This paper unearths solitary wave in a simplified electrical model practical of the nonlinear left-handed transmission electrical line having series capacitance and shunt inductance. The detailed study of the practical model digs out that by taken a good parameter of the lattice, the obtained equation from the evolution of the wave envelope can be reduced to a one-dimensional fractional nonlinear Schrödinger equation, which gives the pipe of the dark solitons propagation. By simulating transient circuits, the evolution of the soliton and modulational instability gain were demonstrated in the areas of perturbation frequency, pulsation, nonlinearity and signal strength with a continuous wave type input signal. It has also been shown that increased perturbation, nonlinearity and signal strength in simplified metamaterial models of short nonlinear transmission lines lead to the formation of Schrödinger dark solitons and increased instability gain in a continuous pulse structure. The establishment of Schrödinger solitons and the progressive distance of a point, i.e. the fractional parameter α = 1, could reduce the width of the signal bandwidth and the breaking of the symmetry of the instability gain in the simplified electrical model practical of the nonlinear left-handed transmission electrical line, so that the evolution of the system cannot be controlled due to the memory effect, which can find important practical applications in communication systems.