Analytical and numerical solutions of the generalized dispersive Swift–Hohenberg equation

The generalization of the Swift–Hohenberg equation is studied. It is shown that the equation does not pass the Kovalevskaya test and does not possess the Painlevé property. Exact solutions of the generalized Swift–Hohenberg equation which are very useful to test numerical algorithms for various boundary value problems are obtained. The numerical algorithm which is based on the Crank–Nicolson–Adams–Bashforth scheme is developed. This algorithm is tested using the exact solutions. The selforganization processes described by the generalization of the Swift–Hohenberg equation are studied.