a comparison between the solutions of the models containing separately conformable and $M-$ truncated derivatives. The solutions are compared in the $2D$ and $3D$ graphics. All computations and representations of the solutions are fulfilled with the help of Mathematica 12. The methods are efficient and easily computable, so they can be applied to get exact solutions of non-linear PDEs (or PDE systems) with the different types of derivatives.