Analysis of Traveling and Standing Waves in the DNA Model by Peyrard-Bishop-Dauxois

Abstract

The model by Peyrard - Bishop - Dauxois (the PBD model), which describes the DNA molecule nonlinear dynamics, is considered. This model represents two chains of rigid disks connected by nonlinear springs. An interaction between opposite disks of different chains is modeled by the Morse potential. Solutions of equations of motion are obtained analytically in two approximations of the small parameter method for two limit cases. The first one is the long-wavelength limit of traveling waves, when frequencies of vibrations are small. Dispersion relations are obtained also for the long-wavelength limit by the small parameter method. The second case is a limit of high frequency standing waves in the form of out-of-phase vibration modes. Two such out-of-phase modes are obtained; it is selected one of them, which has the larger frequency. In both cases systems of nonlinear ODEs are obtained. Nonlinear terms are presented by the Tailor series expansion, where terms up to third degree by displacement are saved. The analytical solutions are compared with checking numerical simulation obtained by the Runge - Kutta method of the 4-th order. The comparison shows a good exactness of these approximate analytical solutions. Stability of the standing localized modes is analyzed by the numerical-analytical approach, which is connected with the Lyapunov definition of stability.