Periodic waves and their stability in competing cubic-quintic nonlinearity

Exact expressions are obtained for stationary periodic waves in cubic-quintic nonlinear media with opposite signs of nonlinearity. It is shown that defocussing/focussing quintic nonlinearity can lead to stabilization of snoidal/(cnoidal and dnoidal) periodic waves depending on whether the Kerr medium is focusing/defocusing. Direct numerical simulations confirm results of the linear stability analysis. Different stable patterns e.g. bright solitons and kinks are also obtained under certain conditions.