Existence and dynamics of modulated solitary waves in the modified Peyrard–Bishop model of DNA

Abstract

In this paper, we study the existence and dynamics of solitary waves in the modified Peyrard–Bishop (PB) model of DNA. Firstly, we introduce the solvent interaction function on the usual model and study its effects on the frequency. In the second place, using the semi-discrete approximation, we show that the dynamics of modulated waves in the network are governed by a quintic nonlinear Schrödinger (QNLS) equation. In the quest to find the exact solitary wave solutions, we introduce an ansatz which leads to a cubic–quintic Duffing oscillator equation. Based on the dynamical system approach, we present all phase portraits of the dynamical system. The obtained results show several new phase portraits that cannot exist without the effect of solvent interaction. The exact representations of the nonlinear localized waves corresponding to the homoclinic and heteroclinic orbits in the phase portrait of the dynamical system are given. These waves include bright soliton, kink and anti-kink solitons, and dark soliton. In addition, the impact of solvent parameters on the wave-shape profile of these solutions is studied. It shows that the solvent parameter considerably affects the amplitude and the width of each of the above-enumerated solitary waves.