Higher order investigation on modulated waves in the Peyrard–Bishop–Dauxois DNA model

Despite the widespread use of transcendental functions in the modeling of the dynamics of DNA, most research efforts are limited in their analytical studies of this enthralling system to cubic order polynomial approximations of the corresponding equations of motion. In this paper, we present an investigation of waves in the Peyrard–Bishop–Dauxois model of DNA; while extending the polynomial approximation of the Morse potential up to the sixth order. We show that, within a generalized version of the reductive perturbation method that we have adopted, the equations governing the envelop consist of the standard cubic nonlinear Schrödinger equation and its non homogeneous linearizations. Exact and explicit analytical solutions that correspond to bright solitary waves are obtained for these coupled amplitude equations. A notable qualitative feature of these solutions is the dependence of their propagation speeds and frequencies on their amplitudes. Our approach additionally unveils that these solutions contain some harmonic terms; which are missed in existing works. A very good agreement is found between our analytical analysis and the numerical simulations of the full discrete nonlinear equation of the lattice which use these solutions as initial conditions.