Stationary non-radial localized patterns in the planar Swift-Hohenberg PDE: Constructive proofs of existence

In this paper, we present a methodology for establishing constructive proofs of existence of smooth, stationary, non-radial localized patterns in the planar Swift-Hohenberg equation. Specifically, given an approximate solution $u_0$, we construct an approximate inverse for the linearization around $u_0$, enabling the development of a Newton-Kantorovich approach. Consequently, we derive a sufficient condition for the existence of a unique localized pattern in the vicinity of $u_0$. The verification of this condition is facilitated through a combination of analytic techniques and rigorous numerical computations. Moreover, an additional condition is derived, establishing that the localized pattern serves as the limit of a family of periodic solutions (in space) as the period tends to infinity. The integration of analytical tools and meticulous numerical analysis ensures a comprehensive validation of this condition. To illustrate the efficacy of the proposed methodology, we present computer-assisted proofs for the existence of three distinct unbounded branches of periodic solutions in the planar Swift-Hohenberg equation, all converging towards a localized planar pattern, whose existence is also proven constructively. All computer-assisted proofs, including the requisite codes, are accessible on GitHub at \cite{julia_cadiot}.