Bifurcation of backward and forward solitary waves in helicoidal Peyrard–Bishop–Dauxois model of DNA

In this work, we investigate the bifurcation of backward and forward solitary waves in helicoidal Peyrard–Bishop–Dauxois model of deoxyribonucleic acid (DNA). We start by showing that this model can adopt right-handed or left-handed behavior depending on the wave frequency that it propagates. Then, using the semi-discrete approximation, we show that the dynamics of modulated waves in the network is governed by a quintic nonlinear Schrödinger (QNLS). Applying the phase imprint technique, this QNLS equation is first reduced into a derivative QNLS, and then to a cubic–quintic Duffing oscillator equation. Based on the phase plane analysis, we present all phase portraits of the dynamical system. The obtained results show a number of new phase portraits which cannot exist without the phase imprint approach. 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, dark soliton and grey solitons. In addition, the effects of phase imprint parameter on the wave-shape profile of these solutions are investigated. We find out that the phase imprint parameter considerably affects the amplitude and the width of each of the above enumerated solitary waves, while the phase imprint method leads to new type solutions.