Research paperEffects of nonlinearity and substrate's deformability on modulation instability in NKG equation

- We address analytically and numerically the modulational instability of an extended nonlinear Schrödinger equation derived from a deformable lattice.
- Depending on the signs and strength of additive terms strongly bound to the variations of the shape of substrate potential, the modulational instability of the lattices are quite affected in both the anomalous and normal dispersion regimes.
- We establish the domains and the gains of modulational instability of the system.
- Investigations are verified and complemented by numerical simulations.

This article investigates combined effects of nonlinearities and substrate's deformability on modulational instability. For that, we consider a lattice model based on the nonlinear Klein-Gordon equation with an on-site potential of deformable shape. Such a consideration enables to broaden the description of energy-localization mechanisms in various physical systems. We consider the strong-coupling limit and employ semi-discrete approximation to show that nonlinear wave modulations can be described by an extended nonlinear Schrödinger equation containing a fourth-order dispersion component. The stability of modulation of carrier waves is scrutinized and the following findings are obtained analytically. The various domains of gains and instabilities are provided based upon various combinations of the parameters of the system. The instability gains strongly depend on nonlinear terms and on the kind of shape of the substrate. According to the system's parameters, our model can lead to different sets of known equations such as those in a negative index material embedded into a Kerr medium, glass fibers, resonant optical fiber and others. Consequently, some of the results obtained here are in agreement with those obtained in previous works. The suitable combination of nonlinear terms with the deformability of the substrate can be utilized to specifically control the amplitude of waves and consequently to stabilize their propagations. The results of analytical investigations are validated and complemented by numerical simulations.