On Radial Solutions of the Swift–Hohenberg Equation

We study radial solutions to the generalized Swift–Hohenberg equation on the plane with an additional quadratic term. We find stationary localized radial solutions that decay at infinity and solutions that tend to constants as the radius increases unboundedly (“droplets”). We formulate existence theorems for droplets and sketch the proofs employing the properties of the limit system as $r\to\infty$. This system is a Hamiltonian system corresponding to a spatially one-dimensional stationary Swift–Hohenberg equation. We analyze the properties of this system and also discuss concentric-wave-type solutions. All the results are obtained by combining the methods of the theory of dynamical systems, in particular, the theory of homo- and heteroclinic orbits, and numerical simulation.