We have deﬁned slowness (or reciprocal velocity) corresponding to velocity (v) as cc/v, where c is the speed of light and v is the corresponding velocity. Velocity and slowness are images of each other. Reciprocal symmetric law of addition of velocities fulﬁls the requirement that the sum (or the difference) of velocities remains unchanged if velocities are replaced by the corresponding slownesses. We have postulated reciprocal symmetry, which states that every valid statement has an equally valid (reciprocal) image statement. The postulate has allowed us to derive.
We investigated on the nonlinear behavior of dust ion acoustic (DIA) double Layers (DL's) in a five component(inertial positive and negative ions, Maxwell's electrons, and arbitrary charged positive and negative stationary dust) dusty plasma system (FCDP) with adiabatic change. The nonlinearity and dynamics of planar DL's have also been observed. To observe the impacts of dust polarity on the DL's the dust particles have been considered arbitrarily charged. The nonlinearity of these DL's waves have been observed under adiabatic treatment. In the beginning, a detail mathematical derivation and numerical analysis has done on the linear properties as well as the dispersion relation for the dusty plasma system. To observe nonlinearities through the entire regime reductive perturbation method has been employed, and observed that Gardner approach is able to analyze DL'S solitons for a large region..