Periodic solutions of Swift–Hohenberg equation

The stationary Swift–Hohenberg equation
$$(\Delta+1)^2u-\alpha u-\beta u^2 +u^3=0$$
is considered in the entire space $\mathbb R^n$, $2\le n\le 7$. By further developing the variational approach proposed in the work by Lerman, Naryshkin and Nazarov (2020), we obtain several periodic solutions with additional symmetries.
The talk is based on joint work with S.B. Kolonitskii and L.M. Lerman.