Rotationally symmetric solutions to the Cahn-Hilliard equation

This paper is devoted to construction of new solutions to the Cahn-Hilliard equation in ${\mathbb{R}}^d$. Staring from a Delaunay unduloid $D_{\tau }$ with parameter $\tau \in (0,{\tau }^{\ast })$ we find for each sufficiently small $\epsilon$ a solution $u$ of this equation which is periodic in the direction of the $x_d$ axis and rotationally symmetric with respect to rotations about this axis. The zero level set of $u$ approaches as $\epsilon \to 0$ the surface $D_{\tau }$. We use a refined version of the Lyapunov-Schmidt reduction method which simplifies very technical aspects of previous constructions for similar problems.