Based on Fermat's principle and the special relativity, the transmission of high-frequency electromagnetics is unified by variational formulation on the moving interface. Applying the theoretical model, we investigate the detailed practices of transmission of high-frequency electromagnetic under relativistic conditions. The deduced results illustrate that the effective estimation of the super high-speed effect on a moving interface supports the valuable frame of reference in controlling precision. The results also show that the theoretical model has potential applications in electromagnetically controlled precision in the quantum information, ray sensor, controllable environment, etc.

Electromagnetically-controlled precision is one of novel topics in the electromagnetics. To realize the precision controlling of the electromagnetically complicated phenomenon, the systematic characteristics of medium environment needs considering. Based on the cancellation of interference caused by quantum coherence in the systematic environment of material, the electromagnetically-induced transparency (EIT) can be achieved. For this nonlinear phenomenon, due to the advancement of quantum spot and well, the controlling of the bounded sate of quantum in various dimensions of semiconductor can be operated. So the solid system presents a clear superiority of controlling EIT. High power electromagnetic field excites the dynamic characteristics in solid material, which is the result of systematic reaction between field and material. Under the excitation of electromagnetic pulse, because of quantum coherence, the dual-well semiconductor has the ability to induce the dark state of solitons. In the study of the complicated system of multiple physical fields, two aspects need investigating further. Firstly, in the induction process of electromagnetic filed and solid material, the features of high dispersion and nonlinear reaction appear increasingly. Thus, due to the environmental restriction on dispersion and nonlinear reaction, electromagnetic dissipation is a crucial point, which needs considering in the electromagnetically-controlled precision of the EIT. Secondly, compared with the formation of soliton, the coupling reaction of solitons under co-sate is much complicated. The relation among these factors is necessary to be investigated in the formulation of soliton excitation. Therefore, a dual-well semiconductor is employed as solid environment to analyze the dynamic characteristics of dark solitons in the EIT. In order to achieve the controlling of precision and regulating of the effect, the environmental features of solid materials ought to be systematically considered. Accordingly, the variational method is utilized, through which the bounded action of dissipation and nonlinear coherence is effectively studied for the dark solitons under co-sate, and under the condition of exciting dark soliton in the system of EIT. Using the density matrix and electric polarization, the spectrum of dynamic transmission deviation of EIT is calculated in the solid environment. With the assistance of relevant action principle, the bounded relation of dark solitons under co-state is practically investigated in the dissipative environment of solid system. In addition, the space-time trajectory is analyzed in the applicable region of characteristic equations of dark solution. The deduced result indicates that the systematical balance between dissipative weakening and coherent coupling supports the valuable approach to controlling the space-time evolution of dark solitons in precision. The results also show that the special effect has the potential applications in electromagnetically-controlled precision in the quantum information, ray sensor, controllable environment, etc.