Rigorous existence of front/back heteroclinic connections in the stationary Swift–Hohenberg equation

Establish a rigorous proof of the existence of heteroclinic connections (fronts and backs) between the trivial rest state U = 0 and the periodic roll orbit in the one-dimensional quadratic–cubic Swift–Hohenberg equation in its stationary form, obtained by setting U_t = 0 and rewriting as the reversible four-dimensional spatial dynamical system u_x = f(u; μ) with u = (U, U_x, U_{xx}, U_{xxx}). Concretely, prove that for μ in an appropriate interval, the unstable manifold W^u(0) of the rest state intersects the center-stable manifold W^{cs}(γ(·; μ)) of the periodic orbit to form a front, and, by reversibility, the corresponding back connection, thereby validating the numerically observed structures that underlie homoclinic snaking.

Background

In the discussion of homoclinic snaking for localized stationary patterns, the authors model the phenomenon using the one-dimensional quadratic–cubic Swift–Hohenberg equation. Localized solutions are constructed by gluing a heteroclinic front from the rest state to a periodic orbit to its reversible back, but the existence of such fronts is typically assumed on the basis of numerical evidence and dimension counting.

The paper explicitly notes that, although fronts can be observed numerically, a formal proof of their existence is currently missing and would likely require advances in rigorous and validated numerics. Establishing these connections would place the snakes-and-ladders picture on firm mathematical footing in this canonical model.