Modulational instability and exact solutions of the nonlinear Schrödinger equation coupled with the nonlinear Klein-Gordon equation

The modulational instability (MI) of the coupled nonlinear Schrödinger and nonlinear Klein-Gordon equations is investigated. It is found that there are a number of possibilities for the MI regions due to the generalized dispersion relation, which relates the frequency and wavenumber of the modulating perturbations. Some exact travelling wave solutions are constructed via the solutions of a φ4 model through a simple mapping relation. Furthermore, we present five different types of solutions representing possible final states of modulationally unstable perturbations. The profiles of solitary wave structures are displayed for some fixed parameters.