In this paper, we study the nonlinear dynamics of DNA, for longitudinal and transverse motions, in the framework of the microscopic model of Peyrard and Bishop. With the help of the fractional Riccati expansion method, the coupled fractional nonlinear partial differential equations for the dynamics of DNA model, which consists of two long elastic homogeneous strands connected with each other by an elastic membrane, have been solved. It is shown that the proposed method is effective, direct and can be used for many other fractional nonlinear evolution equations.

Riemann-Liouville calculus, nonlinear fractional differential equations, fractional Riccati expansion method.