A shooting method for the Swift-Hohenberg equation. (Q1861000)

The authors consider the differential equation \[ u^{(4)}+ 2u''+ (1- k)u+ u^3= 0.\tag{\(*\)} \] They prove that for \(0< k< 1\), equation \((*)\) has a periodic solution with the same energy as the solution \(u= 0\), and that for \(1< k<{3\over 2}\), equation \((*)\) has a periodic solution with the same energy as the solutions \(u=\pm \sqrt{k-1}\). The proof is based on the shooting technique and on the fact that \((*)\) has a first integral.