Anharmonic effects on the dynamic behavior’s of Klein Gordon model’s

This work completes and extends the Ref. Tchakoutio Nguetcho et al. (2017), in which we have focused our attention only on the dynamic behavior of gap soliton solutions of the anharmonic Klein-Gordon model immersed in a parameterized on-site substrate potential. We expand our work now inside the permissible frequency band. These considerations have crucial effects on the response of nonlinear excitations that can propagate along this model. Moreover, working in the allowed frequency band is not only interesting from a physical point of view, it also provides an extraordinary mathematical model, a new class of differential equations possessing vital parameters and vertical singular straight lines. The dynamics around these singularities gives new informations of great interest, such as to better understand the break up (rupture) of a stretched polymer chain by pulling and its relation to soliton destruction and which until now had no mathematical explanations yet. The mathematical model of the differential equation we obtain corresponds, as an equivalent experimental model, to the nonlinear transmission electrical line introduced in 2009 in Ref. Yemélé and Kenmogné(2009), whose equations were corrected but not entirely resolved in 2016 in Ref. Yamgoué and Pelap(2016). Our dynamic study thus presents a theoretical prediction for their experimental set up. An extensive account of the bifurcation theory for modulated-wave propagation is given. By investigating the dynamical behavior and bifurcations of solutions of the planar dynamical systems, we not only make a total inventory of the solutions that can include such differential equations, but we derive a variety of exotic solutions corresponding to each phase trajectories as well as their different conditions of existence.