The general Klein-Gordon-Schrödinger system: Modulational instability and exact solutions

The general Klein-Gordon-Schrödinger (gKGS) system is studied where the cubic auto-interactions are introduced in both the nonlinear Schrödinger and the nonlinear Klein-Gordon fields. We first investigate the modulational instability (MI) of the system, and thus derive the general dispersion relation between the frequency and wavenumber of the modulating perturbations, which demonstrates many possibilities for the MI regions. Using the travelling wave reduction, the gKGS system is greatly simplified. Via a simple function expansion method, we obtain some exact travelling wave solutions. Under some special parameter values, some representative wave structures are graphically displayed including the kink, anti-kink, bright, dark, grey and periodic solitons.