Monoatomic chain: modulational instability and exact traveling wave solutions

The objective of this work is to study a monoatomic chain which is described by the Klein-Gordon model with first and second neighbors anharmonic interactions. We consider specifically the case where both of these interactions are of cubic-quartic type and show, through the rotating wave approximation, that envelop waves in our model are governed by a modified nonlinear Schrodinger equation. The modulational instability is investigated on the latter, with a particular attention on the impact of the second neighbors. It is then revealed that the second neighbors interaction increases the regions of instability in the space of parameters and, hence, expands the possibility of propagating solitary waves in the network. More importantly, we derive the exact solutions of our modified nonlinear Schrodinger equation and show the influence of the second neighbors on them. For example, the width of these solutions can be controlled by adjusting the parameters relative to the second neighbors. Numerical simulations are done in order to confirm the analytical studies. Equally, the methods used in this study can also be implemented on other nonlinear equations.