In this presentation, we review many of the theoretical models that have been proposed to describe the nonlinear dynamics of DNA in geometric structures as double-stranded. In each model, the discrete and continuum modeling equations are constructed. Traveling transformation of continuum modeling for transverse and longitudinal motions is obtained. The unified method UM is used to find exact solutions to the Peyrard and Bishop PB model with longitudinal motions. This model admits solitary, soliton, periodic, or chirped wave solutions. It is justified that the most admissible physical solution is the soliton or chirped wave solution. The stability analysis of all these solutions is performed by using topological invariance. It is found that soliton is unstable. So small disturbance in the unbounded amplitude may be due to damage to DNA membranes or bases in pairs.